Fractal dimension in nonhyperbolic chaotic scattering
NASA Technical Reports Server (NTRS)
Lau, Yun-Tung; Finn, John M.; Ott, Edward
1991-01-01
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.
NASA Astrophysics Data System (ADS)
Burkholder, Michael B.; Litster, Shawn
2016-05-01
In this study, we analyze the stability of two-phase flow regimes and their transitions using chaotic and fractal statistics, and we report new measurements of dynamic two-phase pressure drop hysteresis that is related to flow regime stability and channel water content. Two-phase flow dynamics are relevant to a variety of real-world systems, and quantifying transient two-phase flow phenomena is important for efficient design. We recorded two-phase (air and water) pressure drops and flow images in a microchannel under both steady and transient conditions. Using Lyapunov exponents and Hurst exponents to characterize the steady-state pressure fluctuations, we develop a new, measurable regime identification criteria based on the dynamic stability of the two-phase pressure signal. We also applied a new experimental technique by continuously cycling the air flow rate to study dynamic hysteresis in two-phase pressure drops, which is separate from steady-state hysteresis and can be used to understand two-phase flow development time scales. Using recorded images of the two-phase flow, we show that the capacitive dynamic hysteresis is related to channel water content and flow regime stability. The mixed-wettability microchannel and in-channel water introduction used in this study simulate a polymer electrolyte fuel cell cathode air flow channel.
Hierarchical fractal Weyl laws for chaotic resonance states in open mixed systems.
Körber, M J; Michler, M; Bäcker, A; Ketzmerick, R
2013-09-13
In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.
Gallery of Chaotic Attractors Generated by Fractal Network
NASA Astrophysics Data System (ADS)
Bouallegue, Kais
During the last decade, fractal processes and chaotic systems were widely studied in many areas of research. Chaotic systems are highly dependent on initial conditions. Small changes in initial conditions can generate widely diverging or converging outcomes for both bifurcation or attraction in chaotic systems. In this work, we present a new method on how to generate a new family of chaotic attractors by combining these with a network of fractal processes. The proposed approach in this article is based upon the construction of a new system of fractal processes.
Study on the Fractal and Chaotic Features of the Shanghai Composite Index
NASA Astrophysics Data System (ADS)
Wen, Fenghua; Li, Zhong; Xie, Chaohua; Shaw, David
2012-06-01
The Hurst exponent derived by the R/S analysis method of Shanghai stock market's logarithmic return series is about 0.6298. This shows that the Shanghai stock market exhibits fractal features, and a long memory cycle of about one-and-a-half years. With the reconstruction of phase space, the Shanghai Stock attractor dimension converges to 1.335, which means that the Shanghai stock market has chaotic features, and constructing a dynamic system of the Shanghai stock market needs at least two variables. The findings from the principal component analysis support the conclusion of the existence of chaotic features of the Shanghai stock market. The fractal and chaotic features of the Shanghai stock market reveal the nonlinear properties of the Chinese stock market, and the nonlinearity perspective will be more conducive to the formulation of countermeasures for the development of the Chinese stock market.
Fractal dynamics of earthquakes
Bak, P.; Chen, K.
1995-05-01
Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).
Launching the chaotic realm of iso-fractals: A short remark
O'Schmidt, Nathan; Katebi, Reza; Corda, Christian
2015-03-10
In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci’s numbers and (the complex moduli of) Mandelbrot’s numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin’s holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli’s iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.
Fractals and dynamics in art and design.
Guastello, Stephen J
2015-01-01
Many styles of visual art that build on fractal imagery and chaotic dynamics in the creative process have been examined in NDPLS in recent years. This article presents a gallery of artwork turned into design that appeared in the promotional products of the Society for Chaos Theory in Psychology & Life Sciences. The gallery showcases a variety of new imaging styles, including photography, that reflect a deepening perspective on nonlinear dynamics and art. The contributing artworks in design formats combine to render the verve that transcends the boundaries between the artistic and scientific communities.
Optimal inference with chaotic dynamics
NASA Technical Reports Server (NTRS)
Harger, R. O.
1983-01-01
Nonlinear mappings that exhibit chaotic, seemingly random, evolution have appeal as models of dynamic systems. Their deterministic evolution, vis-a-vis Markov evolutions, results in much simpler optimal detection and estimation algorithms. The variation of a chaotic parameter (mu) results in diverse evolutions, suggesting a simple but rich source of model variations. For the specific mapping examined, this latter possibility is problematic due to the extreme sensitivity on mu of the evolution in the chaotic regime.
NASA Astrophysics Data System (ADS)
Freeman, Walter J.
2013-01-01
The first step of the sensory systems is to construct the meaning of the information they receive from the senses. They do this by generating random noise and then filtering the noise with adaptive filters. We simulate the operation with the solutions of matrices of ordinary differential equations that predict subcritical Hopf bifurcations between point and limit cycle attractors. The second step is integration of the outputs from the several sensory systems into a multisensory percept, called a gestalt, which in the third step is consolidated and stored as knowledge. Simulation of the second step requires use of landscapes of nonconvergent chaotic attractors. This is not deterministic chaos, which is much too brittle owing to the infinite sensitivity to initial conditions. It is a hybrid form we call stochastic chaos, which is stabilized by additive noise modeled on noise sources in the sensory systems. Thus bifurcation and chaos theory provides tools for succinct empirical models of cortical dynamics performing the most basic cognitive operations: generalization, abstraction, and categorization in constructing knowledge. The descriptions are in a form that is suitable for more advanced modeling using analog VLSI, neuropercolation from random graph theory, non-equilibrium dissipative thermodynamics, and macroscopic many-body physics. This review concludes with a summary of the applications of stochastic chaos in pattern classification and some prescriptions for neurobiologists on what to look for in large-scale anatomical formations.
Fractals, Coherence and Brain Dynamics
NASA Astrophysics Data System (ADS)
Vitiello, Giuseppe
2010-11-01
I show that the self-similarity property of deterministic fractals provides a direct connection with the space of the entire analytical functions. Fractals are thus described in terms of coherent states in the Fock-Bargmann representation. Conversely, my discussion also provides insights on the geometrical properties of coherent states: it allows to recognize, in some specific sense, fractal properties of coherent states. In particular, the relation is exhibited between fractals and q-deformed coherent states. The connection with the squeezed coherent states is also displayed. In this connection, the non-commutative geometry arising from the fractal relation with squeezed coherent states is discussed and the fractal spectral properties are identified. I also briefly discuss the description of neuro-phenomenological data in terms of squeezed coherent states provided by the dissipative model of brain and consider the fact that laboratory observations have shown evidence that self-similarity characterizes the brain background activity. This suggests that a connection can be established between brain dynamics and the fractal self-similarity properties on the basis of the relation discussed in this report between fractals and squeezed coherent states. Finally, I do not consider in this paper the so-called random fractals, namely those fractals obtained by randomization processes introduced in their iterative generation. Since self-similarity is still a characterizing property in many of such random fractals, my conjecture is that also in such cases there must exist a connection with the coherent state algebraic structure. In condensed matter physics, in many cases the generation by the microscopic dynamics of some kind of coherent states is involved in the process of the emergence of mesoscopic/macroscopic patterns. The discussion presented in this paper suggests that also fractal generation may provide an example of emergence of global features, namely long range
Cryptosystems based on chaotic dynamics
McNees, R.A.; Protopopescu, V.; Santoro, R.T.; Tolliver, J.S.
1993-08-01
An encryption scheme based on chaotic dynamics is presented. This scheme makes use of the efficient and reproducible generation of cryptographically secure pseudo random numbers from chaotic maps. The result is a system which encrypts quickly and possesses a large keyspace, even in small precision implementations. This system offers an excellent solution to several problems including the dissemination of key material, over the air rekeying, and other situations requiring the secure management of information.
Fractal dynamics of bioconvective patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1991-01-01
Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.
Studies in Chaotic adiabatic dynamics
Jarzynski, C.
1994-01-01
Chaotic adiabatic dynamics refers to the study of systems exhibiting chaotic evolution under slowly time-dependent equations of motion. In this dissertation the author restricts his attention to Hamiltonian chaotic adiabatic systems. The results presented are organized around a central theme, namely, that the energies of such systems evolve diffusively. He begins with a general analysis, in which he motivates and derives a Fokker-Planck equation governing this process of energy diffusion. He applies this equation to study the {open_quotes}goodness{close_quotes} of an adiabatic invariant associated with chaotic motion. This formalism is then applied to two specific examples. The first is that of a gas of noninteracting point particles inside a hard container that deforms slowly with time. Both the two- and three-dimensional cases are considered. The results are discussed in the context of the Wall Formula for one-body dissipation in nuclear physics, and it is shown that such a gas approaches, asymptotically with time, an exponential velocity distribution. The second example involves the Fermi mechanism for the acceleration of cosmic rays. Explicit evolution equations are obtained for the distribution of cosmic ray energies within this model, and the steady-state energy distribution that arises when this equation is modified to account for the injection and removal of cosmic rays is discussed. Finally, the author re-examines the multiple-time-scale approach as applied to the study of phase space evolution under a chaotic adiabatic Hamiltonian. This leads to a more rigorous derivation of the above-mentioned Fokker-Planck equation, and also to a new term which has relevance to the problem of chaotic adiabatic reaction forces (the forces acting on slow, heavy degrees of freedom due to their coupling to light, fast chaotic degrees).
The Chaotic Dynamics of Jamming
NASA Astrophysics Data System (ADS)
Egolf, David A.; Banigan, Edward J.; Illich, Matthew K.; Stace-Naughton, Derick J.
2013-03-01
Despite the appearance of simplicity, much of the behavior of granular materials remains mysterious. One intriguing puzzle is the dynamical mechanism underlying the ``jamming'' transition, in which disordered grains become rigid at high density. By applying nonlinear dynamical techniques to simulated 2D shear cells, we reveal the mechanisms of jamming and find they conflict with the prevailing picture of growing cooperative regions. Additionally, at the density corresponding to random close packing, we find a dynamical transition from chaotic to non-chaotic states accompanied by diverging dynamical length and time scales. Furthermore, we find that the dominant cooperative dynamical modes are strongly correlated with particle rearrangements and become increasingly unstable before stress jumps, providing a way to predict the times and locations of these earthquake-like stress-release events. This work was supported by the U.S. National Science Foundation (DMR-0094178) and Research Corporation.
The chaotic dynamics of jamming
NASA Astrophysics Data System (ADS)
Banigan, Edward J.; Illich, Matthew K.; Stace-Naughton, Derick J.; Egolf, David A.
2013-05-01
Granular materials are collections of discrete, macroscopic particles characterized by relatively straightforward interactions. Despite their apparent simplicity, these systems exhibit a number of intriguing phenomena, including the jamming transition, in which a disordered collection of grains becomes rigid when its density exceeds a critical value. Many aspects of this transition have been explored, but an explanation of the underlying dynamical mechanisms for the transition remains elusive. Here, applying nonlinear dynamical techniques to simulated two-dimensional Couette shear cells, we reveal the mechanisms of jamming and find that they conflict with the prevailing picture of growing cooperative regions. In addition, at the density corresponding to random close packing, we find a dynamical transition from chaotic to non-chaotic states accompanied by diverging dynamical length- and timescales. Furthermore, we find that the dominant cooperative dynamical modes are strongly correlated with particle rearrangements and become increasingly unstable before stress jumps, providing a way to predict the times and locations of these striking stress-release events in our simulations.
Stochastic Erosion of Fractal Structure in Nonlinear Dynamical Systems
NASA Astrophysics Data System (ADS)
Agarwal, S.; Wettlaufer, J. S.
2014-12-01
We analyze the effects of stochastic noise on the Lorenz-63 model in the chaotic regime to demonstrate a set of general issues arising in the interpretation of data from nonlinear dynamical systems typical in geophysics. The model is forced using both additive and multiplicative, white and colored noise and it is shown that, through a suitable choice of the noise intensity, both additive and multiplicative noise can produce similar dynamics. We use a recently developed measure, histogram distance, to show the similarity between the dynamics produced by additive and multiplicative forcing. This phenomenon, in a nonlinear fractal structure with chaotic dynamics can be explained by understanding how noise affects the Unstable Periodic Orbits (UPOs) of the system. For delta-correlated noise, the UPOs erode the fractal structure. In the presence of memory in the noise forcing, the time scale of the noise starts to interact with the period of some UPO and, depending on the noise intensity, stochastic resonance may be observed. This also explains the mixing in dissipative dynamical systems in presence of white noise; as the fractal structure is smoothed, the decay of correlations is enhanced, and hence the rate of mixing increases with noise intensity.
Chaotic dynamics from interspike intervals.
Pavlov, A N; Sosnovtseva, O V; Mosekilde, E; Anishchenko, V S
2001-03-01
Considering two different mathematical models describing chaotic spiking phenomena, namely, an integrate-and-fire and a threshold-crossing model, we discuss the problem of extracting dynamics from interspike intervals (ISIs) and show that the possibilities of computing the largest Lyapunov exponent (LE) from point processes differ between the two models. We also consider the problem of estimating the second LE and the possibility to diagnose hyperchaotic behavior by processing spike trains. Since the second exponent is quite sensitive to the structure of the ISI series, we investigate the problem of its computation. PMID:11308739
Chaotic dynamics from interspike intervals
NASA Astrophysics Data System (ADS)
Pavlov, Alexey N.; Sosnovtseva, Olga V.; Mosekilde, Erik; Anishchenko, Vadim S.
2001-03-01
Considering two different mathematical models describing chaotic spiking phenomena, namely, an integrate-and-fire and a threshold-crossing model, we discuss the problem of extracting dynamics from interspike intervals (ISIs) and show that the possibilities of computing the largest Lyapunov exponent (LE) from point processes differ between the two models. We also consider the problem of estimating the second LE and the possibility to diagnose hyperchaotic behavior by processing spike trains. Since the second exponent is quite sensitive to the structure of the ISI series, we investigate the problem of its computation.
A practical test for noisy chaotic dynamics
NASA Astrophysics Data System (ADS)
BenSaïda, Ahmed
2015-12-01
This code computes the largest Lyapunov exponent and tests for the presence of a chaotic dynamics, as opposed to stochastic dynamics, in a noisy scalar series. The program runs under MATLAB® programming language.
Chaos and fractals in dynamical models of transport and reaction.
Gaspard, P; Claus, I
2002-03-15
This paper contains a discussion of dynamical randomness among the different methods of simulation of a fluid and its characterization by the concept of Kolmogorov-Sinai entropy per unit time. Moreover, a renormalization-group method is presented in order to construct the hydrodynamic and reactive modes of relaxation in chaotic models. The renormalization-group construction allows us to obtain the dispersion relation of these modes, i.e. their damping rate versus the wavenumber. Besides, these modes are characterized by a fractal dimension given in terms of a diffusion coefficient and a Lyapunov exponent.
Regular transport dynamics produce chaotic travel times
NASA Astrophysics Data System (ADS)
Villalobos, Jorge; Muñoz, Víctor; Rogan, José; Zarama, Roberto; Johnson, Neil F.; Toledo, Benjamín; Valdivia, Juan Alejandro
2014-06-01
In the hope of making passenger travel times shorter and more reliable, many cities are introducing dedicated bus lanes (e.g., Bogota, London, Miami). Here we show that chaotic travel times are actually a natural consequence of individual bus function, and hence of public transport systems more generally, i.e., chaotic dynamics emerge even when the route is empty and straight, stops and lights are equidistant and regular, and loading times are negligible. More generally, our findings provide a novel example of chaotic dynamics emerging from a single object following Newton's laws of motion in a regularized one-dimensional system.
Regular transport dynamics produce chaotic travel times.
Villalobos, Jorge; Muñoz, Víctor; Rogan, José; Zarama, Roberto; Johnson, Neil F; Toledo, Benjamín; Valdivia, Juan Alejandro
2014-06-01
In the hope of making passenger travel times shorter and more reliable, many cities are introducing dedicated bus lanes (e.g., Bogota, London, Miami). Here we show that chaotic travel times are actually a natural consequence of individual bus function, and hence of public transport systems more generally, i.e., chaotic dynamics emerge even when the route is empty and straight, stops and lights are equidistant and regular, and loading times are negligible. More generally, our findings provide a novel example of chaotic dynamics emerging from a single object following Newton's laws of motion in a regularized one-dimensional system.
Chaotic dynamics near steep transition states
NASA Astrophysics Data System (ADS)
Green, Jason R.; Hofer, Thomas S.; Wales, David J.; Berry, R. Stephen
2012-08-01
Classical molecular motion near potential energy saddles can be more or less chaotic relative to motion near minima. The relative degree of chaos depends on the extent of coupling between the degrees of freedom and on the curvature of the potential energy landscape. Here, we explore these effects using constant energy molecular dynamics simulations and independent criteria associated with locally chaotic behavior - namely, the constancy of the local mode action and the magnitude of finite-time Lyapunov exponents. These criteria reconcile the chaotic basins and relatively ordered saddles of the Lennard-Jones trimer, with the chaotic saddles and ordered basins for reactive, all-atom H2O described by the Garofalini H2O potential. By modifying the Garofalini and Lennard-Jones models we separate the compounding effects of nonlinear three-body interactions and steep reaction path curvature on the local degree of chaos near saddles and minima.
Characterization of chaotic dynamics in the human menstrual cycle
NASA Astrophysics Data System (ADS)
Derry, Gregory; Derry, Paula
2010-03-01
The human menstrual cycle exhibits much unexplained variability, which is typically dismissed as random variation. Given the many delayed nonlinear feedbacks in the reproductive endocrine system, however, the menstrual cycle might well be a nonlinear dynamical system in a chaotic trajectory, and that this instead accounts for the observed variability. Here, we test this hypothesis by performing a time series analysis on data for 7438 menstrual cycles from 38 women in the 20-40 year age range, using the database maintained by the Tremin Research Program on Women's Health. Using phase space reconstruction techniques with a maximum embedding dimension of 6, we find appropriate scaling behavior in the correlation sums for this data, indicating low dimensional deterministic dynamics. A correlation dimension of 2.6 is measured in this scaling regime, and this result is confirmed by recalculation using the Takens estimator. These results may be interpreted as offering an approximation to the fractal dimension of a strange attractor governing the chaotic dynamics of the menstrual cycle.
Aggregation and fragmentation dynamics of inertial particles in chaotic flows.
Zahnow, Jens C; Vilela, Rafael D; Feudel, Ulrike; Tél, Tamás
2008-05-01
Inertial particles advected in chaotic flows often accumulate in strange attractors. While moving in these fractal sets they usually approach each other and collide. Here we consider inertial particles aggregating upon collision. The new particles formed in this process are larger and follow the equation of motion with a new parameter. These particles can in turn fragment when they reach a certain size or shear forces become sufficiently large. The resulting system consists of a large set of coexisting dynamical systems with a varying number of particles. We find that the combination of aggregation and fragmentation leads to an asymptotic steady state. The asymptotic particle size distribution depends on the mechanism of fragmentation. The size distributions resulting from this model are consistent with those found in raindrop statistics and in stirring tank experiments.
Fractal boundaries in magnetotail particle dynamics
NASA Technical Reports Server (NTRS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-01-01
It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.
Fractal dimension and nonlinear dynamical processes
NASA Astrophysics Data System (ADS)
McCarty, Robert C.; Lindley, John P.
1993-11-01
Mandelbrot, Falconer and others have demonstrated the existence of dimensionally invariant geometrical properties of non-linear dynamical processes known as fractals. Barnsley defines fractal geometry as an extension of classical geometry. Such an extension, however, is not mathematically trivial Of specific interest to those engaged in signal processing is the potential use of fractal geometry to facilitate the analysis of non-linear signal processes often referred to as non-linear time series. Fractal geometry has been used in the modeling of non- linear time series represented by radar signals in the presence of ground clutter or interference generated by spatially distributed reflections around the target or a radar system. It was recognized by Mandelbrot that the fractal geometries represented by man-made objects had different dimensions than the geometries of the familiar objects that abound in nature such as leaves, clouds, ferns, trees, etc. The invariant dimensional property of non-linear processes suggests that in the case of acoustic signals (active or passive) generated within a dispersive medium such as the ocean environment, there exists much rich structure that will aid in the detection and classification of various objects, man-made or natural, within the medium.
Dynamic structure factor of vibrating fractals.
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-02-10
Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor S(k,t) of large vibrating fractal networks at large wave numbers k. We show that the decay of S(k,t) is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, ((u[over →](i)(t)-u[over →](i)(0))(2))∼t(ν), where ν depends on the spectral dimension d(s) and fractal dimension d(f). As a result, S(k,t) decays as a stretched exponential S(k,t)≈S(k)e(-(Γ(k)t)(ν)) with Γ(k)∼k(2/ν). Applications to a variety of fractal-like systems are elucidated.
Quantifying chaotic dynamics from interspike intervals
NASA Astrophysics Data System (ADS)
Pavlov, A. N.; Pavlova, O. N.; Mohammad, Y. K.; Shihalov, G. M.
2015-03-01
We address the problem of characterization of chaotic dynamics at the input of a threshold device described by an integrate-and-fire (IF) or a threshold crossing (TC) model from the output sequences of interspike intervals (ISIs). We consider the conditions under which quite short sequences of spiking events provide correct identification of the dynamical regime characterized by the single positive Lyapunov exponent (LE). We discuss features of detecting the second LE for both types of the considered models of events generation.
Virtual Libraries: Interactive Support Software and an Application in Chaotic Models.
ERIC Educational Resources Information Center
Katsirikou, Anthi; Skiadas, Christos; Apostolou, Apostolos; Rompogiannakis, Giannis
This paper begins with a discussion of the characteristics and the singularity of chaotic systems, including dynamic systems theory, chaotic orbit, fractals, chaotic attractors, and characteristics of chaotic systems. The second section addresses the digital libraries (DL) concept and the appropriateness of chaotic models, including definition and…
Observation of chaotic dynamics of coupled nonlinear oscillators
Van Buskirk, R.; Jeffries, C.
1985-05-01
The nonlinear charge storage property of driven Si p-n junction passive resonators gives rise to chaotic dynamics: period doubling, chaos, periodic windows, and an extended period-adding sequence corresponding to entrainment of the resonator by successive subharmonics of the driving frequency. The physical system is described; equations of motion and iterative maps are reviewed. Computed behavior is compared to data, with reasonable agreement for Poincare sections, bifurcation diagrams, and phase diagrams in parameter space (drive voltage, drive frequency). N = 2 symmetrically coupled resonators are found to display period doubling, Hopf bifurcations, entrainment horns (''Arnol'd tongues''), breakup of the torus, and chaos. This behavior is in reasonable agreement with theoretical models based on the characteristics of single-junction resonators. The breakup of the torus is studied in detail, by Poincare sections and by power spectra. Also studied are oscillations of the torus and cyclic crises. A phase diagram of the coupled resonators can be understood from the model. Poincare sections show self-similarity and fractal structure, with measured values of fractal dimension d = 2.03 and d = 2.23 for N = 1 and N = 2 resonators, respectively. Two line-coupled resonators display first a Hopf bifurcation as the drive parameter is increased, in agreement with the model. For N = 4 and N = 12 line-coupled resonators complex quasiperiodic behavior is observed with up to 3 and 4 incommensurate frequencies, respectively.
Sharma, Vijay
2009-01-01
Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts. PMID:19812706
Chaotic dynamics in dense fluids
Posch, H.A.; Hoover, W.G.
1987-09-01
We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential both in equilibrium and nonequilibrium steady states. Lyapunov characteristic exponents lambda/sub n/ describe the mean exponential rates of divergence and convergence of neighbouring trajectories in phase-space. They are useful in characterizing the stochastic properties of a dynamical system. A new algorithm for their calculation is presented which incorporates ideas from control theory and constraint nonequilibrium molecular dynamics. 4 refs., 1 fig.
Chaotic Pattern Dynamics in Spatially Ramped Turbulence
NASA Astrophysics Data System (ADS)
Wiener, R. J.; Ashbaker, E.; Olsen, T.; Bodenschatz, E.
2003-11-01
In previous experiments(Richard J. Wiener et al), Phys. Rev. E 55, 5489 (1997)., Taylor vortex flow in an hourglass geometry has demonstrated a period-doubling cascade to chaotic pattern dynamics. A spatial ramp exists in the Reynolds number. For low reduced Reynolds numbesr \\varepsilon, supercritical vortex flow occurs between regions of subcritical structureless flow with soft boundaries that allow for pattern dynamics. At \\varepsilon ≈ 0.5, the pattern exhibits phase slips that occur irregularly in time. At \\varepsilon ≈ 1.0 the entire system is supercritical, and the pattern is stabilized against phase slips. At \\varepsilon > 15, shear flow creates a spatial ramp in turbulence. Remarkably, the phase slip instability reoccurs. Vortex pairs are created chaotically, possibly due to the spatial variation of the turbulence. The variance and Fourier spectra of time series of light scattered off Kalliroscope tracer were measured. These indicate that a region of turbulence exists, within which phase slips occur, bounded by regions of laminar flow which may provide soft boundaries that allow for the phase dynamics. Despite the presence of turbulence, the dynamics might be describable by a phase equation.
Influence of noise on chaotic laser dynamics
Liu, C.; Abarbanel, H.D.I.; Nunes, K.,; Roy, R.; Gills, Z.,; Abarbanel, H.D.I.,; Nunes K.,
1997-06-01
The Nd:YAG laser with an intracavity second harmonic generating crystal is a versatile test bed for concepts of nonlinear time series analysis as well as for techniques that have been developed for control of chaotic systems. Quantitative comparisons of experimentally measured time series of the infrared light intensity are made with numerically computed time series from a model derived here from basic principles. These comparisons utilize measures that help to distinguish between low and high dimensional dynamics and thus enhance our understanding of the influence of noise sources on the emitted laser light. {copyright} {ital 1997} {ital The American Physical Society}
Nonlinear Dynamics, Chaotic and Complex Systems
NASA Astrophysics Data System (ADS)
Infeld, E.; Zelazny, R.; Galkowski, A.
2011-04-01
Part I. Dynamic Systems Bifurcation Theory and Chaos: 1. Chaos in random dynamical systems V. M. Gunldach; 2. Controlling chaos using embedded unstable periodic orbits: the problem of optimal periodic orbits B. R. Hunt and E. Ott; 3. Chaotic tracer dynamics in open hydrodynamical flows G. Karolyi, A. Pentek, T. Tel and Z. Toroczkai; 4. Homoclinic chaos L. P. Shilnikov; Part II. Spatially Extended Systems: 5. Hydrodynamics of relativistic probability flows I. Bialynicki-Birula; 6. Waves in ionic reaction-diffusion-migration systems P. Hasal, V. Nevoral, I. Schreiber, H. Sevcikova, D. Snita, and M. Marek; 7. Anomalous scaling in turbulence: a field theoretical approach V. Lvov and I. Procaccia; 8. Abelian sandpile cellular automata M. Markosova; 9. Transport in an incompletely chaotic magnetic field F. Spineanu; Part III. Dynamical Chaos Quantum Physics and Foundations Of Statistical Mechanics: 10. Non-equilibrium statistical mechanics and ergodic theory L. A. Bunimovich; 11. Pseudochaos in statistical physics B. Chirikov; 12. Foundations of non-equilibrium statistical mechanics J. P. Dougherty; 13. Thermomechanical particle simulations W. G. Hoover, H. A. Posch, C. H. Dellago, O. Kum, C. G. Hoover, A. J. De Groot and B. L. Holian; 14. Quantum dynamics on a Markov background and irreversibility B. Pavlov; 15. Time chaos and the laws of nature I. Prigogine and D. J. Driebe; 16. Evolutionary Q and cognitive systems: dynamic entropies and predictability of evolutionary processes W. Ebeling; 17. Spatiotemporal chaos information processing in neural networks H. Szu; 18. Phase transitions and learning in neural networks C. Van den Broeck; 19. Synthesis of chaos A. Vanecek and S. Celikovsky; 20. Computational complexity of continuous problems H. Wozniakowski; Part IV. Complex Systems As An Interface Between Natural Sciences and Environmental Social and Economic Sciences: 21. Stochastic differential geometry in finance studies V. G. Makhankov; Part V. Conference Banquet
Robust optimization with transiently chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Sumi, R.; Molnár, B.; Ercsey-Ravasz, M.
2014-05-01
Efficiently solving hard optimization problems has been a strong motivation for progress in analog computing. In a recent study we presented a continuous-time dynamical system for solving the NP-complete Boolean satisfiability (SAT) problem, with a one-to-one correspondence between its stable attractors and the SAT solutions. While physical implementations could offer great efficiency, the transiently chaotic dynamics raises the question of operability in the presence of noise, unavoidable on analog devices. Here we show that the probability of finding solutions is robust to noise intensities well above those present on real hardware. We also developed a cellular neural network model realizable with analog circuits, which tolerates even larger noise intensities. These methods represent an opportunity for robust and efficient physical implementations.
NASA Astrophysics Data System (ADS)
Zeyer, K.-P.; Münster, A. F.; Hauser, M. J. B.; Schneider, F. W.
1994-09-01
We extend previous work describing the passive electrical coupling of two periodic chemical states to include quasiperiodic and chaotic states. Our setup resembles an electrochemical concentration cell (a battery) whose half cells [continuous-flow stirred tank reactors (CSTRs)] each contain the Belousov-Zhabotinsky (BZ) reaction. For a closed electrical circuit the two half cells are weakly coupled by an external variable resistance and by a constant low mass flow. This battery may produce either periodic, quasiperiodic, or chaotic alternating current depending on the dynamic BZ states chosen in the half cells. A lower fractal dimensionality is calculated from the electrical potential of a single chaotic CSTR than from the difference potential (relative potential) of the two chaotic half cell potentials. A similar situation is observed in model calculations of a chaotic spatiotemporal system (the driven Brusselator in one space dimension) where the dimensionality derived from a local time series is lower than the dimensionality of the global trajectory calculated from the Karhunen-Loeve coefficients.
Fractal and Chaos Analysis for Dynamics of Radon Exhalation from Uranium Mill Tailings
NASA Astrophysics Data System (ADS)
Li, Yongmei; Tan, Wanyu; Tan, Kaixuan; Liu, Zehua; Xie, Yanshi
2016-08-01
Tailings from mining and milling of uranium ores potentially are large volumes of low-level radioactive materials. A typical environmental problem associated with uranium tailings is radon exhalation, which can significantly pose risks to environment and human health. In order to reduce these risks, it is essential to study the dynamical nature and underlying mechanism of radon exhalation from uranium mill tailings. This motivates the conduction of this study, which is based on the fractal and chaotic methods (e.g. calculating the Hurst exponent, Lyapunov exponent and correlation dimension) and laboratory experiments of the radon exhalation rates. The experimental results show that the radon exhalation rate from uranium mill tailings is highly oscillated. In addition, the nonlinear analyses of the time series of radon exhalation rate demonstrate the following points: (1) the value of Hurst exponent much larger than 0.5 indicates non-random behavior of the radon time series; (2) the positive Lyapunov exponent and non-integer correlation dimension of the time series imply that the radon exhalation from uranium tailings is a chaotic dynamical process; (3) the required minimum number of variables should be five to describe the time evolution of radon exhalation. Therefore, it can be concluded that the internal factors, including heterogeneous distribution of radium, and randomness of radium decay, as well as the fractal characteristics of the tailings, can result in the chaotic evolution of radon exhalation from the tailings.
Chaotic dynamics of a candle oscillator
NASA Astrophysics Data System (ADS)
Lee, Mary Elizabeth; Byrne, Greg; Fenton, Flavio
The candle oscillator is a simple, fun experiment dating to the late nineteenth century. It consists of a candle with a rod that is transverse to its long axis, around which it is allowed to pivot. When both ends of the candle are lit, an oscillatory motion will initiate due to different mass loss as a function of the flame angle. Stable oscillations can develop due to damping when the system has friction between the rod and the base where the rod rests. However, when friction is minimized, it is possible for chaos to develop. In this talk we will show periodic orbits found in the system as well as calculated, maximal Lyapunov exponents. We show that the system can be described by three ordinary differential equations (one each for angle, angular velocity and mass loss) that can reproduce the experimental data and the transition from stable oscillations to chaotic dynamics as a function of damping.
Nuclear mass dependence of chaotic dynamics in the Ginocchio model
Yoshinaga, N. ); Yoshida, N. , Wako-shi, Saitama 351-01 ); Shigehara, T. ); Cheon, T. )
1993-08-01
The chaotic dynamics in nuclear collective motion is studied in the framework of a schematic shell model which has only monopole and quadrupole degrees of freedom. The model is shown to reproduce the experimentally observed global trend toward less chaotic motion in heavier nuclei. The relation between the current approach and the earlier studies with bosonic models is discussed.
Forecasting catastrophe by exploiting chaotic dynamics
Stewart, H.B.; Lansbury, A.N.
1990-01-01
Our purpose here is to introduce a variation on the theme of short term forecasting from a chaotic time series. We show that for the lowest-dimensional chaotic attractors, it is possible to predict incipient catastrophes, or crises, by examining time series data taken near the catastrophic bifurcation threshold, but always remaining on the safe side of the threshold.
Traffic chaotic dynamics modeling and analysis of deterministic network
NASA Astrophysics Data System (ADS)
Wu, Weiqiang; Huang, Ning; Wu, Zhitao
2016-07-01
Network traffic is an important and direct acting factor of network reliability and performance. To understand the behaviors of network traffic, chaotic dynamics models were proposed and helped to analyze nondeterministic network a lot. The previous research thought that the chaotic dynamics behavior was caused by random factors, and the deterministic networks would not exhibit chaotic dynamics behavior because of lacking of random factors. In this paper, we first adopted chaos theory to analyze traffic data collected from a typical deterministic network testbed — avionics full duplex switched Ethernet (AFDX, a typical deterministic network) testbed, and found that the chaotic dynamics behavior also existed in deterministic network. Then in order to explore the chaos generating mechanism, we applied the mean field theory to construct the traffic dynamics equation (TDE) for deterministic network traffic modeling without any network random factors. Through studying the derived TDE, we proposed that chaotic dynamics was one of the nature properties of network traffic, and it also could be looked as the action effect of TDE control parameters. A network simulation was performed and the results verified that the network congestion resulted in the chaotic dynamics for a deterministic network, which was identical with expectation of TDE. Our research will be helpful to analyze the traffic complicated dynamics behavior for deterministic network and contribute to network reliability designing and analysis.
About Chaotic Dynamics in the Twisted Horseshoe Map
NASA Astrophysics Data System (ADS)
Sovrano, Elisa
2016-06-01
The twisted horseshoe map was developed in order to study a class of density dependent Leslie population models with two age classes. From the beginning, scientists have tried to prove that this map presents chaotic dynamics. Some demonstrations that have appeared in mathematical literature present some difficulties or delicate issues. In this paper, we give a simple and rigorous proof based on a different approach. We also highlight the possibility of getting chaotic dynamics for a broader class of maps.
Chaotic behavior in nonlinear polarization dynamics
David, D.; Holm, D.D.; Tratnik, M.V. )
1989-01-01
We analyze the problem of two counterpropagating optical laser beams in a slightly nonlinear medium from the point of view of Hamiltonian systems; the one-beam subproblem is also investigated as a special case. We are interested in these systems as integrable dynamical systems which undergo chaotic behavior under various types of perturbations. The phase space for the two-beam problem is C{sup 2} {times} C{sup 2} when we restricted the the regime of travelling-wave solutions. We use the method of reduction for Hamiltonian systems invariant under one-parameter symmetry groups to demonstrate that the phase space reduces to the two-sphere S{sup 2} and is therefore completely integrable. The phase portraits of the system are classified and we also determine the bifurcations that modify these portraits; some new degenerate bifurcations are presented in this context. Finally, we introduce various physically relevant perturbations and use the Melnikov method to prove that horseshoe chaos and Arnold diffusion occur as consequences of these perturbations. 10 refs., 7 figs., 1 tab.
Entrainment to a real time fractal visual stimulus modulates fractal gait dynamics.
Rhea, Christopher K; Kiefer, Adam W; D'Andrea, Susan E; Warren, William H; Aaron, Roy K
2014-08-01
Fractal patterns characterize healthy biological systems and are considered to reflect the ability of the system to adapt to varying environmental conditions. Previous research has shown that fractal patterns in gait are altered following natural aging or disease, and this has potential negative consequences for gait adaptability that can lead to increased risk of injury. However, the flexibility of a healthy neurological system to exhibit different fractal patterns in gait has yet to be explored, and this is a necessary step toward understanding human locomotor control. Fifteen participants walked for 15min on a treadmill, either in the absence of a visual stimulus or while they attempted to couple the timing of their gait with a visual metronome that exhibited a persistent fractal pattern (contained long-range correlations) or a random pattern (contained no long-range correlations). The stride-to-stride intervals of the participants were recorded via analog foot pressure switches and submitted to detrended fluctuation analysis (DFA) to determine if the fractal patterns during the visual metronome conditions differed from the baseline (no metronome) condition. DFA α in the baseline condition was 0.77±0.09. The fractal patterns in the stride-to-stride intervals were significantly altered when walking to the fractal metronome (DFA α=0.87±0.06) and to the random metronome (DFA α=0.61±0.10) (both p<.05 when compared to the baseline condition), indicating that a global change in gait dynamics was observed. A variety of strategies were identified at the local level with a cross-correlation analysis, indicating that local behavior did not account for the consistent global changes. Collectively, the results show that a gait dynamics can be shifted in a prescribed manner using a visual stimulus and the shift appears to be a global phenomenon.
Chaotic dynamics in optimal monetary policy
NASA Astrophysics Data System (ADS)
Gomes, O.; Mendes, V. M.; Mendes, D. A.; Sousa Ramos, J.
2007-05-01
There is by now a large consensus in modern monetary policy. This consensus has been built upon a dynamic general equilibrium model of optimal monetary policy as developed by, e.g., Goodfriend and King [ NBER Macroeconomics Annual 1997 edited by B. Bernanke and J. Rotemberg (Cambridge, Mass.: MIT Press, 1997), pp. 231 282], Clarida et al. [J. Econ. Lit. 37, 1661 (1999)], Svensson [J. Mon. Econ. 43, 607 (1999)] and Woodford [ Interest and Prices: Foundations of a Theory of Monetary Policy (Princeton, New Jersey, Princeton University Press, 2003)]. In this paper we extend the standard optimal monetary policy model by introducing nonlinearity into the Phillips curve. Under the specific form of nonlinearity proposed in our paper (which allows for convexity and concavity and secures closed form solutions), we show that the introduction of a nonlinear Phillips curve into the structure of the standard model in a discrete time and deterministic framework produces radical changes to the major conclusions regarding stability and the efficiency of monetary policy. We emphasize the following main results: (i) instead of a unique fixed point we end up with multiple equilibria; (ii) instead of saddle-path stability, for different sets of parameter values we may have saddle stability, totally unstable equilibria and chaotic attractors; (iii) for certain degrees of convexity and/or concavity of the Phillips curve, where endogenous fluctuations arise, one is able to encounter various results that seem intuitively correct. Firstly, when the Central Bank pays attention essentially to inflation targeting, the inflation rate has a lower mean and is less volatile; secondly, when the degree of price stickiness is high, the inflation rate displays a larger mean and higher volatility (but this is sensitive to the values given to the parameters of the model); and thirdly, the higher the target value of the output gap chosen by the Central Bank, the higher is the inflation rate and its
Quantifying chaotic dynamics from integrate-and-fire processes
Pavlov, A. N.; Pavlova, O. N.; Mohammad, Y. K.; Kurths, J.
2015-01-15
Characterizing chaotic dynamics from integrate-and-fire (IF) interspike intervals (ISIs) is relatively easy performed at high firing rates. When the firing rate is low, a correct estimation of Lyapunov exponents (LEs) describing dynamical features of complex oscillations reflected in the IF ISI sequences becomes more complicated. In this work we discuss peculiarities and limitations of quantifying chaotic dynamics from IF point processes. We consider main factors leading to underestimated LEs and demonstrate a way of improving numerical determining of LEs from IF ISI sequences. We show that estimations of the two largest LEs can be performed using around 400 mean periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.
Nonlinear dynamics, fractals, cardiac physiology and sudden death
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.
1987-01-01
The authors propose a diametrically opposite viewpoint to the generally accepted tendency of equating healthy function with order and disease with chaos. With regard to the question of sudden cardiac death and chaos, it is suggested that certain features of dynamical chaos related to fractal structure and fractal dynamics may be important organizing principles in normal physiology and that certain pathologies, including ventricular fibrillation, represent a class of 'pathological periodicities'. Some laboratory work bearing on the relation of nonlinear analysis to physiological and pathophysiological data is briefly reviewed, with tentative theories and models described in reference to the mechanism of ventricular fibrillation.
Wave dynamics of regular and chaotic rays
McDonald, S.W.
1983-09-01
In order to investigate general relationships between waves and rays in chaotic systems, I study the eigenfunctions and spectrum of a simple model, the two-dimensional Helmholtz equation in a stadium boundary, for which the rays are ergodic. Statistical measurements are performed so that the apparent randomness of the stadium modes can be quantitatively contrasted with the familiar regularities observed for the modes in a circular boundary (with integrable rays). The local spatial autocorrelation of the eigenfunctions is constructed in order to indirectly test theoretical predictions for the nature of the Wigner distribution corresponding to chaotic waves. A portion of the large-eigenvalue spectrum is computed and reported in an appendix; the probability distribution of successive level spacings is analyzed and compared with theoretical predictions. The two principal conclusions are: 1) waves associated with chaotic rays may exhibit randomly situated localized regions of high intensity; 2) the Wigner function for these waves may depart significantly from being uniformly distributed over the surface of constant frequency in the ray phase space.
Chaotic dynamics of magnetic domain walls in nanowires
NASA Astrophysics Data System (ADS)
Pivano, A.; Dolocan, V. O.
2016-04-01
The nonlinear dynamics of a transverse domain wall (TDW) in permalloy and nickel nanostrips with two artificially patterned pinning centers is studied numerically up to rf frequencies. The phase diagram frequency-driving amplitude shows a rich variety of dynamical behaviors depending on the material parameters and the type and shape of pinning centers. We find that T-shaped traps (antinotches) create a classical double well Duffing potential that leads to a small chaotic region in the case of nickel and a large one for Py. In contrast, the rectangular constrictions (notches) create an exponential potential that leads to larger chaotic regions interspersed with periodic windows for both Py and Ni. The influence of temperature manifests itself by enlarging the chaotic region and activating thermal jumps between the pinning sites while reducing the depinning field at low frequency in the notched strips.
Fermi resonance in dynamical tunneling in a chaotic billiard
NASA Astrophysics Data System (ADS)
Yi, Chang-Hwan; Kim, Ji-Hwan; Yu, Hyeon-Hye; Lee, Ji-Won; Kim, Chil-Min
2015-08-01
We elucidate that Fermi resonance ever plays a decisive role in dynamical tunneling in a chaotic billiard. Interacting with each other through an avoided crossing, a pair of eigenfunctions are coupled through tunneling channels for dynamical tunneling. In this case, the tunneling channels are an islands chain and its pair unstable periodic orbit, which equals the quantum number difference of the eigenfunctions. This phenomenon of dynamical tunneling is confirmed in a quadrupole billiard in relation with Fermi resonance.
Active synchronization between two different chaotic dynamical system
NASA Astrophysics Data System (ADS)
Maheri, M.; Arifin, N. Md; Ismail, F.
2015-05-01
In this paper we investigate on the synchronization problem between two different chaotic dynamical system based on the Lyapunov stability theorem by using nonlinear control functions. Active control schemes are used for synchronization Liu system as drive and Rossler system as response. Numerical simulation by using Maple software are used to show effectiveness of the proposed schemes.
Active synchronization between two different chaotic dynamical system
Maheri, M.; Arifin, N. Md; Ismail, F.
2015-05-15
In this paper we investigate on the synchronization problem between two different chaotic dynamical system based on the Lyapunov stability theorem by using nonlinear control functions. Active control schemes are used for synchronization Liu system as drive and Rossler system as response. Numerical simulation by using Maple software are used to show effectiveness of the proposed schemes.
Namazi, Hamidreza; Kulish, Vladimir V; Akrami, Amin
2016-01-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the 'complex' visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders. PMID:27217194
NASA Astrophysics Data System (ADS)
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-05-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-01-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders. PMID:27217194
On the vortex dynamics in fractal Fourier turbulence.
Lanotte, Alessandra S; Malapaka, Shiva Kumar; Biferale, Luca
2016-04-01
Incompressible, homogeneous and isotropic turbulence is studied by solving the Navier-Stokes equations on a reduced set of Fourier modes, belonging to a fractal set of dimension D . By tuning the fractal dimension parameter, we study the dynamical effects of Fourier decimation on the vortex stretching mechanism and on the statistics of the velocity and the velocity gradient tensor. In particular, we show that as we move from D = 3 to D ∼ 2.8 , the statistics gradually turns into a purely Gaussian one. This result suggests that even a mild fractal mode reduction strongly depletes the stretching properties of the non-linear term of the Navier-Stokes equations and suppresses anomalous fluctuations. PMID:27125678
Chaotic dynamics of red blood cells in oscillating shear flow
NASA Astrophysics Data System (ADS)
Bagchi, Prosenjit; Cordasco, Daniel
2015-11-01
A 3D computational study of deformable red blood cells in dilute suspension and subject to sinusoidally oscillating shear flow is considered. It is observed that the cell exhibits either a periodic motion or a chaotic motion. In the periodic motion, the cell reverses its orientation either about the flow direction or about the flow gradient, depending on the initial conditions. In certain parameter range, the initial conditions are forgotten and the cells become entrained in the same sequence of horizontal reversals. The chaotic dynamics is characterized by a nonperiodic sequence of horizontal and vertical reversals, and swings. The study provides the first conclusive evidence of the chaotic dynamics of fully deformable cells in oscillating flow using a deterministic numerical model without the introduction of any stochastic noise. An analysis of the chaotic dynamics shows that chaos is only possible in certain frequency bands when the cell membrane can rotate by a certain amount allowing the cells to swing near the maximum shear rate. We make a novel observation that the occurrence of the vertical or horizontal reversal depends only on whether a critical angle, that is independent of the flow frequency, is exceeded at the instant of flow reversal.
Discriminating chaotic and stochastic dynamics through the permutation spectrum test.
Kulp, C W; Zunino, L
2014-09-01
In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism.
Discriminating chaotic and stochastic dynamics through the permutation spectrum test
NASA Astrophysics Data System (ADS)
Kulp, C. W.; Zunino, L.
2014-09-01
In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism.
El Nino: a chaotic dynamical system
Vallis, G.K.
1986-04-11
Most of the principal qualitative features of the El Nino-Southern Oscillation phenomenon can be explained by a simple but physically motivated theory. These features are the occurrence of sea-surface warmings in the eastern equatorial Pacific and the associated trade wind reversal; the aperiodicity of these events; the preferred onset time with respect to the seasonal cycle; and the much weaker events in the Atlantic and Indian oceans. The theory, in its simplest form, is a conceptual model for the interaction of just three variables, namely near-surface temperatures in the east and west equatorial ocean and a wind-driven current advecting the temperature field. For a large range of parameters, the model is naturally chaotic and aperiodically produces El Nino-like events. For a smaller basin, representing a smaller ocean, the events are proportionally less intense.
Chaotic dynamics of corotating magnetospheric convection
NASA Technical Reports Server (NTRS)
Summers, Danny; Mu, Jian-Lin
1994-01-01
The corotating plasma convection system of the Jovian magnetosphere is analyzed. The macroscopic (mhd) model introduced by Summers and Mu, (1992) that incorporates the effects of microdiffusion is extended by including previously neglected density effects. We reduce the governing partial differential equations to a third-order ordinary differential system by the Galerkin technique of mode truncation. We carry out such a severe truncation partly in the interests of tractability, and leave open the question of the efficacy of adding additional modes. Exhaustive numerical integrations are carried out to calculate the long-term solutions, and we discover that a rich array of plasma motions is possible, dependent on the value of the height-integrated ionospheric Pederson conductivity Sigma. If Sigma is less than a certain critical value Sigma(sub c), then plasma motion can be expected to be chaotic (or periodic), while if Sigma is greater than Sigma(sub c), then steady state convection is expected. In the former case, whether the plasma motion is chaotic or periodic (and, if periodic, the magnitude of the period) can be very sensitive to the value of Sigma. The value of Sigma(sub c), which is a function of a parameter q that occurs in the assumed form of the stationary radial profile (varies as L(exp -q) of the plasma mass per unit magnetic flux, lies well within the accepted range of values of Sigma for Jupiter, i.e. Sigma greater than or equal to 0.1 mho and less than or equal to 10 mho.
Generalized correlation integral vectors: A distance concept for chaotic dynamical systems
Haario, Heikki; Kalachev, Leonid; Hakkarainen, Janne
2015-06-15
Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. We modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.
Chaotic dynamics in cardiac aggregates induced by potassium channel block
NASA Astrophysics Data System (ADS)
Quail, Thomas; McVicar, Nevin; Aguilar, Martin; Kim, Min-Young; Hodge, Alex; Glass, Leon; Shrier, Alvin
2012-09-01
Chaotic rhythms in deterministic models can arise as a consequence of changes in model parameters. We carried out experimental studies in which we induced a variety of complex rhythms in aggregates of embryonic chick cardiac cells using E-4031 (1.0-2.5 μM), a drug that blocks the hERG potassium channel. Following the addition of the drug, the regular rhythm evolved to display a spectrum of complex dynamics: irregular rhythms, bursting oscillations, doublets, and accelerated rhythms. The interbeat intervals of the irregular rhythms can be described by one-dimensional return maps consistent with chaotic dynamics. A Hodgkin-Huxley-style cardiac ionic model captured the different types of complex dynamics following blockage of the hERG mediated potassium current.
Jung, Jinwoo; Lee, Jewon; Song, Hanjung
2011-03-01
This paper presents a fully integrated circuit implementation of an operational amplifier (op-amp) based chaotic neuron model with a bipolar output function, experimental measurements, and analyses of its chaotic behavior. The proposed chaotic neuron model integrated circuit consists of several op-amps, sample and hold circuits, a nonlinear function block for chaotic signal generation, a clock generator, a nonlinear output function, etc. Based on the HSPICE (circuit program) simulation results, approximated empirical equations for analyses were formulated. Then, the chaotic dynamical responses such as bifurcation diagrams, time series, and Lyapunov exponent were calculated using these empirical equations. In addition, we performed simulations about two chaotic neuron systems with four synapses to confirm neural network connections and got normal behavior of the chaotic neuron such as internal state bifurcation diagram according to the synaptic weight variation. The proposed circuit was fabricated using a 0.8-μm single poly complementary metal-oxide semiconductor technology. Measurements of the fabricated single chaotic neuron with ± 2.5 V power supplies and a 10 kHz sampling clock frequency were carried out and compared with the simulated results.
Jung, Jinwoo; Lee, Jewon; Song, Hanjung
2011-03-15
This paper presents a fully integrated circuit implementation of an operational amplifier (op-amp) based chaotic neuron model with a bipolar output function, experimental measurements, and analyses of its chaotic behavior. The proposed chaotic neuron model integrated circuit consists of several op-amps, sample and hold circuits, a nonlinear function block for chaotic signal generation, a clock generator, a nonlinear output function, etc. Based on the HSPICE (circuit program) simulation results, approximated empirical equations for analyses were formulated. Then, the chaotic dynamical responses such as bifurcation diagrams, time series, and Lyapunov exponent were calculated using these empirical equations. In addition, we performed simulations about two chaotic neuron systems with four synapses to confirm neural network connections and got normal behavior of the chaotic neuron such as internal state bifurcation diagram according to the synaptic weight variation. The proposed circuit was fabricated using a 0.8-{mu}m single poly complementary metal-oxide semiconductor technology. Measurements of the fabricated single chaotic neuron with {+-}2.5 V power supplies and a 10 kHz sampling clock frequency were carried out and compared with the simulated results.
Chaotic dynamics of flexible Euler-Bernoulli beams.
Awrejcewicz, J; Krysko, A V; Kutepov, I E; Zagniboroda, N A; Dobriyan, V; Krysko, V A
2013-12-01
Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c(2)) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q(0) and frequency ω(p) of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.
Conformal dynamics of fractal growth patterns without randomness
Davidovitch; Feigenbaum; Hentschel; Procaccia
2000-08-01
Many models of fractal growth patterns (such as diffusion limited aggregation and dielectric breakdown models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we introduce a wide class of fractal growth models with highly complex geometry but without any randomness in their growth rules. The models are defined in terms of deterministic itineraries of iterated conformal maps, generating the function Phi((n))(omega) which maps the exterior of the unit circle to the exterior of an n-particle growing aggregate. The complexity of the evolving interfaces is fully contained in the deterministic dynamics of the conformal map Phi((n))(omega). We focus attention on a class of growth models in which the itinerary is quasiperiodic. Such itineraries can be approached via a series of rational approximants. The analytic power gained is used to introduce a scaling theory of the fractal growth patterns and to identify the exponent that determines the fractal dimension.
Social opinion dynamics is not chaotic
NASA Astrophysics Data System (ADS)
Lim, Chjan; Zhang, Weituo
2016-08-01
Motivated by the research on social opinion dynamics over large and dense networks, a general framework for verifying the monotonicity property of multi-agent dynamics is introduced. This allows a derivation of sociologically meaningful sufficient conditions for monotonicity that are tailor-made for social opinion dynamics, which typically have high nonlinearity. A direct consequence of monotonicity is that social opinion dynamics is nonchaotic. A key part of this framework is the definition of a partial order relation that is suitable for a large class of social opinion dynamics such as the generalized naming games. Comparisons are made to previous techniques to verify monotonicity. Using the results obtained, we extend many of the consequences of monotonicity to this class of social dynamics, including several corollaries on their asymptotic behavior, such as global convergence to consensus and tipping points of a minority fraction of zealots or leaders.
Forward and adjoint sensitivity computation of chaotic dynamical systems
Wang, Qiqi
2013-02-15
This paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems, such as the Lorenz attractor. The algorithms compute the derivative of long time averaged “statistical” quantities to infinitesimal perturbations of the system parameters. The algorithms are demonstrated on the Lorenz attractor. We show that sensitivity derivatives of statistical quantities can be accurately estimated using a single, short trajectory (over a time interval of 20) on the Lorenz attractor.
Nonlinear analysis of anesthesia dynamics by Fractal Scaling Exponent.
Gifani, P; Rabiee, H R; Hashemi, M R; Taslimi, P; Ghanbari, M
2006-01-01
The depth of anesthesia estimation has been one of the most research interests in the field of EEG signal processing in recent decades. In this paper we present a new methodology to quantify the depth of anesthesia by quantifying the dynamic fluctuation of the EEG signal. Extraction of useful information about the nonlinear dynamic of the brain during anesthesia has been proposed with the optimum Fractal Scaling Exponent. This optimum solution is based on the best box sizes in the Detrended Fluctuation Analysis (DFA) algorithm which have meaningful changes at different depth of anesthesia. The Fractal Scaling Exponent (FSE) Index as a new criterion has been proposed. The experimental results confirm that our new Index can clearly discriminate between aware to moderate and deep anesthesia levels. Moreover, it significantly reduces the computational complexity and results in a faster reaction to the transients in patients' consciousness levels in relations with the other algorithms.
Modeling and control of evolving, noisy chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Weber, Nicholas Noel
We study the modeling and control of evolving dynamical systems. In particular we model the dynamics of an evolving noisy iterative map, we study the extraction of the control parameter of the same map using a sparse time series from it, and we also study the dynamics of an evolving electronic circuit whose control parameter changes in proportion to low-pass filtering of one of its dynamical variables. First, we study a noisy one-dimensional iterative map whose system parameter evolves randomly in time. We find that there is an optimal number of model parameters and that there is an optimal number of data points to be retained as input for the model. Second, we study modeling based on a sparsely sampled time series. We compare three different methods of extracting the system parameter from a sparse set of the map's data. Two of these methods employ an ensemble of test trajectories in order to determine if statistical properties, such as the law of large numbers, facilitate the search for the system parameter. The third extraction method uses a single test trajectory. Third, we study self-adjusting dynamical systems. We study the logistic map and a chaotic electronic circuit, the Chua oscillator. We find that when these systems begin in a chaotic region of phase space, they self-adjust their own dynamics to the edge of chaos or a periodic window neighboring chaos. In addition, we study a self-adjusting system which has both low-pass filtered feedback and linear feedback control applied to its system parameter. The objective of the linear feedback control component is to drive the parameter to a target value in the presence of the low-pass component which behaves as described earlier. We find that the system parameter stays close to the target parameter value if the dynamics is non-chaotic. (Abstract shortened by UMI.)
Dynamics and Synchronization of Semiconductor Lasers for Chaotic Optical Communications
NASA Astrophysics Data System (ADS)
Liu, Jia-Ming; Chen, How-Foo; Tang, Shuo
The objective of this chapter is to provide a complete picture of the nonlinear dynamics and chaos synchronization of single-mode semiconductor lasers for chaotic optical communications. Basic concepts and theoretical framework are reviewed. Experimental results are presented to demonstrate the fundamental concepts. Numerical computations are employed for mapping the dynamical states and for illustrating certain detailed characteristics of the chaotic states. Three different semiconductor laser systems, namely, the optical injection system, the optical feedback system, and the optoelectronic feedback system, that are of most interest for high-bit-rate chaotic optical communications are considered. The optical injection system is a nonautonomous system that follows a period-doubling route to chaos. The optical feedback system is a phase-sensitive delayed-feedback autonomous system for which all three known routes, namely, period-doubling, quasiperiodicity, and intermittency, to chaos can be found. The optical feedback system is a phase-insensitive delayed-feedback autonomous system that follows a quasiperiodicity route to chaotic pulsing. Identical synchronization in unidirectionally coupled configurations is the focus of discussions for chaotic communications. For optical injection and optical feedback systems, the frequency, phase, and amplitude of the optical fields of both transmitter and receiver lasers are all locked in synchronism when complete synchronization is accomplished. For the optoelectronic feedback system, chaos synchronization involves neither the locking of the optical frequency nor the synchronization of the optical phase. For both optical feedback and optoelectronic feedback systems, where the transmitter is configured with a delayed feedback loop, anticipated and retarded synchronization can be observed as the difference between the feedback delay time and the propagation time from the transmitter laser to the receiver laser is varied. For a
Global and Chaotic Dynamics for a Parametrically Excited Thin Plate
NASA Astrophysics Data System (ADS)
ZHANG, W.
2001-02-01
The global bifurcations and chaotic dynamics of a parametrically excited, simply supported rectangular thin plate are analyzed. The formulas of the thin plate are derived by von Karman-type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, theory of normal form is used to give the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, global bifurcation analysis of the parametrically excited rectangular thin plate is given by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is found by numerical simulation.
Enhancement of Magma Mixing Efficiency by Chaotic Dynamics: an Experimental Study
NASA Astrophysics Data System (ADS)
Perugini, D.; de Campos, C. P.; Ertel, W.; Dingwell, D. B.; Poli, G.
2010-12-01
Magma mixing is common in the Earth. Understanding the dynamics of the mixing process is necessary for dealing with the likely consequences of mixing events in the petrogenesis of igneous rocks and the physics of volcanic eruptive triggers. We present a new apparatus to perform chaotic mixing experiments in systems of melts with high viscosity contrast. The apparatus consists of an outer and an inner cylinder, which can be independently rotated at finite strains to generate chaotic streamlines. The two cylinder axes are offset. Two end-member silicate melt compositions were synthesized from oxide and carbonate components and used in the experiments: (1) a peralkaline haplogranite and (2) a haplobasalt. The viscosity ratio between these two melts was of the order of 103. Experiments have been performed for ca. 2 h, at 1,400°C under laminar fluid dynamic conditions [Re ~ 10^(-7)]. Optical analysis of post-experimental samples revealed a complex pattern of mingled filaments forming a scale-invariant (i.e. fractal) distribution down to the μm-scale, as commonly observed in natural samples. This is due to the development of stretching and folding of the two melts in space and time. Chemical analysis showed that the original end-member compositions had nearly entirely disappeared from the filaments generated by the chaotic flow field. In addition, strong non-linear correlations in inter-elemental plots were observed. The generation of thin layers of compositionally widely contrasting interfaces strongly enhanced chemical diffusion producing a remarkable modulation of compositional fields over a short-length scale. Notably, diffusive fractionation generated highly heterogeneous pockets of melt, in which depletion or enrichment of chemical elements occurred, depending on their potential to spread within the magma mixing system. Results presented in this work offer new insights into the complexity of processes expected to be operating during magma mixing and may have
Chaotic expression dynamics implies pluripotency: when theory and experiment meet
Furusawa, Chikara; Kaneko, Kunihiko
2009-01-01
Background During normal development, cells undergo a unidirectional course of differentiation that progressively decreases the number of cell types they can potentially become. Pluripotent stem cells can differentiate into several types of cells, but terminally differentiated cells cannot differentiate any further. A fundamental problem in stem cell biology is the characterization of the difference in cellular states, e.g., gene expression profiles, between pluripotent stem cells and terminally differentiated cells. Presentation of the hypothesis To address the problem, we developed a dynamical systems model of cells with intracellular protein expression dynamics and interactions with each other. According to extensive simulations, cells with irregular (chaotic) oscillations in gene expression dynamics have the potential to differentiate into other cell types. During development, such complex oscillations are lost successively, leading to a loss of pluripotency. These simulation results, together with recent single-cell-level measurements in stem cells, led us to the following hypothesis regarding pluripotency: Chaotic oscillation in the expression of some genes leads to cell pluripotency and affords cellular state heterogeneity, which is supported by itinerancy over quasi-stable states. Differentiation stabilizes these states, leading to a loss of pluripotency. Testing the hypothesis To test the hypothesis, it is crucial to measure the time course of gene expression levels at the single-cell level by fluorescence microscopy and fluorescence-activated cell sorting (FACS) analysis. By analyzing the time series of single-cell-level expression data, one can distinguish whether the variation in protein expression level over time is due only to stochasticity in expression dynamics or originates from the chaotic dynamics inherent to cells, as our hypothesis predicts. By further analyzing the expression in differentiated cell types, one can examine whether the loss of
A Brief Historical Introduction to Fractals and Fractal Geometry
ERIC Educational Resources Information Center
Debnath, Lokenath
2006-01-01
This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and…
Chaotic dynamics of flexible Euler-Bernoulli beams
Awrejcewicz, J.; Kutepov, I. E. Zagniboroda, N. A. Dobriyan, V. Krysko, V. A.
2013-12-15
Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c{sup 2}) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q{sub 0} and frequency ω{sub p} of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.
Zhou Weihang; Chen Zhanghai; Zhang Bo; Yu, C. H.; Lu Wei; Shen, S. C.
2010-07-09
We report magnetic field control of the quantum chaotic dynamics of hydrogen analogues in an anisotropic solid state environment. The chaoticity of the system dynamics was quantified by means of energy level statistics. We analyzed the magnetic field dependence of the statistical distribution of the impurity energy levels and found a smooth transition between the Poisson limit and the Wigner limit, i.e., transition between regular Poisson and fully chaotic Wigner dynamics. The effect of the crystal field anisotropy on the quantum chaotic dynamics, which manifests itself in characteristic transitions between regularity and chaos for different field orientations, was demonstrated.
Urey Prize Lecture - Chaotic dynamics in the solar system
NASA Technical Reports Server (NTRS)
Wisdom, Jack
1987-01-01
Attention is given to solar system cases in which chaotic solutions of Newton's equations are important, as in chaotic rotation and orbital evolution. Hyperion is noted to be tumbling chaotically; chaotic orbital evolution is suggested to be of fundamental importance to an accounting for the Kirkwood gaps in asteroid distribution and for the phase space boundary of the chaotic zone at the 3/1 mean-motion commensurability with Jupiter. In addition, chaotic trajectories in the 2/1 chaotic zone reach very high eccentricities by a route that carries them to high inclinations temporarily.
Fractal analysis on human dynamics of library loans
NASA Astrophysics Data System (ADS)
Fan, Chao; Guo, Jin-Li; Zha, Yi-Long
2012-12-01
In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The values of the Hurst exponent and length of non-periodic cycle calculated through rescaled range analysis indicate that the time series of human behaviors and their sub-series are fractal with self-similarity and long-range dependence. Then the time series are converted into complex networks by the visibility algorithm. The topological properties of the networks such as scale-free property and small-world effect imply that there is a close relationship among the numbers of repetitious behaviors performed by people during certain periods of time. Our work implies that there is intrinsic regularity in the human collective repetitious behaviors. The conclusions may be helpful to develop some new approaches to investigate the fractal feature and mechanism of human dynamics, and provide some references for the management and forecast of human collective behaviors.
A fractal approach to dynamic inference and distribution analysis
van Rooij, Marieke M. J. W.; Nash, Bertha A.; Rajaraman, Srinivasan; Holden, John G.
2013-01-01
Event-distributions inform scientists about the variability and dispersion of repeated measurements. This dispersion can be understood from a complex systems perspective, and quantified in terms of fractal geometry. The key premise is that a distribution's shape reveals information about the governing dynamics of the system that gave rise to the distribution. Two categories of characteristic dynamics are distinguished: additive systems governed by component-dominant dynamics and multiplicative or interdependent systems governed by interaction-dominant dynamics. A logic by which systems governed by interaction-dominant dynamics are expected to yield mixtures of lognormal and inverse power-law samples is discussed. These mixtures are described by a so-called cocktail model of response times derived from human cognitive performances. The overarching goals of this article are twofold: First, to offer readers an introduction to this theoretical perspective and second, to offer an overview of the related statistical methods. PMID:23372552
Chaotic dynamics in erbium-doped fiber ring lasers
Abarbanel, H.D.; Kennel, M.B.; Buhl, M.; Lewis, C.T. )
1999-09-01
Chaotically oscillating rare-earth-doped fiber ring lasers (DFRLs) may provide an attractive way to exploit the broad bandwidth available in an optical communications system. Recent theoretical and experimental investigations have successfully shown techniques to modulate information onto the wide-band chaotic oscillations, transmit that signal along an optical fiber, and demodulate the information at the receiver. We develop a theoretical model of a DFRL and discuss an efficient numerical simulation which includes intrinsic linear and nonlinear induced birefringence, both transverse polarizations, group velocity dispersion, and a finite gain bandwidth. We analyze first a configuration with a single loop of optical fiber containing the doped fiber amplifier, and then, as suggested by Roy and VanWiggeren, we investigate a system with two rings of optical fiber[emdash]one made of passive fiber alone. The typical round-trip time for the passive optical ring connecting the erbium-doped amplifier to itself is 200 ns, so [approx]10[sup 5] round-trips are required to see the slow effects of the population inversion dynamics in this laser system. Over this large number of round-trips, physical effects like GVD and the Kerr nonlinearity, which may appear small at our frequencies and laser powers via conventional estimates, may accumulate and dominate the dynamics. We demonstrate from our model that chaotic oscillations of the ring laser with parameters relevant to erbium-doped fibers arises from the nonlinear Kerr effect and not from interplay between the atomic population inversion and radiation dynamics. thinsp [copyright] [ital 1999] [ital The American Physical Society
A review of sigma models for quantum chaotic dynamics.
Altland, Alexander; Gnutzmann, Sven; Haake, Fritz; Micklitz, Tobias
2015-07-01
We review the construction of the supersymmetric sigma model for unitary maps, using the color-flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization. PMID:26181515
A review of sigma models for quantum chaotic dynamics.
Altland, Alexander; Gnutzmann, Sven; Haake, Fritz; Micklitz, Tobias
2015-07-01
We review the construction of the supersymmetric sigma model for unitary maps, using the color-flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.
A review of sigma models for quantum chaotic dynamics
NASA Astrophysics Data System (ADS)
Altland, Alexander; Gnutzmann, Sven; Haake, Fritz; Micklitz, Tobias
2015-07-01
We review the construction of the supersymmetric sigma model for unitary maps, using the color-flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.
Chaotic dynamics of flexible beams driven by external white noise
NASA Astrophysics Data System (ADS)
Awrejcewicz, J.; Krysko, A. V.; Papkova, I. V.; Zakharov, V. M.; Erofeev, N. P.; Krylova, E. Yu.; Mrozowski, J.; Krysko, V. A.
2016-10-01
Mathematical models of continuous structural members (beams, plates and shells) subjected to an external additive white noise are studied. The structural members are considered as systems with infinite number of degrees of freedom. We show that in mechanical structural systems external noise can not only lead to quantitative changes in the system dynamics (that is obvious), but also cause the qualitative, and sometimes surprising changes in the vibration regimes. Furthermore, we show that scenarios of the transition from regular to chaotic regimes quantified by Fast Fourier Transform (FFT) can lead to erroneous conclusions, and a support of the wavelet analysis is needed. We have detected and illustrated the modifications of classical three scenarios of transition from regular vibrations to deterministic chaos. The carried out numerical experiment shows that the white noise lowers the threshold for transition into spatio-temporal chaotic dynamics. A transition into chaos via the proposed modified scenarios developed in this work is sensitive to small noise and significantly reduces occurrence of periodic vibrations. Increase of noise intensity yields decrease of the duration of the laminar signal range, i.e., time between two successive turbulent bursts decreases. Scenario of transition into chaos of the studied mechanical structures essentially depends on the control parameters, and it can be different in different zones of the constructed charts (control parameter planes). Furthermore, we found an interesting phenomenon, when increase of the noise intensity yields surprisingly the vibrational characteristics with a lack of noisy effect (chaos is destroyed by noise and windows of periodicity appear).
A challenge to chaotic itinerancy from brain dynamics
NASA Astrophysics Data System (ADS)
Kay, Leslie M.
2003-09-01
Brain hermeneutics and chaotic itinerancy proposed by Tsuda are attractive characterizations of perceptual dynamics in the mammalian olfactory system. This theory proposes that perception occurs at the interface between itinerant neural representation and interaction with the environment. Quantifiable application of these dynamics has been hampered by the lack of definable history and action processes which characterize the changes induced by behavioral state, attention, and learning. Local field potentials measured from several brain areas were used to characterize dynamic activity patterns for their use as representations of history and action processes. The signals were recorded from olfactory areas (olfactory bulb, OB, and pyriform cortex) and hippocampal areas (entorhinal cortex and dentate gyrus, DG) in the brains of rats. During odor-guided behavior the system shows dynamics at three temporal scales. Short time-scale changes are system-wide and can occur in the space of a single sniff. They are predictable, associated with learned shifts in behavioral state and occur periodically on the scale of the intertrial interval. These changes occupy the theta (2-12 Hz), beta (15-30 Hz), and gamma (40-100 Hz) frequency bands within and between all areas. Medium time-scale changes occur relatively unpredictably, manifesting in these data as alterations in connection strength between the OB and DG. These changes are strongly correlated with performance in associated trial blocks (5-10 min) and may be due to fluctuations in attention, mood, or amount of reward received. Long time-scale changes are likely related to learning or decline due to aging or disease. These may be modeled as slow monotonic processes that occur within or across days or even weeks or years. The folding of different time scales is proposed as a mechanism for chaotic itinerancy, represented by dynamic processes instead of static connection strengths. Thus, the individual maintains continuity of
Coexisting chaotic and periodic dynamics in clock escapements.
Moon, Francis C; Stiefel, Preston D
2006-09-15
This paper addresses the nature of noise in machines. As a concrete example, we examine the dynamics of clock escapements from experimental, historical and analytical points of view. Experiments on two escapement mechanisms from the Reuleaux kinematic collection at Cornell University are used to illustrate chaotic-like noise in clocks. These vibrations coexist with the periodic dynamics of the balance wheel or pendulum. A mathematical model is presented that shows how self-generated chaos in clocks can break the dry friction in the gear train. This model is shown to exhibit a strange attractor in the structural vibration of the clock. The internal feedback between the oscillator and the escapement structure is similar to anti-control of chaos models.
Escape dynamics and fractal basins boundaries in the three-dimensional Earth-Moon system
NASA Astrophysics Data System (ADS)
Zotos, Euaggelos E.
2016-03-01
The orbital dynamics of a spacecraft, or a comet, or an asteroid in the Earth-Moon system in a scattering region around the Moon using the three dimensional version of the circular restricted three-body problem is numerically investigated. The test particle can move in bounded orbits around the Moon or escape through the openings around the Lagrange points L1 and L2 or even collide with the surface of the Moon. We explore in detail the first four of the five possible Hill's regions configurations depending on the value of the Jacobi constant which is of course related with the total orbital energy. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits in several two-dimensional types of planes and distinguishing between four types of motion: (i) ordered bounded, (ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we locate the different basins and we relate them with the corresponding spatial distributions of the escape and collision times. Our outcomes reveal the high complexity of this planetary system. Furthermore, the numerical analysis suggests a strong dependence of the properties of the considered basins with both the total orbital energy and the initial value of the z coordinate, with a remarkable presence of fractal basin boundaries along all the regimes. Our results are compared with earlier ones regarding the planar version of the Earth-Moon system.
Generalized Gaussian wave packet dynamics: Integrable and chaotic systems
NASA Astrophysics Data System (ADS)
Pal, Harinder; Vyas, Manan; Tomsovic, Steven
2016-01-01
The ultimate semiclassical wave packet propagation technique is a complex, time-dependent Wentzel-Kramers-Brillouin method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle-point trajectories at its foundation are found using a multidimensional Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions that are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of ℏ that comes with using the saddle-point trajectories.
Chaotic dynamics and diffusion in a piecewise linear equation
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-15
Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
Generalized Gaussian wave packet dynamics: Integrable and chaotic systems.
Pal, Harinder; Vyas, Manan; Tomsovic, Steven
2016-01-01
The ultimate semiclassical wave packet propagation technique is a complex, time-dependent Wentzel-Kramers-Brillouin method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle-point trajectories at its foundation are found using a multidimensional Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions that are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of ℏ that comes with using the saddle-point trajectories.
Chaotic dynamics and diffusion in a piecewise linear equation
NASA Astrophysics Data System (ADS)
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-01
Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
RAPID DYNAMICAL MASS SEGREGATION AND PROPERTIES OF FRACTAL STAR CLUSTERS
Yu Jincheng; Chen Li; De Grijs, Richard
2011-05-01
We investigate the evolution of young star clusters using N-body simulations. We confirm that subvirial and fractal-structured clusters will dynamically mass segregate on a short timescale (within 0.5 Myr). We adopt a modified minimum-spanning-tree method to measure the degree of mass segregation, demonstrating that the stars escaping from a cluster's potential are important for the temporal dependence of mass segregation in the cluster. The form of the initial velocity distribution will also affect the degree of mass segregation. If it depends on radius, the outer parts of the cluster would expand without undergoing collapse. In velocity space, we find 'inverse mass segregation', which indicates that massive stars have higher velocity dispersions than their lower-mass counterparts.
Regular and chaotic dynamics of a piecewise smooth bouncer
Langer, Cameron K. Miller, Bruce N.
2015-07-15
The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with constant restitution coefficient, and (iii) inelastic with a velocity-dependent restitution function. We confirm the existence of distinct unbounded orbits (Fermi acceleration) in the elastic model, and investigate regular and chaotic behavior in the inelastic cases. We also examine in the constant restitution model trajectories wherein the particle experiences an infinite number of collisions in a finite time, i.e., the phenomenon of inelastic collapse. We address these so-called “sticking solutions” and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find the non-smooth nature of the system produces novel bifurcation phenomena not seen in the sinusoidal model, including border-collision bifurcations. The analytical and numerical investigations reveal that although our piecewise linear bouncer is a simplified version of the sinusoidal model, the former not only captures essential features of the latter but also exhibits behavior unique to the discontinuous dynamics.
NASA Astrophysics Data System (ADS)
Zhang, Zhe; Xun, Zhi-Peng; Wu, Ling; Chen, Yi-Li; Xia, Hui; Hao, Da-Peng; Tang, Gang
2016-06-01
In order to study the effects of the microscopic details of fractal substrates on the scaling behavior of the growth model, a generalized linear fractal Langevin-type equation, ∂h / ∂t =(- 1) m + 1 ν∇ mzrw h (zrw is the dynamic exponent of random walk on substrates), driven by nonconserved and conserved noise is proposed and investigated theoretically employing scaling analysis. Corresponding dynamic scaling exponents are obtained.
Universal behavior in the parametric evolution of chaotic saddles.
Lai, Y C; Zyczkowski, K; Grebogi, C
1999-05-01
Chaotic saddles are nonattracting dynamical invariant sets that physically lead to transient chaos. As a system parameter changes, chaotic saddles can evolve via an infinite number of homoclinic or heteroclinic tangencies of their stable and unstable manifolds. Based on previous numerical evidence and a rigorous analysis of a class of representative models, we show that dynamical invariants such as the topological entropy and the fractal dimension of chaotic saddles obey a universal behavior: they exhibit a devil-staircase characteristic as a function of the system parameter.
Group theoretic reduction of Laplacian dynamical problems on fractal lattices
Schwalm, W.A.; Schwalm, M.K.; Giona, M.
1997-06-01
Discrete forms of the Schr{umlt o}dinger equation, the diffusion equation, the linearized Landau-Ginzburg equation, and discrete models for vibrations and spin dynamics belong to a class of Laplacian-based finite difference models. Real-space renormalization of such models on finitely ramified regular fractals is known to give exact recursion relations. It is shown that these recursions commute with Lie groups representing continuous symmetries of the discrete models. Each such symmetry reduces the order of the renormalization recursions by one, resulting in a system of recursions with one fewer variable. Group trajectories are obtained from inverse images of fixed and invariant sets of the recursions. A subset of the Laplacian finite difference models can be mapped by change of boundary conditions and time dependence to a diffusion problem with closed boundaries. In such cases conservation of mass simplifies the group flow and obtaining the groups becomes easier. To illustrate this, the renormalization recursions for Green functions on four standard examples are decoupled. The examples are (1) the linear chain, (2) an anisotropic version of Dhar{close_quote}s 3-simplex, similar to a model dealt with by Hood and Southern, (3) the fourfold coordinated Sierpi{acute n}ski lattice of Rammal and of Domany {ital et al.}, and (4) a form of the Vicsek lattice. Prospects for applying the group theoretic method to more general dynamical systems are discussed. {copyright} {ital 1997} {ital The American Physical Society}
NASA Astrophysics Data System (ADS)
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.
2012-09-01
, this version of the package only deals with systems of first-order differential equations. Unusual features This package provides user-friendly software tools for analyzing the character of a dynamical system, whether it displays chaotic behaviour, and so on. Options within the package allow the user to specify characteristics that separate the trajectories into families of curves. In conjunction with the facilities for altering the user's viewpoint, this provides a graphical interface for the speedy and easy identification of regions with interesting dynamics. An unusual characteristic of the package is its interface for performing the numerical integrations in C using a fifth-order Runge-Kutta method (default). This potentially improves the speed of the numerical integration by some orders of magnitude and, in cases where it is necessary to calculate thousands of graphs in regions of difficult integration, this feature is very desirable. Besides that tool, somewhat more experienced users can produce their own C integrator and, by using the commands available in the package, use it as the C integrator provided with the package as long as the new integrator manages the input and output in the same format as the default one does. Running time This depends strongly on the dynamical system. With an Intel® Core™ i3 CPU M330 @ 2.13 GHz, the integration of 50 graphs, for a system of two first-order equations, typically takes less than a second to run (with the C integration interface). Without the C interface, it takes a few seconds. In order to calculate the fractal dimension, where we typically use 10,000 points to integrate, using the C interface it takes from 20 to 30 s. Without the C interface, it becomes really impractical, taking, sometimes, for the same case, almost an hour. For some cases, it takes many hours.
Chaotic features of nuclear structure and dynamics: selected topics
NASA Astrophysics Data System (ADS)
Zelevinsky, Vladimir; Volya, Alexander
2016-03-01
Quantum chaos has become an important element of our knowledge about physics of complex systems. In typical mesoscopic systems of interacting particles the dynamics invariably become chaotic when the level density, growing by combinatorial reasons, leads to the increasing probability of mixing simple mean-field (particle-hole) configurations. The resulting stationary states have exceedingly complicated structures that are comparable to those in random matrix theory. We discuss the main properties of mesoscopic quantum chaos and show that it can serve as a justification for application of statistical mechanics to mesoscopic systems. We show that quantum chaos becomes a powerful instrument for experimental, theoretical and computational work. The generalization to open systems and effects in the continuum are discussed with the help of the effective non-Hermitian Hamiltonian; it is shown how to formulate this approach for numerous problems of quantum signal transmission. The artificially introduced randomness can also be helpful for a deeper understanding of physics. We indicate the problems that require more investigation so as to be understood further.
Efficient sensitivity analysis method for chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Liao, Haitao
2016-05-01
The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.
OPEN PROBLEM: Orbits' statistics in chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Arnold, V.
2008-07-01
This paper shows how the measurement of the stochasticity degree of a finite sequence of real numbers, published by Kolmogorov in Italian in a journal of insurances' statistics, can be usefully applied to measure the objective stochasticity degree of sequences, originating from dynamical systems theory and from number theory. Namely, whenever the value of Kolmogorov's stochasticity parameter of a given sequence of numbers is too small (or too big), one may conclude that the conjecture describing this sequence as a sample of independent values of a random variables is highly improbable. Kolmogorov used this strategy fighting (in a paper in 'Doklady', 1940) against Lysenko, who had tried to disprove the classical genetics' law of Mendel experimentally. Calculating his stochasticity parameter value for the numbers from Lysenko's experiment reports, Kolmogorov deduced, that, while these numbers were different from the exact fulfilment of Mendel's 3 : 1 law, any smaller deviation would be a manifestation of the report's number falsification. The calculation of the values of the stochasticity parameter would be useful for many other generators of pseudorandom numbers and for many other chaotically looking statistics, including even the prime numbers distribution (discussed in this paper as an example).
Direct chaotic dynamics to any desired orbits via a closed-loop control
NASA Astrophysics Data System (ADS)
Chen, Chien-Chong
1996-02-01
A modification of the chaotic entrainment method by superimposing a feedback control term can successfully control dynamical systems to any desired orbits (goal dynamics). Numerical studies show that autonomous Lorenz systems can be controlled to arbitrary goal dynamics such as flat profiles, simple and highly modulated oscillations. Also a non-autonomous Duffing-Holmes oscillator can be stabilized to a chosen dynamics.
Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans
Skardal, Per Sebastian; Restrepo, Juan G.
2014-12-15
The spatiotemporal dynamics of cardiac tissue is an active area of research for biologists, physicists, and mathematicians. Of particular interest is the study of period-doubling bifurcations and chaos due to their link with cardiac arrhythmogenesis. In this paper, we study the spatiotemporal dynamics of a recently developed model for calcium-driven alternans in a one dimensional cable of tissue. In particular, we observe in the cable coexistence of regions with chaotic and multi-periodic dynamics over wide ranges of parameters. We study these dynamics using global and local Lyapunov exponents and spatial trajectory correlations. Interestingly, near nodes—or phase reversals—low-periodic dynamics prevail, while away from the nodes, the dynamics tend to be higher-periodic and eventually chaotic. Finally, we show that similar coexisting multi-periodic and chaotic dynamics can also be observed in a detailed ionic model.
Fractal dynamics of heartbeat time series of young persons with metabolic syndrome
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; Alonso-Martínez, A.; Ramírez-Hernández, L.; Martínez-Hernández, G.
2012-10-01
Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.
Kinematic variability, fractal dynamics and local dynamic stability of treadmill walking
2011-01-01
Background Motorized treadmills are widely used in research or in clinical therapy. Small kinematics, kinetics and energetics changes induced by Treadmill Walking (TW) as compared to Overground Walking (OW) have been reported in literature. The purpose of the present study was to characterize the differences between OW and TW in terms of stride-to-stride variability. Classical (Standard Deviation, SD) and non-linear (fractal dynamics, local dynamic stability) methods were used. In addition, the correlations between the different variability indexes were analyzed. Methods Twenty healthy subjects performed 10 min TW and OW in a random sequence. A triaxial accelerometer recorded trunk accelerations. Kinematic variability was computed as the average SD (MeanSD) of acceleration patterns among standardized strides. Fractal dynamics (scaling exponent α) was assessed by Detrended Fluctuation Analysis (DFA) of stride intervals. Short-term and long-term dynamic stability were estimated by computing the maximal Lyapunov exponents of acceleration signals. Results TW did not modify kinematic gait variability as compared to OW (multivariate T2, p = 0.87). Conversely, TW significantly modified fractal dynamics (t-test, p = 0.01), and both short and long term local dynamic stability (T2 p = 0.0002). No relationship was observed between variability indexes with the exception of significant negative correlation between MeanSD and dynamic stability in TW (3 × 6 canonical correlation, r = 0.94). Conclusions Treadmill induced a less correlated pattern in the stride intervals and increased gait stability, but did not modify kinematic variability in healthy subjects. This could be due to changes in perceptual information induced by treadmill walking that would affect locomotor control of the gait and hence specifically alter non-linear dependencies among consecutive strides. Consequently, the type of walking (i.e. treadmill or overground) is important to consider in each protocol
Dynamics, Analysis and Implementation of a Multiscroll Memristor-Based Chaotic Circuit
NASA Astrophysics Data System (ADS)
Alombah, N. Henry; Fotsin, Hilaire; Ngouonkadi, E. B. Megam; Nguazon, Tekou
This article introduces a novel four-dimensional autonomous multiscroll chaotic circuit which is derived from the actual simplest memristor-based chaotic circuit. A fourth circuit element — another inductor — is introduced to generate the complex behavior observed. A systematic study of the chaotic behavior is performed with the help of some nonlinear tools such as Lyapunov exponents, phase portraits, and bifurcation diagrams. Multiple scroll attractors are observed in Matlab, Pspice environments and also experimentally. We also observe the phenomenon of antimonotonicity, periodic and chaotic bubbles, multiple periodic-doubling bifurcations, Hopf bifurcations, crises and the phenomenon of intermittency. The chaotic dynamics of this circuit is realized by laboratory experiments, Pspice simulations, numerical and analytical investigations. It is observed that the results from the three environments agree to a great extent. This topology is likely convenient to be used to intentionally generate chaos in memristor-based chaotic circuit applications, given the fact that multiscroll chaotic systems have found important applications as broadband signal generators, pseudorandom number generators for communication engineering and also in biometric authentication.
New Dynamical Insights on the Global Behavior of Chaotic Attractors
NASA Astrophysics Data System (ADS)
Jones, Timothy Douglas
A paraphrase of Tolstoy that has become popular in the field of nonlinear dynamics is that while all linear systems are linear in the same way, all nonlinear systems are nonlinear in their own ways. Despite this being quite true, there can be found a number of universal features in nonlinear systems which unify them in ways that enhance our understanding of their behavior. That nature is replete with nonlinear systems has proven to be a great challenge to our scientific understanding of the world. And while mathematics has proven to be apt at describing a multitude of physical phenomenon in the form of deterministic equations which describe future behavior based on a system's current state, it in and of itself held a rather shocking surprise which is now called Chaos. In Chaos we find deterministic systems which, due to our lack of omniscience, and the physical impossibility of building computers with infinite precision, become wildly unpredictable as they evolve in time. A number of new tools were developed to understand these systems, including a powerful program of topological analysis which has been completed for three dimensions. Yet, there still remains a number of unanswered dynamical questions about chaotic systems. Two such questions are the primary focus of this thesis. The first question we will address is regarding the general shape of the strange attractor. Specifically, what can we learn about the shape of strange attractor from the dynamical equations without numerically integrating them? For example, the Rossler and Lorenz attractors have remarkably similar dynamical equations, and yet are topologically very distinct. There is no self-evident relation between the dynamical equations that describe a strange attractor and its shape in phase space. Previously, we only had the fixed points to act as general guides as to the shape of the attractor, but these point sets are not exceedingly descriptive. We will outline work done to find more interesting
Nonlinear dynamics of drops and bubbles and chaotic phenomena
NASA Technical Reports Server (NTRS)
Trinh, Eugene H.; Leal, L. G.; Feng, Z. C.; Holt, R. G.
1994-01-01
Nonlinear phenomena associated with the dynamics of free drops and bubbles are investigated analytically, numerically and experimentally. Although newly developed levitation and measurement techniques have been implemented, the full experimental validation of theoretical predictions has been hindered by interfering artifacts associated with levitation in the Earth gravitational field. The low gravity environment of orbital space flight has been shown to provide a more quiescent environment which can be utilized to better match the idealized theoretical conditions. The research effort described in this paper is a closely coupled collaboration between predictive and guiding theoretical activities and a unique experimental program involving the ultrasonic and electrostatic levitation of single droplets and bubbles. The goal is to develop and to validate methods based on nonlinear dynamics for the understanding of the large amplitude oscillatory response of single drops and bubbles to both isotropic and asymmetric pressure stimuli. The first specific area on interest has been the resonant coupling between volume and shape oscillatory modes isolated gas or vapor bubbles in a liquid host. The result of multiple time-scale asymptotic treatment, combined with domain perturbation and bifurcation methods, has been the prediction of resonant and near-resonant coupling between volume and shape modes leading to stable as well as chaotic oscillations. Experimental investigations of the large amplitude shape oscillation modes of centimeter-size single bubbles trapped in water at 1 G and under reduced hydrostatic pressure, have suggested the possibility of a low gravity experiment to study the direct coupling between these low frequency shape modes and the volume pulsation, sound-radiating mode. The second subject of interest has involved numerical modeling, using the boundary integral method, of the large amplitude shape oscillations of charged and uncharged drops in the presence
NASA Astrophysics Data System (ADS)
Mariño, Inés P.; Míguez, Joaquín; Meucci, Riccardo
2009-05-01
We propose a Monte Carlo methodology for the joint estimation of unobserved dynamic variables and unknown static parameters in chaotic systems. The technique is sequential, i.e., it updates the variable and parameter estimates recursively as new observations become available, and, hence, suitable for online implementation. We demonstrate the validity of the method by way of two examples. In the first one, we tackle the estimation of all the dynamic variables and one unknown parameter of a five-dimensional nonlinear model using a time series of scalar observations experimentally collected from a chaotic CO2 laser. In the second example, we address the estimation of the two dynamic variables and the phase parameter of a numerical model commonly employed to represent the dynamics of optoelectronic feedback loops designed for chaotic communications over fiber-optic links.
Periodic and Chaotic Dynamics of the Ehrhard-Müller System
NASA Astrophysics Data System (ADS)
Park, Junho; Lee, Hyunho; Baik, Jong-Jin
2016-06-01
This paper investigates nonlinear ordinary differential equations of the Ehrhard-Müller system which describes natural convection in a single-phase loop in the presence of nonsymmetric heating. Stability and dynamics of periodic and chaotic behaviors of the equations are investigated and the periodicity diagram is obtained in wide ranges of parameters. Regimes of both periodic and chaotic solutions are observed with complex behaviors such that the periodic regimes enclose the chaotic regime while they are also immersed inside the chaotic regime with various shapes. An asymptotic analysis is performed for sufficiently large parameters to understand the enclosure by the periodic regimes and asymptotic limit cycles are obtained to compare with limit cycles obtained from numerical results.
Richness of chaotic dynamics in nonholonomic models of a celtic stone
NASA Astrophysics Data System (ADS)
Gonchenko, Alexander S.; Gonchenko, Sergey V.; Kazakov, Alexey O.
2013-09-01
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Dynamic behaviors of fractal-like domains in monolayers
NASA Astrophysics Data System (ADS)
Wang, Mu; Sun, Cheng; Peng, Ru-Wen; Ming, Nai-Ben; Esch, Jan Van; Ringsdorf, Helmut; Bennema, Piet
1996-06-01
In this paper we report our recent investigations on the morphological evolution of fractal-like domains of the liquid-condensed (LC) phase in lipid monolayers. It is demonstrated that the dimension of the LC domains increases upon continuous illumination of microscope light. The experimental data indicate that the increasing rate of fractal dimension of the LC domains depends on the concentration of fluorescence probes. By analyzing the domain growth process we find that the self-similarity of the pattern disappears gradually during its growth. The possible mechanism behind the observed phenomena is discussed.
Jamming and chaotic dynamics in different granular systems
NASA Astrophysics Data System (ADS)
Maghsoodi, Homayoon; Luijten, Erik
Although common in nature and industry, the jamming transition has long eluded a concrete, mechanistic explanation. Recently, Banigan et al. (Nat. Phys. 9, 288-292, 2013) proposed a method for characterizing this transition in a granular system in terms of the system's chaotic properties, as quantified by the largest Lyapunov exponent. They demonstrated that in a two-dimensional shear cell the jamming transition coincides with the bulk density at which the system's largest Lyapunov exponent changes sign, indicating a transition between chaotic and non-chaotic regimes. To examine the applicability of this observation to realistic granular systems, we study a model that includes frictional forces within an expanded phase space. Furthermore, we test the generality of the relation between chaos and jamming by investigating the relationship between jamming and the chaotic properties of several other granular systems, notably sheared systems (Howell, D., Behringer R. P., Veje C., Phys. Rev. Lett. 82, 5241-5244, 1999) and systems with a free boundary. Finally, we quantify correlations between the largest Lyapunov vector and collective rearrangements of the system to demonstrate the predictive capabilities enabled by adopting this perspective of jamming.
Resistive magnetohydrodynamic reconnection: Resolving long-term, chaotic dynamics
Keppens, R.; Restante, A. L.; Lapenta, G.; Porth, O.; Galsgaard, K.; Frederiksen, J. T.; Parnell, C.
2013-09-15
In this paper, we address the long-term evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (grid-adaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shock-capturing, conservative, grid-adaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco-)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shock-capturing, high order finite differences, and particle in cell-like methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.
Recurrence Quantification of Fractal Structures
Webber, Charles L.
2012-01-01
By definition, fractal structures possess recurrent patterns. At different levels repeating patterns can be visualized at higher magnifications. The purpose of this chapter is threefold. First, general characteristics of dynamical systems are addressed from a theoretical mathematical perspective. Second, qualitative and quantitative recurrence analyses are reviewed in brief, but the reader is directed to other sources for explicit details. Third, example mathematical systems that generate strange attractors are explicitly defined, giving the reader the ability to reproduce the rich dynamics of continuous chaotic flows or discrete chaotic iterations. The challenge is then posited for the reader to study for themselves the recurrent structuring of these different dynamics. With a firm appreciation of the power of recurrence analysis, the reader will be prepared to turn their sights on real-world systems (physiological, psychological, mechanical, etc.). PMID:23060808
Chaotic pendulum: The complete attractor
NASA Astrophysics Data System (ADS)
DeSerio, Robert
2003-03-01
A commercial chaotic pendulum is modified to study nonlinear dynamics, including the determination of Poincaré sections, fractal dimensions, and Lyapunov exponents. The apparatus is driven by a simple oscillating mechanism powered by a 200 pulse per revolution stepper motor running at constant angular velocity. A computer interface generates the uniform pulse train needed to run the stepper motor and, with each pulse, reads a rotary encoder attached to the pendulum axle. Ten million readings from overnight runs of 50 000 drive cycles were smoothed and differentiated to obtain the pendulum angle θ and the angular velocity ω at each pulse of the drive. A plot of the 50 000 (θ,ω) phase points corresponding to one phase of the drive system produces a single Poincaré section. Thus, 200 Poincaré sections are experimentally available, one at each step of the drive. Viewed separately, any one of them strikingly illustrates the fractal geometry of the underlying chaotic attractor. Viewed sequentially in a repeating loop, they demonstrate the stretching and folding of phase point density typical of chaotic dynamics. Results for four pendulum damping conditions are presented and compared.
Chaotic ray dynamics in an optical cavity with a beam splitter.
Puentes, Graciana; Aiello, Andrea; Woerdman, J P
2004-05-01
We investigate the ray dynamics in an optical cavity when a ray-splitting mechanism is present. The cavity is a conventional two-mirror stable resonator, and the ray splitting is achieved by inserting an optical beam splitter perpendicular to the cavity axis. Using Hamiltonian optics, we show that such a simple device presents surprisingly rich chaotic ray dynamics.
NASA Astrophysics Data System (ADS)
Martienssen, W.; Hübinger, B.; Doerner, R.
A method to transfer secret information using chaotic dynamical systems is proposed. It is based on modulating a chaotic system with the message such that its time evolution contains the hidden information. Decryption of the cipher is achieved by chaos control. Operation of the scheme is demonstrated by en- and decoding a short german text.
Sunada, Satoshi; Harayama, Takahisa; Davis, Peter; Tsuzuki, Ken; Arai, Ken-Ichi; Yoshimura, Kazuyuki; Uchida, Atsushi
2012-12-01
We present an experimental method for directly observing the amplification of microscopic intrinsic noise in a high-dimensional chaotic laser system, a laser with delayed feedback. In the experiment, the chaotic laser system is repeatedly switched from a stable lasing state to a chaotic state, and the time evolution of an ensemble of chaotic states starting from the same initial state is measured. It is experimentally demonstrated that intrinsic noises amplified by the chaotic dynamics are transformed into macroscopic fluctuating signals, and the probability density of the output light intensity actually converges to a natural invariant probability density in a strongly chaotic regime. Moreover, with the experimental method, we discuss the application of the chaotic laser systems to physical random bit generators. It is experimentally shown that the convergence to the invariant density plays an important role in nondeterministic random bit generation, which could be desirable for future ultimate secure communication systems.
Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays
NASA Astrophysics Data System (ADS)
Suzuki, Yoko; Lu, Mingyang; Ben-Jacob, Eshel; Onuchic, José N.
2016-02-01
Regulatory gene circuit motifs play crucial roles in performing and maintaining vital cellular functions. Frequently, theoretical studies of gene circuits focus on steady-state behaviors and do not include time delays. In this study, the inclusion of time delays is shown to entirely change the time-dependent dynamics for even the simplest possible circuits with one and two gene elements with self and cross regulations. These elements can give rise to rich behaviors including periodic, quasi-periodic, weak chaotic, strong chaotic and intermittent dynamics. We introduce a special power-spectrum-based method to characterize and discriminate these dynamical modes quantitatively. Our simulation results suggest that, while a single negative feedback loop of either one- or two-gene element can only have periodic dynamics, the elements with two positive/negative feedback loops are the minimalist elements to have chaotic dynamics. These elements typically have one negative feedback loop that generates oscillations, and another unit that allows frequent switches among multiple steady states or between oscillatory and non-oscillatory dynamics. Possible dynamical features of several simple one- and two-gene elements are presented in details. Discussion is presented for possible roles of the chaotic behavior in the robustness of cellular functions and diseases, for example, in the context of cancer.
Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays
Suzuki, Yoko; Lu, Mingyang; Ben-Jacob, Eshel; Onuchic, José N.
2016-01-01
Regulatory gene circuit motifs play crucial roles in performing and maintaining vital cellular functions. Frequently, theoretical studies of gene circuits focus on steady-state behaviors and do not include time delays. In this study, the inclusion of time delays is shown to entirely change the time-dependent dynamics for even the simplest possible circuits with one and two gene elements with self and cross regulations. These elements can give rise to rich behaviors including periodic, quasi-periodic, weak chaotic, strong chaotic and intermittent dynamics. We introduce a special power-spectrum-based method to characterize and discriminate these dynamical modes quantitatively. Our simulation results suggest that, while a single negative feedback loop of either one- or two-gene element can only have periodic dynamics, the elements with two positive/negative feedback loops are the minimalist elements to have chaotic dynamics. These elements typically have one negative feedback loop that generates oscillations, and another unit that allows frequent switches among multiple steady states or between oscillatory and non-oscillatory dynamics. Possible dynamical features of several simple one- and two-gene elements are presented in details. Discussion is presented for possible roles of the chaotic behavior in the robustness of cellular functions and diseases, for example, in the context of cancer. PMID:26876008
Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays.
Suzuki, Yoko; Lu, Mingyang; Ben-Jacob, Eshel; Onuchic, José N
2016-02-15
Regulatory gene circuit motifs play crucial roles in performing and maintaining vital cellular functions. Frequently, theoretical studies of gene circuits focus on steady-state behaviors and do not include time delays. In this study, the inclusion of time delays is shown to entirely change the time-dependent dynamics for even the simplest possible circuits with one and two gene elements with self and cross regulations. These elements can give rise to rich behaviors including periodic, quasi-periodic, weak chaotic, strong chaotic and intermittent dynamics. We introduce a special power-spectrum-based method to characterize and discriminate these dynamical modes quantitatively. Our simulation results suggest that, while a single negative feedback loop of either one- or two-gene element can only have periodic dynamics, the elements with two positive/negative feedback loops are the minimalist elements to have chaotic dynamics. These elements typically have one negative feedback loop that generates oscillations, and another unit that allows frequent switches among multiple steady states or between oscillatory and non-oscillatory dynamics. Possible dynamical features of several simple one- and two-gene elements are presented in details. Discussion is presented for possible roles of the chaotic behavior in the robustness of cellular functions and diseases, for example, in the context of cancer.
Estimating the level of dynamical noise in time series by using fractal dimensions
NASA Astrophysics Data System (ADS)
Sase, Takumi; Ramírez, Jonatán Peña; Kitajo, Keiichi; Aihara, Kazuyuki; Hirata, Yoshito
2016-03-01
We present a method for estimating the dynamical noise level of a 'short' time series even if the dynamical system is unknown. The proposed method estimates the level of dynamical noise by calculating the fractal dimensions of the time series. Additionally, the method is applied to EEG data to demonstrate its possible effectiveness as an indicator of temporal changes in the level of dynamical noise.
Robust PRNG based on homogeneously distributed chaotic dynamics
NASA Astrophysics Data System (ADS)
Garasym, Oleg; Lozi, René; Taralova, Ina
2016-02-01
This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two topologically coupled maps are studied: TTL rc non-alternate, and TTL SC alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim two new nonlinear CPRNG: MTTL 2 sc and NTTL 2 are proposed. The maps successfully passed numerous statistical, graphical and numerical tests, due to proposed ring coupling and injection mechanisms.
NASA Astrophysics Data System (ADS)
Chen, Yun; Yang, Hui
2016-08-01
Engineered and natural systems often involve irregular and self-similar geometric forms, which is called fractal geometry. For instance, precision machining produces a visually flat surface, while which looks like a rough mountain in the nanometer scale under the microscope. Human heart consists of a fractal network of muscle cells, Purkinje fibers, arteries and veins. Cardiac electrical activity exhibits highly nonlinear and fractal behaviors. Although space-time dynamics occur on the fractal geometry, e.g., chemical etching on the surface of machined parts and electrical conduction in the heart, most of existing works modeled space-time dynamics (e.g., reaction, diffusion and propagation) on the Euclidean geometry (e.g., flat planes and rectangular volumes). This brings inaccurate approximation of real-world dynamics, due to sensitive dependence of nonlinear dynamical systems on initial conditions. In this paper, we developed novel methods and tools for the numerical simulation and pattern recognition of spatiotemporal dynamics on fractal surfaces of complex systems, which include (1) characterization and modeling of fractal geometry, (2) fractal-based simulation and modeling of spatiotemporal dynamics, (3) recognizing and quantifying spatiotemporal patterns. Experimental results show that the proposed methods outperform traditional modeling approaches based on the Euclidean geometry, and provide effective tools to model and characterize space-time dynamics on fractal surfaces of complex systems.
Combinatorial Optimization by Amoeba-Based Neurocomputer with Chaotic Dynamics
NASA Astrophysics Data System (ADS)
Aono, Masashi; Hirata, Yoshito; Hara, Masahiko; Aihara, Kazuyuki
We demonstrate a computing system based on an amoeba of a true slime mold Physarum capable of producing rich spatiotemporal oscillatory behavior. Our system operates as a neurocomputer because an optical feedback control in accordance with a recurrent neural network algorithm leads the amoeba's photosensitive branches to search for a stable configuration concurrently. We show our system's capability of solving the traveling salesman problem. Furthermore, we apply various types of nonlinear time series analysis to the amoeba's oscillatory behavior in the problem-solving process. The results suggest that an individual amoeba might be characterized as a set of coupled chaotic oscillators.
Fractal dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
The fractal dimension of the protein aggregates can be estimated by dynamic mechanical methods when the particle aggregates are imbedded in a polymer matrix. Nanocomposites were formed by mixing hydrolyzed soy protein isolate (HSPI) nanoparticle aggregates with styrene-butadiene (SB) latex, followe...
Chaotic behavior in the dynamical system of a continuous stirred tank reactor
NASA Astrophysics Data System (ADS)
Retzloff, D. G.; Chan, P. C.-H.; Chicone, C.; Offin, D.; Mohamed, R.
1987-03-01
The dynamical system describing a continuous stirred tank reactor (CSTR) for the reactions A→B→C and A→C, B→D is considered. A circulating attractor with accompanying circulating orbits is shown to exist when the critical point of the system is unique and unstable. The orbit structure has been numerically found to consist of periodic orbits and chaotic behavior.
[Regular and chaotic dynamics with applications in nonlinear optics]. Final report
Kovacic, G.
1998-10-12
The following major pieces of work were completed under the sponsorship of this grant: (1) singular perturbation theory for dynamical systems; (2) homoclinic orbits and chaotic dynamics in second-harmonic generating, optically pumped, passive optical cavities; (3) chaotic dynamics in short ring-laser cavities; (4) homoclinic orbits in moderately-long ring-laser cavities; (5) finite-dimensional attractor in ring-laser cavities; (6) turbulent dynamics in long ring-laser cavities; (7) bifurcations in a model for a free-boundary problem for the heat equation; (8) weakly nonlinear dynamics of interface propagation; (9) slowly periodically forced planar Hamiltonian systems; and (10) soliton spectrum of the solutions of the nonlinear Schroedinger equation. A brief summary of the research is given for each project.
Dynamic synchronization of a time-evolving optical network of chaotic oscillators.
Cohen, Adam B; Ravoori, Bhargava; Sorrentino, Francesco; Murphy, Thomas E; Ott, Edward; Roy, Rajarshi
2010-12-01
We present and experimentally demonstrate a technique for achieving and maintaining a global state of identical synchrony of an arbitrary network of chaotic oscillators even when the coupling strengths are unknown and time-varying. At each node an adaptive synchronization algorithm dynamically estimates the current strength of the net coupling signal to that node. We experimentally demonstrate this scheme in a network of three bidirectionally coupled chaotic optoelectronic feedback loops and we present numerical simulations showing its application in larger networks. The stability of the synchronous state for arbitrary coupling topologies is analyzed via a master stability function approach.
Minati, Ludovico
2014-09-01
In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals. PMID:25273190
Minati, Ludovico E-mail: ludovico.minati@unitn.it
2014-09-01
In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals.
Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics
NASA Astrophysics Data System (ADS)
Altmann, Eduardo G.
2009-01-01
The chaotic ray dynamics inside dielectric cavities is described by the properties of an invariant chaotic saddle. The localization of the far-field emission in specific directions, recently observed in different experiments and wave simulations, is found to be a consequence of the filamentary pattern of the saddle’s unstable manifold. For cavities with mixed phase space, the chaotic saddle is divided in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential (t
Verification of chaotic behavior in an experimental loudspeaker.
Reiss, Joshua D; Djurek, Ivan; Petosic, Antonio; Djurek, Danijel
2008-10-01
The dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time series analysis. Delay time, embedding dimension, fractal dimension, and other empirical quantities are determined from experimental data. Particular attention is paid to issues of stationarity in a system in order to identify sources of uncertainty. Lyapunov exponents and fractal dimension are measured using several independent techniques. Results are compared in order to establish independent confirmation of low dimensional dynamics and a positive dominant Lyapunov exponent. We thus show that the loudspeaker may function as a chaotic system suitable for low dimensional modeling and the application of chaos control techniques.
Distribution of resonance strengths in microwave billiards of mixed and chaotic dynamics
NASA Astrophysics Data System (ADS)
Dembowski, C.; Dietz, B.; Friedrich, T.; Gräf, H.-D.; Harney, H. L.; Heine, A.; Miski-Oglu, M.; Richter, A.
2005-04-01
A new measure for statistical properties of the wave function components of quantum systems, the distribution of the product of two partial widths, is introduced. It is tested with data obtained in analog experiments with microwave billiards, where the product of two partial widths equals the resonance strengths in the microwave spectra. The billiards are from the family of the Limaçons, one with chaotic and two with mixed classical dynamics. For completely chaotic systems the partial widths generically obey a Porter-Thomas distribution. We show that in this case the distribution of their product equals a K0 distribution. While we find deviations of the experimental strength distribution from the K0 distribution for the billiards with mixed dynamics, the distributions agree perfectly for the chaotic billiard, when taking into account the experimental threshold of detection in the theoretical description. Hence, the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory.
Generalized correlation integral vectors: A distance concept for chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Hakkarainen, Janne; Haario, Heikki; Kalachev, Leonid
2016-04-01
It is difficult to distinguish systematic trends from natural variability of data in many important applications, such as weather or climate systems. The practical challenge of estimating parameters in chaotic systems is related to the fact that a fixed model parameter does not correspond to a unique model integration, but to a set of quite different solutions as obtained for example by setting slightly different initial values. But while all such trajectories are different, they approximate the same underlying attractor and should be considered in this sense equivalent. In this paper, we propose a statistical approach to quantify such "sameness" of trajectories, and to distinguish trajectories that are significantly different. Various formulations of fractal dimensions have been developed to characterize the geometry of such attractors. The aim of this paper is to modify one of these, the so-called correlation dimension, to develop a way to quantify the variability of samples of an attractor by mapping the respective phase space trajectories onto vectors, whose statistical distribution can be empirically estimated. The distributions turn out to be Gaussian, which provide us a well-defined statistical tool to compare the trajectories. We use the approach for the task of parameter estimation of chaotic systems. The methodology is illustrated using computational examples for both low and high dimensional systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.
Carbone, Francesco; De Luca, Antonio; Barna, Valentin; Ferjani, Sameh; Vena, Carlo; Versace, Carlo; Strangi, Giuseppe
2009-08-01
An important effect of dynamical localization of light waves in liquid crystal electro-hydrodynamic instabilities is reported by investigating coherent backscattering effects. Recurrent multiple scattering in dynamic and chaotic complex fluids lead to a cone of enhanced backscattered light. The cone width and the related mean free path dependence on the dynamic scattering regimes emphasize the diverse light localization scales related to the internal structures present in the sample. The systems investigated up to now were mainly nano-powdered solutions or biological tissues, without any external control on the disorder. Here, an anisotropic complex fluid is "driven" throughout chaotic regimes by an external electric field, giving rise to dynamics that evolve through several spatio-temporal patterns.
Desktop chaotic systems: Intuition and visualization
NASA Technical Reports Server (NTRS)
Bright, Michelle M.; Melcher, Kevin J.; Qammar, Helen K.; Hartley, Tom T.
1993-01-01
This paper presents a dynamic study of the Wildwood Pendulum, a commercially available desktop system which exhibits a strange attractor. The purpose of studying this chaotic pendulum is twofold: to gain insight in the paradigmatic approach of modeling, simulating, and determining chaos in nonlinear systems; and to provide a desktop model of chaos as a visual tool. For this study, the nonlinear behavior of this chaotic pendulum is modeled, a computer simulation is performed, and an experimental performance is measured. An assessment of the pendulum in the phase plane shows the strange attractor. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined for both the model and the experimental pendulum systems. Correlation dimension results indicate that the pendulum and the model are chaotic and their fractal dimensions are similar.
An analytical and numerical study of chaotic dynamics in a simple bouncing ball model
NASA Astrophysics Data System (ADS)
Okninski, Andrzej; Radziszewski, Bogusław
2011-02-01
Dynamics of a ball moving in gravitational field and colliding with a moving table is studied in this paper. The motion of the limiter is assumed as periodic with piecewise constant velocity—it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity. The Poincaré map, describing evolution from an impact to the next impact, is derived and scenarios of transition to chaotic dynamics are investigated analytically and numerically.
Multifractality and the effect of turbulence on the chaotic dynamics of a HeNe laser
NASA Astrophysics Data System (ADS)
Gulich, Damián.; Zunino, Luciano; Pérez, Darío.; Garavaglia, Mario
2013-09-01
We propose the use of multifractal detrended fluctuation analysis (MF-DFA) to measure the influence of atmospheric turbulence on the chaotic dynamics of a HeNe laser. Fit ranges for MF-DFA are obtained with goodness of linear fit (GoLF) criterion. The chaotic behavior is generated by means of a simple interferometric setup with a feedback to the cavity of the gas laser. Such dynamics have been studied in the past and modeled as a function of the feedback level. Different intensities of isotropic turbulence have been generated with a turbulator device, allowing a structure constant for the index of refraction of air adjustable by means of a temperature difference parameter in the unit. Considering the recent interest in message encryption with this kind of setups, the study of atmospheric turbulence effects plays a key role in the field of secure laser communication through the atmosphere. In principle, different intensities of turbulence may be interpreted as different levels of white noise on the original chaotic series. These results can be of utility for performance optimization in chaotic free-space laser communication systems.
Structure of the Asteroid Belt from the Gas Giants' Growth and Chaotic Dynamics
NASA Astrophysics Data System (ADS)
Izidoro, André; Raymond, Sean N.; Pierens, Arnaud; Morbidelli, Alessandro; Winter, Othon; Nesvorny, David
2016-05-01
The structure of the asteroid belt holds a record of the Solar System's dynamical history. The current belt only contains 10-3 Earth masses yet the asteroids' orbits are dynamically excited, with a large spread in eccentricity and inclination. The belt is also chemically segregated: the inner belt is dominated by dry S-types and the outer belt by hydrated C-types. Here we propose a new model in which the asteroid belt was always low-mass and was partially populated and sculpted by the giant planets on chaotic, resonant orbits. We first show that the compositional dichotomy of the asteroid belt is a simple consequence of Jupiter's growth in the gaseous protoplanetary disk. As Jupiter's core rapidly grew by accreting gas, orbits of nearby planetesimals were perturbed onto Jupiter-crossing trajectories. A significant fraction (~10%) of objects in the neighborhood exterior of Jupiter's orbit were implanted by gas drag into the outer parts of the asteroid belt as C-types. While the gas giants were likely in mean motion resonance at the end of the gaseous disk phase, we show that small perturbations may have driven them into a chaotic but stable state. After the dissipation of the gaseous disk, stochastic variations in the gas giants orbits caused resonances to chaotically jump across the main belt and excite the asteroids' orbits. Our results suggest that the early Solar System was chaotic and introduce a simple framework to understand the origins of the asteroid belt.
Quantum chaotic scattering in graphene systems in the absence of invariant classical dynamics.
Wang, Guang-Lei; Ying, Lei; Lai, Ying-Cheng; Grebogi, Celso
2013-05-01
Quantum chaotic scattering is referred to as the study of quantum behaviors of open Hamiltonian systems that exhibit transient chaos in the classical limit. Traditionally a central issue in this field is how the elements of the scattering matrix or their functions fluctuate as a system parameter, e.g., the electron Fermi energy, is changed. A tacit hypothesis underlying previous works was that the underlying classical phase-space structure remains invariant as the parameter varies, so semiclassical theory can be used to explain various phenomena in quantum chaotic scattering. There are, however, experimental situations where the corresponding classical chaotic dynamics can change characteristically with some physical parameter. Multiple-terminal quantum dots are one such example where, when a magnetic field is present, the classical chaotic-scattering dynamics can change between being nonhyperbolic and being hyperbolic as the Fermi energy is changed continuously. For such systems semiclassical theory is inadequate to account for the characteristics of conductance fluctuations with the Fermi energy. To develop a general framework for quantum chaotic scattering associated with variable classical dynamics, we use multi-terminal graphene quantum-dot systems as a prototypical model. We find that significant conductance fluctuations occur with the Fermi energy even for fixed magnetic field strength, and the characteristics of the fluctuation patterns depend on the energy. We propose and validate that the statistical behaviors of the conductance-fluctuation patterns can be understood by the complex eigenvalue spectrum of the generalized, complex Hamiltonian of the system which includes self-energies resulted from the interactions between the device and the semi-infinite leads. As the Fermi energy is increased, complex eigenvalues with extremely smaller imaginary parts emerge, leading to sharp resonances in the conductance.
Quantum chaotic scattering in graphene systems in the absence of invariant classical dynamics
NASA Astrophysics Data System (ADS)
Wang, Guang-Lei; Ying, Lei; Lai, Ying-Cheng; Grebogi, Celso
2013-05-01
Quantum chaotic scattering is referred to as the study of quantum behaviors of open Hamiltonian systems that exhibit transient chaos in the classical limit. Traditionally a central issue in this field is how the elements of the scattering matrix or their functions fluctuate as a system parameter, e.g., the electron Fermi energy, is changed. A tacit hypothesis underlying previous works was that the underlying classical phase-space structure remains invariant as the parameter varies, so semiclassical theory can be used to explain various phenomena in quantum chaotic scattering. There are, however, experimental situations where the corresponding classical chaotic dynamics can change characteristically with some physical parameter. Multiple-terminal quantum dots are one such example where, when a magnetic field is present, the classical chaotic-scattering dynamics can change between being nonhyperbolic and being hyperbolic as the Fermi energy is changed continuously. For such systems semiclassical theory is inadequate to account for the characteristics of conductance fluctuations with the Fermi energy. To develop a general framework for quantum chaotic scattering associated with variable classical dynamics, we use multi-terminal graphene quantum-dot systems as a prototypical model. We find that significant conductance fluctuations occur with the Fermi energy even for fixed magnetic field strength, and the characteristics of the fluctuation patterns depend on the energy. We propose and validate that the statistical behaviors of the conductance-fluctuation patterns can be understood by the complex eigenvalue spectrum of the generalized, complex Hamiltonian of the system which includes self-energies resulted from the interactions between the device and the semi-infinite leads. As the Fermi energy is increased, complex eigenvalues with extremely smaller imaginary parts emerge, leading to sharp resonances in the conductance.
New developments in classical chaotic scattering.
Seoane, Jesús M; Sanjuán, Miguel A F
2013-01-01
Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described. PMID:23242261
New developments in classical chaotic scattering.
Seoane, Jesús M; Sanjuán, Miguel A F
2013-01-01
Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described.
Short-time dynamics of an Ising system on fractal structures
Zheng, Guang-Ping; Li, Mo
2000-11-01
The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents {beta} and {nu} of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.
Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series
Bollt; Stanford; Lai; Zyczkowski
2000-10-16
A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to what extent the symbolic dynamics so obtained can faithfully represent the phase-space dynamics. Our principal result is that such a practice can lead to a severe misrepresentation of the dynamical system. The measured topological entropy is a Devil's staircase-like, but surprisingly nonmonotone, function of a parameter characterizing the amount of misplacement of the partition.
Detecting abrupt dynamic change based on changes in the fractal properties of spatial images
NASA Astrophysics Data System (ADS)
Liu, Qunqun; He, Wenping; Gu, Bin; Jiang, Yundi
2016-08-01
Many abrupt climate change events often cannot be detected timely by conventional abrupt detection methods until a few years after these events have occurred. The reason for this lag in detection is that abundant and long-term observational data are required for accurate abrupt change detection by these methods, especially for the detection of a regime shift. So, these methods cannot help us understand and forecast the evolution of the climate system in a timely manner. Obviously, spatial images, generated by a coupled spatiotemporal dynamical model, contain more information about a dynamic system than a single time series, and we find that spatial images show the fractal properties. The fractal properties of spatial images can be quantitatively characterized by the Hurst exponent, which can be estimated by two-dimensional detrended fluctuation analysis (TD-DFA). Based on this, TD-DFA is used to detect an abrupt dynamic change of a coupled spatiotemporal model. The results show that the TD-DFA method can effectively detect abrupt parameter changes in the coupled model by monitoring the changing in the fractal properties of spatial images. The present method provides a new way for abrupt dynamic change detection, which can achieve timely and efficient abrupt change detection results.
The Retrospective Iterated Analysis Scheme for Nonlinear Chaotic Dynamics
NASA Technical Reports Server (NTRS)
Todling, Ricardo
2002-01-01
Atmospheric data assimilation is the name scientists give to the techniques of blending atmospheric observations with atmospheric model results to obtain an accurate idea of what the atmosphere looks like at any given time. Because two pieces of information are used, observations and model results, the outcomes of data assimilation procedure should be better than what one would get by using one of these two pieces of information alone. There is a number of different mathematical techniques that fall under the data assimilation jargon. In theory most these techniques accomplish about the same thing. In practice, however, slight differences in the approaches amount to faster algorithms in some cases, more economical algorithms in other cases, and even give better overall results in yet some other cases because of practical uncertainties not accounted for by theory. Therefore, the key is to find the most adequate data assimilation procedure for the problem in hand. In our Data Assimilation group we have been doing extensive research to try and find just such data assimilation procedure. One promising possibility is what we call retrospective iterated analysis (RIA) scheme. This procedure has recently been implemented and studied in the context of a very large data assimilation system built to help predict and study weather and climate. Although the results from that study suggest that the RIA scheme produces quite reasonable results, a complete evaluation of the scheme is very difficult due to the complexity of that problem. The present work steps back a little bit and studies the behavior of the RIA scheme in the context of a small problem. The problem is small enough to allow full assessment of the quality of the RIA scheme, but it still has some of the complexity found in nature, namely, its chaotic-type behavior. We find that the RIA performs very well for this small but still complex problem which is a result that seconds the results of our early studies.
Electric field driven fractal growth dynamics in polymeric medium
NASA Astrophysics Data System (ADS)
Dawar, Anit; Chandra, Amita
2014-08-01
This paper reports the extension of earlier work (Dawar and Chandra, 2012) [27] by including the influence of low values of electric field on diffusion limited aggregation (DLA) patterns in polymer electrolyte composites. Subsequently, specified cut-off value of voltage has been determined. Below the cut-off voltage, the growth becomes direction independent (i.e., random) and gives rise to ramified DLA patterns while above the cut-off, growth is governed by diffusion, convection and migration. These three terms (i.e., diffusion, convection and migration) lead to structural transition that varies from dense branched morphology (DBM) to chain-like growth to dendritic growth, i.e., from high field region (A) to constant field region (B) to low field region (C), respectively. The paper further explores the growth under different kinds of electrode geometries (circular and square electrode geometry). A qualitative explanation for fractal growth phenomena at applied voltage based on Nernst-Planck equation has been proposed.
Numerical test for hyperbolicity of chaotic dynamics in time-delay systems.
Kuptsov, Pavel V; Kuznetsov, Sergey P
2016-07-01
We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.
Numerical test for hyperbolicity of chaotic dynamics in time-delay systems
NASA Astrophysics Data System (ADS)
Kuptsov, Pavel V.; Kuznetsov, Sergey P.
2016-07-01
We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.
Numerical test for hyperbolicity of chaotic dynamics in time-delay systems.
Kuptsov, Pavel V; Kuznetsov, Sergey P
2016-07-01
We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos. PMID:27575062
Cycles, randomness, and transport from chaotic dynamics to stochastic processes.
Gaspard, Pierre
2015-09-01
An overview of advances at the frontier between dynamical systems theory and nonequilibrium statistical mechanics is given. Sensitivity to initial conditions is a mechanism at the origin of dynamical randomness-alias temporal disorder-in deterministic dynamical systems. In spatially extended systems, sustaining transport processes, such as diffusion, relationships can be established between the characteristic quantities of dynamical chaos and the transport coefficients, bringing new insight into the second law of thermodynamics. With methods from dynamical systems theory, the microscopic time-reversal symmetry can be shown to be broken at the statistical level of description in nonequilibrium systems. In this way, the thermodynamic entropy production turns out to be related to temporal disorder and its time asymmetry away from equilibrium. PMID:26428559
Dynamics of the stochastic Lorenz chaotic system with long memory effects
Zeng, Caibin Yang, Qigui
2015-12-15
Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution.
Regular and chaotic dynamics of a chain of magnetic dipoles with moments of inertia
Shutyi, A. M.
2009-05-15
The nonlinear dynamic modes of a chain of coupled spherical bodies having dipole magnetic moments that are excited by a homogeneous ac magnetic field are studied using numerical analysis. Bifurcation diagrams are constructed and used to find conditions for the presence of several types of regular, chaotic, and quasi-periodic oscillations. The effect of the coupling of dipoles on the excited dynamics of the system is revealed. The specific features of the Poincare time sections are considered for the cases of synchronous chaos with antiphase synchronization and asynchronous chaos. The spectrum of Lyapunov exponents is calculated for the dynamic modes of an individual dipole.
NASA Astrophysics Data System (ADS)
Mohammad, Yasir K.; Pavlova, Olga N.; Pavlov, Alexey N.
2016-04-01
We discuss the problem of quantifying chaotic dynamics at the input of the "integrate-and-fire" (IF) model from the output sequences of interspike intervals (ISIs) for the case when the fluctuating threshold level leads to the appearance of noise in ISI series. We propose a way to detect an ability of computing dynamical characteristics of the input dynamics and the level of noise in the output point processes. The proposed approach is based on the dependence of the largest Lyapunov exponent from the maximal orientation error used at the estimation of the averaged rate of divergence of nearby phase trajectories.
A New Image Encryption Scheme Based on Dynamic S-Boxes and Chaotic Maps
NASA Astrophysics Data System (ADS)
Rehman, Atique Ur; Khan, Jan Sher; Ahmad, Jawad; Hwang, Soeng Oun
2016-03-01
Substitution box is a unique and nonlinear core component of block ciphers. A better designing technique of substitution box can boost up the quality of ciphertexts. In this paper, a new encryption method based on dynamic substitution boxes is proposed via using two chaotic maps. To break the correlation in an original image, pixels values of the original plaintext image are permuted row- and column-wise through random sequences. The aforementioned random sequences are generated by 2-D Burgers chaotic map. For the generation of dynamic substitution boxes, Logistic chaotic map is employed. In the process of diffusion, the permuted image is divided into blocks and each block is substituted via different dynamic substitution boxes. In contrast to conventional encryption schemes, the proposed scheme does not undergo the fixed block cipher and hence the security level can be enhanced. Extensive security analysis including histogram test is applied on the proposed image encryption technique. All experimental results reveal that the proposed scheme has a high level of security and robustness for transmission of digital images on insecure communication channels.
Chaotic and ballistic dynamics in time-driven quasiperiodic lattices
NASA Astrophysics Data System (ADS)
Wulf, Thomas; Schmelcher, Peter
2016-04-01
We investigate the nonequilibrium dynamics of classical particles in a driven quasiperiodic lattice based on the Fibonacci sequence. An intricate transient dynamics of extraordinarily long ballistic flights at distinct velocities is found. We argue how these transients are caused and can be understood by a hierarchy of block decompositions of the quasiperiodic lattice. A comparison to the cases of periodic and fully randomized lattices is performed.
Chaotic and ballistic dynamics in time-driven quasiperiodic lattices.
Wulf, Thomas; Schmelcher, Peter
2016-04-01
We investigate the nonequilibrium dynamics of classical particles in a driven quasiperiodic lattice based on the Fibonacci sequence. An intricate transient dynamics of extraordinarily long ballistic flights at distinct velocities is found. We argue how these transients are caused and can be understood by a hierarchy of block decompositions of the quasiperiodic lattice. A comparison to the cases of periodic and fully randomized lattices is performed. PMID:27176301
NASA Technical Reports Server (NTRS)
Makikallio, T. H.; Ristimae, T.; Airaksinen, K. E.; Peng, C. K.; Goldberger, A. L.; Huikuri, H. V.
1998-01-01
Dynamic analysis techniques may uncover abnormalities in heart rate (HR) behavior that are not easily detectable with conventional statistical measures. However, the applicability of these new methods for detecting possible abnormalities in HR behavior in various cardiovascular disorders is not well established. Conventional measures of HR variability were compared with short-term (< or = 11 beats, alpha1) and long-term (> 11 beats, alpha2) fractal correlation properties and with approximate entropy of RR interval data in 38 patients with stable angina pectoris without previous myocardial infarction or cardiac medication at the time of the study and 38 age-matched healthy controls. The short- and long-term fractal scaling exponents (alpha1, alpha2) were significantly higher in the coronary patients than in the healthy controls (1.34 +/- 0.15 vs 1.11 +/- 0.12 [p <0.001] and 1.10 +/- 0.08 vs 1.04 +/- 0.06 [p <0.01], respectively), and they also had lower approximate entropy (p <0.05), standard deviation of all RR intervals (p <0.01), and high-frequency spectral component of HR variability (p <0.05). The short-term fractal scaling exponent performed better than other heart rate variability parameters in differentiating patients with coronary artery disease from healthy subjects, but it was not related to the clinical or angiographic severity of coronary artery disease or any single nonspectral or spectral measure of HR variability in this retrospective study. Patients with stable angina pectoris have altered fractal properties and reduced complexity in their RR interval dynamics relative to age-matched healthy subjects. Dynamic analysis may complement traditional analyses in detecting altered HR behavior in patients with stable angina pectoris.
Fractal structures and processes
Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M.
1996-06-01
Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much of the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts. {copyright} {ital 1996 American Institute of Physics.}
Statistical properties of chaotic dynamical systems which exhibit strange attractors
Jensen, R.V.; Oberman, C.R.
1981-07-01
A path integral method is developed for the calculation of the statistical properties of turbulent dynamical systems. The method is applicable to conservative systems which exhibit a transition to stochasticity as well as dissipative systems which exhibit strange attractors. A specific dissipative mapping is considered in detail which models the dynamics of a Brownian particle in a wave field with a broad frequency spectrum. Results are presented for the low order statistical moments for three turbulent regimes which exhibit strange attractors corresponding to strong, intermediate, and weak collisional damping.
Exact coherent structures and chaotic dynamics in a model of cardiac tissue.
Byrne, Greg; Marcotte, Christopher D; Grigoriev, Roman O
2015-03-01
Unstable nonchaotic solutions embedded in the chaotic attractor can provide significant new insight into chaotic dynamics of both low- and high-dimensional systems. In particular, in turbulent fluid flows, such unstable solutions are referred to as exact coherent structures (ECS) and play an important role in both initiating and sustaining turbulence. The nature of ECS and their role in organizing spatiotemporally chaotic dynamics, however, is reasonably well understood only for systems on relatively small spatial domains lacking continuous Euclidean symmetries. Construction of ECS on large domains and in the presence of continuous translational and/or rotational symmetries remains a challenge. This is especially true for models of excitable media which display spiral turbulence and for which the standard approach to computing ECS completely breaks down. This paper uses the Karma model of cardiac tissue to illustrate a potential approach that could allow computing a new class of ECS on large domains of arbitrary shape by decomposing them into a patchwork of solutions on smaller domains, or tiles, which retain Euclidean symmetries locally. PMID:25833430
Exact coherent structures and chaotic dynamics in a model of cardiac tissue
Byrne, Greg; Marcotte, Christopher D.; Grigoriev, Roman O.
2015-03-15
Unstable nonchaotic solutions embedded in the chaotic attractor can provide significant new insight into chaotic dynamics of both low- and high-dimensional systems. In particular, in turbulent fluid flows, such unstable solutions are referred to as exact coherent structures (ECS) and play an important role in both initiating and sustaining turbulence. The nature of ECS and their role in organizing spatiotemporally chaotic dynamics, however, is reasonably well understood only for systems on relatively small spatial domains lacking continuous Euclidean symmetries. Construction of ECS on large domains and in the presence of continuous translational and/or rotational symmetries remains a challenge. This is especially true for models of excitable media which display spiral turbulence and for which the standard approach to computing ECS completely breaks down. This paper uses the Karma model of cardiac tissue to illustrate a potential approach that could allow computing a new class of ECS on large domains of arbitrary shape by decomposing them into a patchwork of solutions on smaller domains, or tiles, which retain Euclidean symmetries locally.
Fractal dynamics of light scattering intensity fluctuation in disordered dusty plasmas
Safaai, S. S.; Muniandy, S. V.; Chew, W. X.; Asgari, H.; Yap, S. L.; Wong, C. S.
2013-10-15
Dynamic light scattering (DLS) technique is a simple and yet powerful technique for characterizing particle properties and dynamics in complex liquids and gases, including dusty plasmas. Intensity fluctuation in DLS experiments often studied using correlation analysis with assumption that the fluctuation is statistically stationary. In this study, the temporal variation of the nonstationary intensity fluctuation is analyzed directly to show the existence of fractal characteristics by employing wavelet scalogram approach. Wavelet based scale decomposition approach is used to separate non-scaling background noise (without dust) from scaling intensity fluctuation from dusty plasma. The Hurst exponents for light intensity fluctuation in dusty plasma at different neutral gas pressures are determined. At low pressures, weaker damping of dust motions via collisions with neutral gases results in stronger persistent behavior in the fluctuation of DLS time series. The fractal scaling Hurst exponent is demonstrated to be useful for characterizing structural phases in complex disordered dusty plasma, especially when particle configuration or sizes are highly inhomogeneous which makes the standard pair-correlation function difficult to interpret. The results from fractal analysis are compared with alternative interpretation of disorder based on approximate entropy and particle transport using mean square displacement.
Study on a new chaotic bitwise dynamical system and its FPGA implementation
NASA Astrophysics Data System (ADS)
Wang, Qian-Xue; Yu, Si-Min; Guyeux, C.; Bahi, J.; Fang, Xiao-Le
2015-06-01
In this paper, the structure of a new chaotic bitwise dynamical system (CBDS) is described. Compared to our previous research work, it uses various random bitwise operations instead of only one. The chaotic behavior of CBDS is mathematically proven according to the Devaney's definition, and its statistical properties are verified both for uniformity and by a comprehensive, reputed and stringent battery of tests called TestU01. Furthermore, a systematic methodology developing the parallel computations is proposed for FPGA platform-based realization of this CBDS. Experiments finally validate the proposed systematic methodology. Project supported by China Postdoctoral Science Foundation (Grant No. 2014M552175), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Chinese Education Ministry, the National Natural Science Foundation of China (Grant No. 61172023), and the Specialized Research Foundation of Doctoral Subjects of Chinese Education Ministry (Grant No. 20114420110003).
Li, Nianqiang; Kim, Byungchil; Chizhevsky, V N; Locquet, A; Bloch, M; Citrin, D S; Pan, Wei
2014-03-24
This paper reports the experimental investigation of two different approaches to random bit generation based on the chaotic dynamics of a semiconductor laser with optical feedback. By computing high-order finite differences of the chaotic laser intensity time series, we obtain time series with symmetric statistical distributions that are more conducive to ultrafast random bit generation. The first approach is guided by information-theoretic considerations and could potentially reach random bit generation rates as high as 160 Gb/s by extracting 4 bits per sample. The second approach is based on pragmatic considerations and could lead to rates of 2.2 Tb/s by extracting 55 bits per sample. The randomness of the bit sequences obtained from the two approaches is tested against three standard randomness tests (ENT, Diehard, and NIST tests), as well as by calculating the statistical bias and the serial correlation coefficients on longer sequences of random bits than those used in the standard tests.
Bells Galore: Oscillations and circle-map dynamics from space-filling fractal functions
Puente, C.E.; Cortis, A.; Sivakumar, B.
2008-10-15
The construction of a host of interesting patterns over one and two dimensions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings parameters) in which there is no limit. Specifically, the one-dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, despite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.
The Analysis of the Influence of Odorant's Complexity on Fractal Dynamics of Human Respiration.
Namazi, Hamidreza; Akrami, Amin; Kulish, Vladimir V
2016-01-01
One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to 'complex' olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases. PMID:27244590
The Analysis of the Influence of Odorant’s Complexity on Fractal Dynamics of Human Respiration
Namazi, Hamidreza; Akrami, Amin; Kulish, Vladimir V.
2016-01-01
One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to ‘complex’ olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases. PMID:27244590
The Analysis of the Influence of Odorant’s Complexity on Fractal Dynamics of Human Respiration
NASA Astrophysics Data System (ADS)
Namazi, Hamidreza; Akrami, Amin; Kulish, Vladimir V.
2016-05-01
One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to ‘complex’ olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases.
Chaotic dynamics in the Maxwell-Bloch equations
Holm, D.D.; Kovacic, G.
1992-03-06
In the slowly varying envelope approximation and the rotating wave approximation for the Maxwell-Bloch equations, we describe how the presence of a small-amplitude probe laser in an excited, two-level, resonant medium leads to homoclinic chaos in the laser-matter dynamics. We also describe a derivation of the Maxwell-Bloch equations from an action principle.
Evolution of collision numbers for a chaotic gas dynamics.
Vidgop, Alexander Jonathan; Fouxon, Itzhak
2011-11-01
We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger numbers of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.
Do the chaotic features of gait change in Parkinson's disease?
Sarbaz, Yashar; Towhidkhah, Farzad; Jafari, Ayyoob; Gharibzadeh, Shahriar
2012-08-21
Some previous studies have focused on chaotic properties of Parkinson's disease (PD). It seems that considering PD from dynamical systems perspective is a relevant method that may lead to better understanding of the disease. There is some ambiguity about chaotic nature in PD symptoms and normal behaviour. Some studies claim that normal gait has somehow a chaotic behaviour and disturbed gait in PD has decreased chaotic nature. However, it is worth noting that the basis of this idea is the difference of fractal behaviour in gait of normal and PD patients, which is concluded from Long Range Correlation (LRC) indices. Our primary calculations show that a large number of normal persons and patients have similar LRC. It seems that chaotic studies on PD need a different view. Because of short time recording of symptoms, accurate calculation of chaotic features is tough. On the other hand, long time recording of symptoms is experimentally difficult. In this research, we have first designed a physiologically plausible model for normal and PD gait. Then, after validating the model with neural network classifier, we used the model for extracting long time simulation of stride in normal and PD persons. These long time simulations were then used for calculating the chaotic features of gait. According to change of phase space behaviour and alteration of three largest lyapunov exponents, it was observed that simulated normal persons act as chaotic systems in stride production, but simulated PD does not have chaotic dynamics and is stochastic. Based on our results, it may be claimed that normal gait has chaotic nature which is disturbed in PD state. Surely, long time real recordings from gait signal in normal persons and PD patients are necessary to warranty this hypothesis.
Billock, V A; Cunningham, D W; Havig, P R; Tsou, B H
2001-10-01
Recent work establishes that static and dynamic natural images have fractal-like l/falpha spatiotemporal spectra. Artifical textures, with randomized phase spectra, and 1/falpha amplitude spectra are also used in studies of texture and noise perception. Influenced by colorimetric principles and motivated by the ubiquity of 1/falpha spatial and temporal image spectra, we treat the spatial and temporal frequency exponents as the dimensions characterizing a dynamic texture space, and we characterize two key attributes of this space, the spatiotemporal appearance map and the spatiotemporal discrimination function (a map of MacAdam-like just-noticeable-difference contours).
NASA Astrophysics Data System (ADS)
Páez, Rocío Isabel; Efthymiopoulos, Christos
2015-02-01
The possibility that giant extrasolar planets could have small Trojan co-orbital companions has been examined in the literature from both viewpoints of the origin and dynamical stability of such a configuration. Here we aim to investigate the dynamics of hypothetical small Trojan exoplanets in domains of secondary resonances embedded within the tadpole domain of motion. To this end, we consider the limit of a massless Trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of additional planets (hereafter referred to as the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the dynamics of the Trojan body. The paper contains a theoretical and a numerical part. In the theoretical part, we develop a Hamiltonian formalism in action-angle variables, which allows us to treat in a unified way resonant dynamics and secular effects on the Trojan body in both the ERTBP or the RMPP. In both cases, our formalism leads to a decomposition of the Hamiltonian in two parts, . , called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom, while contains only terms depending trigonometrically on slow (secular) angles. is formally identical in the ERTBP and the RMPP, apart from a re-definition of some angular variables. An important physical consequence of this analysis is that the slow chaotic diffusion along resonances proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism. We found that this is best approximated by the paradigm of `modulational diffusion'. In the paper's numerical part, we then focus on the ERTBP in order to make a detailed numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we first provide a survey of the resonant web for characteristic mass parameter values of the primary, in which the
Nonequilibrium dynamics in lattice ecosystems: Chaotic stability and dissipative structures
NASA Astrophysics Data System (ADS)
Solé, Ricard V.; Bascompte, Jordi; Valls, Joaquim
1992-07-01
A generalized coupled map lattice (CML) model of ecosystem dynamics is presented. We consider the spatiotemporal behavior of a prey-predator map, a model of host-parasitoid interactions, and two-species competition. The latter model can show phase separation of domains (Turing-like structures) even when chaos is present. We also use this CML model to explore the time evolution and structural properties of ecological networks built with a set of N competing species. The May-Wigner criterion is applied as a measure of stability, and some regularities in the stable networks observed are discussed.
Length scale of a chaotic element in Rayleigh-Bénard convection.
Karimi, A; Paul, M R
2012-12-01
We describe an approach to quantify the length scale of a chaotic element of a Rayleigh-Bénard convection layer exhibiting spatiotemporal chaos. The length scale of a chaotic element is determined by simultaneously evolving the dynamics of two convection layers with a unidirectional coupling that involves only the time-varying values of the fluid velocity and temperature on the lateral boundaries of the domain. In our results we numerically simulate the full Boussinesq equations for the precise conditions of experiment. By varying the size of the boundary used for the coupling we identify a length scale that describes the size of a chaotic element. The length scale of the chaotic element is of the same order of magnitude, and exhibits similar trends, as the natural chaotic length scale that is based upon the fractal dimension.
The role of model dynamics in ensemble Kalman filter performance for chaotic systems
Ng, G.-H.C.; McLaughlin, D.; Entekhabi, D.; Ahanin, A.
2011-01-01
The ensemble Kalman filter (EnKF) is susceptible to losing track of observations, or 'diverging', when applied to large chaotic systems such as atmospheric and ocean models. Past studies have demonstrated the adverse impact of sampling error during the filter's update step. We examine how system dynamics affect EnKF performance, and whether the absence of certain dynamic features in the ensemble may lead to divergence. The EnKF is applied to a simple chaotic model, and ensembles are checked against singular vectors of the tangent linear model, corresponding to short-term growth and Lyapunov vectors, corresponding to long-term growth. Results show that the ensemble strongly aligns itself with the subspace spanned by unstable Lyapunov vectors. Furthermore, the filter avoids divergence only if the full linearized long-term unstable subspace is spanned. However, short-term dynamics also become important as non-linearity in the system increases. Non-linear movement prevents errors in the long-term stable subspace from decaying indefinitely. If these errors then undergo linear intermittent growth, a small ensemble may fail to properly represent all important modes, causing filter divergence. A combination of long and short-term growth dynamics are thus critical to EnKF performance. These findings can help in developing practical robust filters based on model dynamics. ?? 2011 The Authors Tellus A ?? 2011 John Wiley & Sons A/S.
Regular and chaotic motions in applied dynamics of a rigid body.
Beletskii, V. V.; Pivovarov, M. L.; Starostin, E. L.
1996-06-01
Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics. PMID:12780243
Regular and chaotic motions in applied dynamics of a rigid body.
Beletskii, V. V.; Pivovarov, M. L.; Starostin, E. L.
1996-06-01
Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics.
Ławniczak, Michał; Białous, Małgorzata; Yunko, Vitalii; Bauch, Szymon; Sirko, Leszek
2015-03-01
We present the results of an experimental study of the elastic enhancement factor W for a microwave rectangular cavity simulating a two-dimensional quantum billiard in a transient region between regular and chaotic dynamics. The cavity was coupled to a vector network analyzer via two microwave antennas. The departure of the system from an integrable one due to the presence of antennas acting as scatterers is characterized by the parameter of chaoticity κ=2.8. The experimental results for the rectangular cavity are compared with those obtained for a microwave rough cavity simulating a chaotic quantum billiard. The experimental results were obtained for the frequency range ν=16-18.5 GHz and moderate absorption strength γ=5.2-7.4. We show that the elastic enhancement factor for the rectangular cavity lies below the theoretical value W=3 predicted for integrable systems, and it is significantly higher than that obtained for the rough cavity. The results obtained for the microwave rough cavity are smaller than those obtained within the framework of random matrix theory, and they lie between them and those predicted within a recently introduced model of the two-channel coupling [V. V. Sokolov and O. V. Zhirov, arXiv:1411.6211 [nucl-th
On the Large Scale Dynamics in the Wake of a Fractal Obstacle
NASA Astrophysics Data System (ADS)
Higham, Jonathan; Brevis, Wernher
2015-11-01
In a water flume three-dimensional Particle Tracking Velocimetry is used to capture the turbulent wake of two full-width and wall-mounted obstacles: The first obstacle is a uniformly spaced array of square cylinders of same length-scale; the second is a three-iteration pre-fractal based on a the deterministic Sierpinski Carpet. Both obstacles emerge from the water surface and had the same porosity. For the description of the instantaneous vortical structures the velocity gradient tensor is analysed. It is found that whilst the largest length scales of the fractal dominated the vorticity field in the wake, the smaller length-scale within the obstacle caused intense vortical structures within the near field of the wake. To further investigate the spatio-temporal behaviour of the wake a simple and integrated use of the Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) is introduced. POD is used to rank the spatial structures relatable to the total variance (i.e. vorticity) while DMD is used to identify their dominant oscillation frequencies and spatial characteristics. From the POD it is clear that the largest length-scale creates spatially dominant structures, whilst the DMD extracts a set of oscillatory frequencies relatable to each fractal length-scale.
A chaotic attractor in timing noise from the Vela pulsar?
NASA Technical Reports Server (NTRS)
Harding, Alice K.; Shinbrot, Troy; Cordes, James M.
1990-01-01
Fourteen years of timing residual data from the Vela pulsar have been analyzed in order to determine if a chaotic dynamical process is the origin of timing noise. Using the correlation sum technique, a dimension of about 1.5 is obtained. This low dimension indicates underlying structure in the phase residuals which may be evidence for a chaotic attractor. It is therefore possible that nonlinear dynamics intrinsic to the spin-down may be the cause of the timing noise in the Vela pulsar. However, it has been found that the stimulated random walks in frequency and frequency derivative often used to model pulsar timing noise also have low fractal dimension, using the same analysis technique. Recent work suggesting that random processes with steep power spectra can mimic strange attractors seems to be confirmed in the case of these random walks. It appears that the correlation sum estimator for dimension is unable to distinguish between chaotic and random processes.
Periodic and chaotic dynamics in a map-based model of tumor-immune interaction.
Moghtadaei, Motahareh; Hashemi Golpayegani, Mohammad Reza; Malekzadeh, Reza
2013-10-01
Clinicians and oncologists believe that tumor growth has unpredictable dynamics. For this reason they encounter many difficulties in the treatment of cancer. Mathematical modeling is a great tool to improve our better understanding of the complicated biological system of tumor growth. Also, it can help to identify states of the disease and as a result help to predict later behaviors of the tumor. Having an insight into the future behaviors of the tumor can be very useful for the oncologists and clinicians to decide on the treatment method and dosage of the administered drug. This paper suggests that a suitable model for the tumor growth system should be a discrete model capable of exhibiting periodic and complex chaotic dynamics. This is the key feature of the proposed model. The model is validated here through experimental data and its potential dynamics are analyzed. The model can explain many biologically observed tumor states and dynamics, such as exponential growth, and periodic and chaotic behaviors in the steady states. The model shows that even an avascular tumor could become invasive under certain conditions. PMID:23770106
Fuzzy fractals, chaos, and noise
Zardecki, A.
1997-05-01
To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.
Chaotic dynamics of Comet 1P/Halley: Lyapunov exponent and survival time expectancy
NASA Astrophysics Data System (ADS)
Muñoz-Gutiérrez, M. A.; Reyes-Ruiz, M.; Pichardo, B.
2015-03-01
The orbital elements of Comet Halley are known to a very high precision, suggesting that the calculation of its future dynamical evolution is straightforward. In this paper we seek to characterize the chaotic nature of the present day orbit of Comet Halley and to quantify the time-scale over which its motion can be predicted confidently. In addition, we attempt to determine the time-scale over which its present day orbit will remain stable. Numerical simulations of the dynamics of test particles in orbits similar to that of Comet Halley are carried out with the MERCURY 6.2 code. On the basis of these we construct survival time maps to assess the absolute stability of Halley's orbit, frequency analysis maps to study the variability of the orbit, and we calculate the Lyapunov exponent for the orbit for variations in initial conditions at the level of the present day uncertainties in our knowledge of its orbital parameters. On the basis of our calculations of the Lyapunov exponent for Comet Halley, the chaotic nature of its motion is demonstrated. The e-folding time-scale for the divergence of initially very similar orbits is approximately 70 yr. The sensitivity of the dynamics on initial conditions is also evident in the self-similarity character of the survival time and frequency analysis maps in the vicinity of Halley's orbit, which indicates that, on average, it is unstable on a time-scale of hundreds of thousands of years. The chaotic nature of Halley's present day orbit implies that a precise determination of its motion, at the level of the present-day observational uncertainty, is difficult to predict on a time-scale of approximately 100 yr. Furthermore, we also find that the ejection of Halley from the Solar system or its collision with another body could occur on a time-scale as short as 10 000 yr.
Dynamics of the Uranian and Saturnian satellite systems - A chaotic route to melting Miranda?
NASA Technical Reports Server (NTRS)
Dermott, Stanley F.; Malhotra, Renu; Murray, Carl D.
1988-01-01
Miranda's anomalously large inclination, in conjunction with the postaccretional resurfacing of both Miranda and Ariel and anomalously large eccentricities characterizing the inner Uranian satellites, are presently held to suggest the disruption of resonant configurations that once existed in this satellite system. Classical analytical methods for the dynamics of resonance are here used to demonstrate how temporary capture into a second- or higher-order resonance can generate large increases in eccentricity and inclination on comparatively short time-scales. Such capture into resonance may result in chaotic motion.
Random matrix theory for mixed regular-chaotic dynamics in the super-extensive regime
El-Hady, A. Abd; Abul-Magd, A. Y.
2011-10-27
We apply Tsallis's q-indexed nonextensive entropy to formulate a random matrix theory (RMT), which may be suitable for systems with mixed regular-chaotic dynamics. We consider the super-extensive regime of q<1. We obtain analytical expressions for the level-spacing distributions, which are strictly valid for 2 X2 random-matrix ensembles, as usually done in the standard RMT. We compare the results with spacing distributions, numerically calculated for random matrix ensembles describing a harmonic oscillator perturbed by Gaussian orthogonal and unitary ensembles.
Subharmonic bifurcations and chaotic dynamics of an air damping completely inelastic bouncing ball
NASA Astrophysics Data System (ADS)
Han, Hong; Jiang, Zehui; Zhang, Rui; Lyu, Jing
2013-12-01
We investigate the dynamics of a plastic ball on a vibrated platform in air by introducing air damping effect into the completely inelastic bouncing ball model. The air damping gives rise to larger saddle-node bifurcation points and a chaos confirmed by the largest Lyapunov exponent of a one-dimensional discrete mapping. The calculated bifurcation point distribution shows that the periodic motion of the ball is suppressed and a chaos emerges earlier for an increasing air damping. When the reset mechanism and the linear stability which cause periodic motion of the ball both collapse, the investigated system is fully chaotic.
Transient Dynamics of Electric Power Systems: Direct Stability Assessment and Chaotic Motions
NASA Astrophysics Data System (ADS)
Chu, Chia-Chi
A power system is continuously experiencing disturbances. Analyzing, predicting, and controlling transient dynamics, which describe transient behaviors of the power system following disturbances, is a major concern in the planning and operation of a power utility. Important conclusions and decisions are made based on the result of system transient behaviors. As today's power network becomes highly interconnected and much more complex, it has become essential to enhance the fundamental understanding of transient dynamics, and to develop fast and reliable computational algorithms. In this thesis, we emphasize mathematical rigor rather than physical insight. Nonlinear dynamical system theory is applied to study two fundamental topics: direct stability assessment and chaotic motions. Conventionally, power system stability is determined by calculating the time-domain transient behaviors for a given disturbance. In contrast, direct methods identify whether or not the system will remain stable once the disturbance is removed by comparing the corresponding energy value of the post-fault system to a calculated threshold value. Direct methods not only avoid the time-consuming numerical integration of the time domain approach, but also provide a quantitative measure of the degree of system stability. We present a general framework for the theoretical foundations of direct methods. Canonical representations of network-reduction models as well as network-preserving models are proposed to facilitate the analysis and the construction of energy functions of various power system models. An advanced and practical method, called the boundary of stability region based controlling unstable equilibrium point method (BCU method), of computing the controlling unstable equilibrium point is proposed along with its theoretical foundation. Numerical solution algorithms capable of supporting on-line applications of direct methods are provided. Further possible improvements and enhancements are
Sigalov, G; Gendelman, O V; AL-Shudeifat, M A; Manevitch, L I; Vakakis, A F; Bergman, L A
2012-03-01
We show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits non-conventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the non-conventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.
Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure.
Reuveni, Shlomi; Granek, Rony; Klafter, Joseph
2010-08-01
Proteins have been shown to exhibit strange/anomalous dynamics displaying non-Debye density of vibrational states, anomalous spread of vibrational energy, large conformational changes, nonexponential decay of correlations, and nonexponential unfolding times. The anomalous behavior may, in principle, stem from various factors affecting the energy landscape under which a protein vibrates. Investigating the origins of such unconventional dynamics, we focus on the structure-dynamics interplay and introduce a stochastic approach to the vibrational dynamics of proteins. We use diffusion, a method sensitive to the structural features of the protein fold and them alone, in order to probe protein structure. Conducting a large-scale study of diffusion on over 500 Protein Data Bank structures we find it to be anomalous, an indication of a fractal-like structure. Taking advantage of known and newly derived relations between vibrational dynamics and diffusion, we demonstrate the equivalence of our findings to the existence of structurally originated anomalies in the vibrational dynamics of proteins. We conclude that these anomalies are a direct result of the fractal-like structure of proteins. The duality between diffusion and vibrational dynamics allows us to make, on a single-molecule level, experimentally testable predictions. The time dependent vibrational mean square displacement of an amino acid is predicted to be subdiffusive. The thermal variance in the instantaneous distance between amino acids is shown to grow as a power law of the equilibrium distance. Mean first passage time analysis is offered as a practical tool that may aid in the identification of amino acid pairs involved in large conformational changes.
Characterization of chaotic dynamics in the vocalization of Cervus elaphus corsicanus (L)
NASA Astrophysics Data System (ADS)
Facchini, Angelo; Bastianoni, Simone; Marchettini, Nadia; Rustici, Mauro
2003-12-01
Chaos, oscillations, instabilities, and intermittency represent only some nonlinear examples apparent in the natural world. These phenomena appear in any field of study, and advances in complex and nonlinear dynamic techniques bring about opportunities to better understand animal signals. In this work an analysis method is suggested based on the characterization of the vocal-fold dynamics by means of the nonlinear time-series analysis, and by the computations of the parameters typical of chaotic oscillations: Attractor reconstruction, spectrum of Lyapunov exponents, and maximum Lyapunov exponent were used to reconstruct the dynamic of the vocal folds. Identifying a sort of vocal fingerprint can be useful in biodiversity monitoring and understanding the health status of a given animal. This method was applied to the vocalization of the Cervus elaphus corsicanus, the Sardinian red deer.
Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics
NASA Technical Reports Server (NTRS)
Iyengar, N.; Peng, C. K.; Morin, R.; Goldberger, A. L.; Lipsitz, L. A.
1996-01-01
We postulated that aging is associated with disruption in the fractallike long-range correlations that characterize healthy sinus rhythm cardiac interval dynamics. Ten young (21-34 yr) and 10 elderly (68-81 yr) rigorously screened healthy subjects underwent 120 min of continuous supine resting electrocardiographic recording. We analyzed the interbeat interval time series using standard time and frequency domain statistics and using a fractal measure, detrended fluctuation analysis, to quantify long-range correlation properties. In healthy young subjects, interbeat intervals demonstrated fractal scaling, with scaling exponents (alpha) from the fluctuation analysis close to a value of 1.0. In the group of healthy elderly subjects, the interbeat interval time series had two scaling regions. Over the short range, interbeat interval fluctuations resembled a random walk process (Brownian noise, alpha = 1.5), whereas over the longer range they resembled white noise (alpha = 0.5). Short (alpha s)- and long-range (alpha 1) scaling exponents were significantly different in the elderly subjects compared with young (alpha s = 1.12 +/- 0.19 vs. 0.90 +/- 0.14, respectively, P = 0.009; alpha 1 = 0.75 +/- 0.17 vs. 0.99 +/- 0.10, respectively, P = 0.002). The crossover behavior from one scaling region to another could be modeled as a first-order autoregressive process, which closely fit the data from four elderly subjects. This implies that a single characteristic time scale may be dominating heartbeat control in these subjects. The age-related loss of fractal organization in heartbeat dynamics may reflect the degradation of integrated physiological regulatory systems and may impair an individual's ability to adapt to stress.
Provata, A; Tsekouras, G A
2003-05-01
Dynamical patterns, in the form of consecutive moving stripes or rings, are shown to develop spontaneously in the cyclic lattice Lotka-Volterra model, when realized on square lattice, at the reaction limited regime. Each stripe consists of different particles (species) and the borderlines between consecutive stripes are fractal. The interface width w between the different species scales as w(L,t) approximately L(alpha)f(t/L(z)), where L is the linear size of the interface, t is the time, and alpha and z are the static and dynamical critical exponents, respectively. The critical exponents were computed as alpha=0.49+/-0.03 and z=1.53+/-0.13 and the propagating fronts show dynamical characteristics similar to those of the Eden growth models.
NASA Technical Reports Server (NTRS)
Pikkujamsa, S. M.; Makikallio, T. H.; Sourander, L. B.; Raiha, I. J.; Puukka, P.; Skytta, J.; Peng, C. K.; Goldberger, A. L.; Huikuri, H. V.
1999-01-01
BACKGROUND: New methods of R-R interval variability based on fractal scaling and nonlinear dynamics ("chaos theory") may give new insights into heart rate dynamics. The aims of this study were to (1) systematically characterize and quantify the effects of aging from early childhood to advanced age on 24-hour heart rate dynamics in healthy subjects; (2) compare age-related changes in conventional time- and frequency-domain measures with changes in newly derived measures based on fractal scaling and complexity (chaos) theory; and (3) further test the hypothesis that there is loss of complexity and altered fractal scaling of heart rate dynamics with advanced age. METHODS AND RESULTS: The relationship between age and cardiac interbeat (R-R) interval dynamics from childhood to senescence was studied in 114 healthy subjects (age range, 1 to 82 years) by measurement of the slope, beta, of the power-law regression line (log power-log frequency) of R-R interval variability (10(-4) to 10(-2) Hz), approximate entropy (ApEn), short-term (alpha(1)) and intermediate-term (alpha(2)) fractal scaling exponents obtained by detrended fluctuation analysis, and traditional time- and frequency-domain measures from 24-hour ECG recordings. Compared with young adults (<40 years old, n=29), children (<15 years old, n=27) showed similar complexity (ApEn) and fractal correlation properties (alpha(1), alpha(2), beta) of R-R interval dynamics despite lower spectral and time-domain measures. Progressive loss of complexity (decreased ApEn, r=-0.69, P<0.001) and alterations of long-term fractal-like heart rate behavior (increased alpha(2), r=0.63, decreased beta, r=-0.60, P<0.001 for both) were observed thereafter from middle age (40 to 60 years, n=29) to old age (>60 years, n=29). CONCLUSIONS: Cardiac interbeat interval dynamics change markedly from childhood to old age in healthy subjects. Children show complexity and fractal correlation properties of R-R interval time series comparable to those
Bałazy, Anna; Podgórski, Albert
2007-07-15
Nonspherical particles, such as fractal-like aggregates emitted by diesel engines, are commonly met in the ambient air. Some of them are believed to be carcinogenic to humans, thus their efficient removal is of crucial practical importance. A fibrous filter is the device commonly used for aerosol purification but the literature lacks experimental data concerning aggregates filtration. Effect of aggregates' parameters (fractal dimension, primary particle radius) as well as fiber diameter and air velocity on the filtration efficiency is investigated theoretically using the modified Brownian dynamics method. Three different expressions for the friction coefficient evaluation for the aggregates were examined. The results obtained indicate that structure of an aggregate, filter structure and process conditions strongly influence the aggregates deposition efficiency, which significantly differs from the values determined for mass-equivalent spherical particles. The results determined using the Brownian dynamics approach were compared with the values calculated using classical single fiber theory and noticeable discrepancy was observed for the most penetrating particles, while both approaches agree for the limiting cases of small or large particles. Peclet number based on the mobility radius and the interception parameter based on the outer radius are the proper criteria to describe diffusional and deterministic deposition of aggregates.
NASA Technical Reports Server (NTRS)
Hausdorff, J. M.; Mitchell, S. L.; Firtion, R.; Peng, C. K.; Cudkowicz, M. E.; Wei, J. Y.; Goldberger, A. L.
1997-01-01
Fluctuations in the duration of the gait cycle (the stride interval) display fractal dynamics and long-range correlations in healthy young adults. We hypothesized that these stride-interval correlations would be altered by changes in neurological function associated with aging and certain disease states. To test this hypothesis, we compared the stride-interval time series of 1) healthy elderly subjects and young controls and of 2) subjects with Huntington's disease and healthy controls. Using detrended fluctuation analysis we computed alpha, a measure of the degree to which one stride interval is correlated with previous and subsequent intervals over different time scales. The scaling exponent alpha was significantly lower in elderly subjects compared with young subjects (elderly: 0.68 +/- 0.14; young: 0.87 +/- 0.15; P < 0.003). The scaling exponent alpha was also smaller in the subjects with Huntington's disease compared with disease-free controls (Huntington's disease: 0.60 +/- 0.24; controls: 0.88 +/-0.17; P < 0.005). Moreover, alpha was linearly related to degree of functional impairment in subjects with Huntington's disease (r = 0.78, P < 0.0005). These findings demonstrate that strike-interval fluctuations are more random (i.e., less correlated) in elderly subjects and in subjects with Huntington's disease. Abnormal alterations in the fractal properties of gait dynamics are apparently associated with changes in central nervous system control.
Fractal structure and the dynamics of aggregation of synthetic melanin in low pH aqueous solutions
Huang, J.S.; Sung, J.; Eisner, M.; Moss, S.C.; Gallas, J.
1989-01-01
We have used static and dynamic light scattering to study the dynamics of aggregation of synthetic melanin, an amorphous biopolymeric substance, in low pH aqueous solution. We have found that, depending on the final pH value of the solutions, there existed two regimes of the aggregation kinetics, one corresponding to diffusion limited aggregation (DLA), and the other corresponding to reaction limited aggregation (RLA). The precipitates formed in these two regimes can be characterized by fractal structures. We have found fractal dimensions of d/sub f/ = 1.8 for the DLA clusters and d/sub f/ = 2.2 for the RLA clusters. These results agree well with the proposed limits of the fractal dimensions of the gold aggregates formed in aqueous solutions by Weitz et al.
Unification of two fractal families
NASA Astrophysics Data System (ADS)
Liu, Ying
1995-06-01
Barnsley and Hurd classify the fractal images into two families: iterated function system fractals (IFS fractals) and fractal transform fractals, or local iterated function system fractals (LIFS fractals). We will call IFS fractals, class 2 fractals and LIFS fractals, class 3 fractals. In this paper, we will unify these two approaches plus another family of fractals, the class 5 fractals. The basic idea is given as follows: a dynamical system can be represented by a digraph, the nodes in a digraph can be divided into two parts: transient states and persistent states. For bilevel images, a persistent node is a black pixel. A transient node is a white pixel. For images with more than two gray levels, a stochastic digraph is used. A transient node is a pixel with the intensity of 0. The intensity of a persistent node is determined by a relative frequency. In this way, the two families of fractals can be generated in a similar way. In this paper, we will first present a classification of dynamical systems and introduce the transformation based on digraphs, then we will unify the two approaches for fractal binary images. We will compare the decoding algorithms of the two families. Finally, we will generalize the discussion to continuous-tone images.
High-frequency chaotic dynamics enabled by optical phase-conjugation
Mercier, Émeric; Wolfersberger, Delphine; Sciamanna, Marc
2016-01-01
Wideband chaos is of interest for applications such as random number generation or encrypted communications, which typically use optical feedback in a semiconductor laser. Here, we show that replacing conventional optical feedback with phase-conjugate feedback improves the chaos bandwidth. In the range of achievable phase-conjugate mirror reflectivities, the bandwidth increase reaches 27% when compared with feedback from a conventional mirror. Experimental measurements of the time-resolved frequency dynamics on nanosecond time-scales show that the bandwidth enhancement is related to the onset of self-pulsing solutions at harmonics of the external-cavity frequency. In the observed regime, the system follows a chaotic itinerancy among these destabilized high-frequency external-cavity modes. The recorded features are unique to phase-conjugate feedback and distinguish it from the long-standing problem of time-delayed feedback dynamics. PMID:26739806
Fractal 1/f Dynamics Suggest Entanglement of Measurement and Human Performance
ERIC Educational Resources Information Center
Holden, John G.; Choi, Inhyun; Amazeen, Polemnia G.; Van Orden, Guy
2011-01-01
Variability of repeated measurements in human performances exhibits fractal 1/f noise. Yet the relative strength of this fractal pattern varies widely across conditions, tasks, and individuals. Four experiments illustrate how subtle details of the conditions of measurement change the fractal patterns observed across task conditions. The results…
NASA Astrophysics Data System (ADS)
Willie, Robert
2016-09-01
In this paper, we study a model system of equations of the time dependent Ginzburg-Landau equations of superconductivity in a Lorentz gauge, in scale of Hilbert spaces E^{α } with initial data in E^{β } satisfying 3α + β ≥ N/2, where N=2,3 is such that the spatial domain of the equations [InlineEquation not available: see fulltext.]. We show in the asymptotic dynamics of the equations, well-posedness of the dynamical system for a global exponential attractor {{U}}subset E^{α } compact in E^{β } if α >β , uniform differentiability of orbits on the attractor in E0\\cong L2, and the existence of an explicit finite bounding estimate on the fractal dimension of the attractor yielding that its Hausdorff dimension is as well finite. Uniform boundedness in (0,∞ )× Ω of solutions in E^{1/2}\\cong H1(Ω ) is in addition investigated.
Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis
NASA Astrophysics Data System (ADS)
Mantica, Giorgio; Perotti, Luca
2016-09-01
Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase–space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical rôle of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non-standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments. Dedicated to the memory of Joseph Ford, on the twentieth anniversary of his departure.
Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis
NASA Astrophysics Data System (ADS)
Mantica, Giorgio; Perotti, Luca
2016-09-01
Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase-space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical rôle of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non-standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments. Dedicated to the memory of Joseph Ford, on the twentieth anniversary of his departure.
Harmonic resonance structure and chaotic dynamics in the earth-vibrator system
Walker, D.
1995-05-01
Source-generated energy in seismic vibrator records includes ultraharmonics, sub-harmonics, ultra-subharmonics and possibly chaotic oscillatory behavior. Non-linear behaviors can be modeled using a ``hard-spring`` form of the Duffing equation. Modeling indicates that a qualitatively similar harmonic resonance structure is present for a broad range of possible mathematical descriptions. Qualitative global system behaviors may be examined without knowledge of actual earth parameters. Non-linear resonances become stronger, relative to fundamental sweep frequencies, as the driving force increases or damping decreases. System response energy levels are highest when non-linear resonances are strong. The presence of chaotic energy can indicate the highest energy state of a system response. Field data examples are consistent with behaviors predicted by modeling. Conventional correlation and stack uses a fraction of the energy produced in the earth-vibrator system. A correlation and filtering process that uses a representation of the source dynamics based on the system response can reduce signal degradation due to non-linear resonance.
Chaotic dynamics in charged-particle beams: Possible analogs of galactic evolution
Bohn, Courtlandt L.; /Northern Illinois U. /Fermilab
2004-12-01
During the last couple of years of his life, Henry Kandrup became intensely interested in using charged-particle beams as a tool for exploring the dynamics of evolving galaxies. He and I recognized that both galaxies and charged-particle beams can exhibit collisionless relaxation on surprisingly short time scales, and that this circumstance can be attributed to phase mixing of chaotic orbits. The chaos is often triggered by resonances caused by time dependence in the bulk potential, which acts almost identically for attractive gravitational forces as for repulsive electrostatic forces superposed on external focusing forces. Together we published several papers concerning evolving beams and galaxies, papers that relate to diverse topics such as the physics of chaotic mixing, the applicability of the Vlasov-Poisson formalism, and the production of diffuse halos. We also teamed with people from the University of Maryland to begin designing controlled experiments to be done at the University of Maryland Electron Ring. This paper highlights our collaborative findings as well as plans for future investigations that the findings have motivated.
Chaotic dynamics outside Saturn’s main rings: The case of Atlas
NASA Astrophysics Data System (ADS)
Renner, Stéfan; Cooper, Nicholas J.; El Moutamid, Maryame; Evans, Mike W.; Murray, Carl D.; Sicardy, Bruno
2014-11-01
We revisit in detail the dynamics of Atlas. From a fit to new Cassini ISS astrometric observations spanning February 2004 to August 2013, we estimate GM_Atlas=0.384+/-0.001 x 10^(-3)km^3s^(-2), a value 13% smaller than the previously published estimate but with an order of magnitude reduction in the uncertainty. Our numerically-derived orbit shows that Atlas is currently librating in both a 54:53 corotation eccentricity resonance (CER) and a 54:53 Lindblad eccentricity resonance (LER) with Prometheus. We demonstrate that the orbit of Atlas is chaotic, with a Lyapunov time of order 10 years, as a direct consequence of the coupled resonant interaction (CER/LER) with Prometheus. The interactions between the two resonances is investigated using the CoraLin analytical model (El Moutamid et al., 2014), showing that the chaotic zone fills almost all the corotation site occupied by the satellite’s orbit. Four 70 :67 apse-type mean motion resonances with Pandora are also overlapping, but these resonances have a much weaker effect on Atlas.We estimate the capture probabilities of Atlas into resonances with Prometheus as the orbits expand through tidal effects, and discuss the implications for the orbital evolution.
Multiparticle sintering dynamics: from fractal-like aggregates to compact structures.
Eggersdorfer, Max L; Kadau, Dirk; Herrmann, Hans J; Pratsinis, Sotiris E
2011-05-17
Multiparticle sintering is encountered in almost all high temperature processes for material synthesis (titania, silica, and nickel) and energy generation (e.g., fly ash formation) resulting in aggregates of primary particles (hard- or sinter-bonded agglomerates). This mechanism of particle growth is investigated quantitatively by mass and energy balances during viscous sintering of amorphous aerosol materials (e.g., SiO(2) and polymers) that typically have a distribution of sizes and complex morphology. This model is validated at limited cases of sintering between two (equally or unequally sized) particles, and chains of particles. The evolution of morphology, surface area and radii of gyration of multiparticle aggregates are elucidated for various sizes and initial fractal dimension. For each of these structures that had been generated by diffusion limited (DLA), cluster-cluster (DLCA), and ballistic particle-cluster agglomeration (BPCA) the surface area evolution is monitored and found to scale differently than that of the radius of gyration (moment of inertia). Expressions are proposed for the evolution of fractal dimension and the surface area of aggregates undergoing viscous sintering. These expressions are important in design of aerosol processes with population balance equations (PBE) and/or fluid dynamic simulations for material synthesis or minimization and even suppression of particle formation. PMID:21488641
Escape dynamics and fractal basin boundaries in the planar Earth-Moon system
NASA Astrophysics Data System (ADS)
de Assis, Sheila C.; Terra, Maisa O.
2014-10-01
The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth-Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points and as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincaré sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions.
Chaotic dynamics of stellar spin in binaries and the production of misaligned hot Jupiters.
Storch, Natalia I; Anderson, Kassandra R; Lai, Dong
2014-09-12
Many exoplanetary systems containing hot Jupiters are observed to have highly misaligned orbital axes relative to the stellar spin axes. Kozai-Lidov oscillations of orbital eccentricity and inclination induced by a binary companion, in conjunction with tidal dissipation, constitute a major channel for the production of hot Jupiters. We demonstrate that gravitational interaction between the planet and its oblate host star can lead to chaotic evolution of the stellar spin axis during Kozai cycles. As parameters such as the planet mass and stellar rotation period are varied, periodic islands can appear in an ocean of chaos, in a manner reminiscent of other dynamical systems. In the presence of tidal dissipation, the complex spin evolution can leave an imprint on the final spin-orbit misalignment angles. PMID:25214623
Chaotic dynamics of stellar spin in binaries and the production of misaligned hot Jupiters.
Storch, Natalia I; Anderson, Kassandra R; Lai, Dong
2014-09-12
Many exoplanetary systems containing hot Jupiters are observed to have highly misaligned orbital axes relative to the stellar spin axes. Kozai-Lidov oscillations of orbital eccentricity and inclination induced by a binary companion, in conjunction with tidal dissipation, constitute a major channel for the production of hot Jupiters. We demonstrate that gravitational interaction between the planet and its oblate host star can lead to chaotic evolution of the stellar spin axis during Kozai cycles. As parameters such as the planet mass and stellar rotation period are varied, periodic islands can appear in an ocean of chaos, in a manner reminiscent of other dynamical systems. In the presence of tidal dissipation, the complex spin evolution can leave an imprint on the final spin-orbit misalignment angles.
Luo, Shaohua
2014-09-01
This paper is concerned with the problem of adaptive fuzzy dynamic surface control (DSC) for the permanent magnet synchronous motor (PMSM) system with chaotic behavior, disturbance and unknown control gain and parameters. Nussbaum gain is adopted to cope with the situation that the control gain is unknown. And the unknown items can be estimated by fuzzy logic system. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. Finally, the numerical simulations indicate that the proposed scheme can suppress the chaos of PMSM and show the effectiveness and robustness of the proposed method.
Luo, Shaohua
2014-09-01
This paper is concerned with the problem of adaptive fuzzy dynamic surface control (DSC) for the permanent magnet synchronous motor (PMSM) system with chaotic behavior, disturbance and unknown control gain and parameters. Nussbaum gain is adopted to cope with the situation that the control gain is unknown. And the unknown items can be estimated by fuzzy logic system. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. Finally, the numerical simulations indicate that the proposed scheme can suppress the chaos of PMSM and show the effectiveness and robustness of the proposed method.
A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system
Kuznetsov, S. P. Seleznev, E. P.
2006-02-15
A nonautonomous nonlinear system is constructed and implemented as an experimental device. As represented by a 4D stroboscopic Poincare map, the system exhibits a Smale-Williams-type strange attractor. The system consists of two coupled van der Pol oscillators whose frequencies differ by a factor of two. The corresponding Hopf bifurcation parameters slowly vary as periodic functions of time in antiphase with one another; i.e., excitation is alternately transferred between the oscillators. The mechanisms underlying the system's chaotic dynamics and onset of chaos are qualitatively explained. A governing system of differential equations is formulated. The existence of a chaotic attractor is confirmed by numerical results. Hyperbolicity is verified numerically by performing a statistical analysis of the distribution of the angle between the stable and unstable subspaces of manifolds of the chaotic invariant set. Experimental results are in qualitative agreement with numerical predictions.
The influence of auditory-motor coupling on fractal dynamics in human gait.
Hunt, Nathaniel; McGrath, Denise; Stergiou, Nicholas
2014-01-01
Humans exhibit an innate ability to synchronize their movements to music. The field of gait rehabilitation has sought to capitalize on this phenomenon by invoking patients to walk in time to rhythmic auditory cues with a view to improving pathological gait. However, the temporal structure of the auditory cue, and hence the temporal structure of the target behavior has not been sufficiently explored. This study reveals the plasticity of auditory-motor coupling in human walking in relation to 'complex' auditory cues. The authors demonstrate that auditory-motor coupling can be driven by different coloured auditory noise signals (e.g. white, brown), shifting the fractal temporal structure of gait dynamics towards the statistical properties of the signals used. This adaptive capability observed in whole-body movement, could potentially be harnessed for targeted neuromuscular rehabilitation in patient groups, depending on the specific treatment goal. PMID:25080936
The influence of auditory-motor coupling on fractal dynamics in human gait
Hunt, Nathaniel; McGrath, Denise; Stergiou, Nicholas
2014-01-01
Humans exhibit an innate ability to synchronize their movements to music. The field of gait rehabilitation has sought to capitalize on this phenomenon by invoking patients to walk in time to rhythmic auditory cues with a view to improving pathological gait. However, the temporal structure of the auditory cue, and hence the temporal structure of the target behavior has not been sufficiently explored. This study reveals the plasticity of auditory-motor coupling in human walking in relation to ‘complex' auditory cues. The authors demonstrate that auditory-motor coupling can be driven by different coloured auditory noise signals (e.g. white, brown), shifting the fractal temporal structure of gait dynamics towards the statistical properties of the signals used. This adaptive capability observed in whole-body movement, could potentially be harnessed for targeted neuromuscular rehabilitation in patient groups, depending on the specific treatment goal. PMID:25080936
Scaling anomalies in the coarsening dynamics of fractal viscous fingering patterns
NASA Astrophysics Data System (ADS)
Conti, Massimo; Lipshtat, Azi; Meerson, Baruch
2004-03-01
We analyze a recent experiment of Sharon et al. (2003) on the coarsening, due to surface tension, of fractal viscous fingering patterns (FVFPs) grown in a radial Hele-Shaw cell. We argue that an unforced Hele-Shaw model, a natural model for that experiment, belongs to the same universality class as model B of phase ordering. Two series of numerical simulations with model B are performed, with the FVFPs grown in the experiment and with diffusion limited aggregates as the initial conditions. We observed Lifshitz-Slyozov scaling t1/3 at intermediate distances and slow convergence to this scaling at small distances. Dynamic scale invariance breaks down at large distances.
Kengne, Jacques; Kenmogne, Fabien
2014-12-15
The nonlinear dynamics of fourth-order Silva-Young type chaotic oscillators with flat power spectrum recently introduced by Tamaseviciute and collaborators is considered. In this type of oscillators, a pair of semiconductor diodes in an anti-parallel connection acts as the nonlinear component necessary for generating chaotic oscillations. Based on the Shockley diode equation and an appropriate selection of the state variables, a smooth mathematical model (involving hyperbolic sine and cosine functions) is derived for a better description of both the regular and chaotic dynamics of the system. The complex behavior of the oscillator is characterized in terms of its parameters by using time series, bifurcation diagrams, Lyapunov exponents' plots, Poincaré sections, and frequency spectra. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the proposed mathematical model to accurately describe/predict both the regular and chaotic behaviors of the oscillator.
Kengne, Jacques; Kenmogne, Fabien
2014-12-01
The nonlinear dynamics of fourth-order Silva-Young type chaotic oscillators with flat power spectrum recently introduced by Tamaseviciute and collaborators is considered. In this type of oscillators, a pair of semiconductor diodes in an anti-parallel connection acts as the nonlinear component necessary for generating chaotic oscillations. Based on the Shockley diode equation and an appropriate selection of the state variables, a smooth mathematical model (involving hyperbolic sine and cosine functions) is derived for a better description of both the regular and chaotic dynamics of the system. The complex behavior of the oscillator is characterized in terms of its parameters by using time series, bifurcation diagrams, Lyapunov exponents' plots, Poincaré sections, and frequency spectra. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the proposed mathematical model to accurately describe/predict both the regular and chaotic behaviors of the oscillator.
Kengne, Jacques; Kenmogne, Fabien
2014-12-01
The nonlinear dynamics of fourth-order Silva-Young type chaotic oscillators with flat power spectrum recently introduced by Tamaseviciute and collaborators is considered. In this type of oscillators, a pair of semiconductor diodes in an anti-parallel connection acts as the nonlinear component necessary for generating chaotic oscillations. Based on the Shockley diode equation and an appropriate selection of the state variables, a smooth mathematical model (involving hyperbolic sine and cosine functions) is derived for a better description of both the regular and chaotic dynamics of the system. The complex behavior of the oscillator is characterized in terms of its parameters by using time series, bifurcation diagrams, Lyapunov exponents' plots, Poincaré sections, and frequency spectra. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the proposed mathematical model to accurately describe/predict both the regular and chaotic behaviors of the oscillator. PMID:25554054
ERIC Educational Resources Information Center
Esbenshade, Donald H., Jr.
1991-01-01
Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)
Chaotic behavior of collective ion dynamics in the presence of an external static magnetic field
NASA Astrophysics Data System (ADS)
Poria, Swarup; Ghosh, Samiran
2016-06-01
The two-dimensional nonlinear collective ion dynamics in the presence of external magnetic field in an electron-ion plasma is investigated. The analysis is performed for traveling plane waves to elucidate the various aspects of the phase-space dynamics. The presence of magnetic field makes the dynamics of the nonlinear wave complex with a complicated phase-space behavior. Thus, the nonlinear wave supports a wide class of nonlinear structures viz., single soliton, multi-soliton, periodic, and quasi-periodic oscillations depending on the values of M (Mach number) and Ω (the ratio of ion gyro-frequency to the ion plasma frequency). The computational results predict the chaotic behavior of the nonlinear wave and the transition to chaos takes place when Ω ≳ 0.35 depending on the direction of propagation and the value of M. The amplitude of the wave depends on the obliqueness of the propagation and Mach number, whereas the magnetic field changes the dispersion properties of the wave.
Richie, D.A.; Zhang, T.; Zachman, J.C.; Tabatabaie, N. . Dept. of Electrical and Computer Engineering); Choquette, K.D. ); Leibenguth, R.E. )
1994-11-01
The authors discuss the dynamics of transverse mode competition in an etched air-post vertical-cavity laser diode under dc excitation using an annular ring contact as a spatial filter. Distinct regions of operation are found for various ranges of fixed bias currents. At 1.5 times threshold, the device enters a region which exhibits chaotic fluctuations between the fundamental and a higher order lasing mode. The dynamics of these fluctuations are studied using the method of delays, and a calculation of the power spectrum and the correlation dimension are reported. It is found that the dynamics of the chaotic fluctuations have a correlation dimension of approximately 2.8. The results are indicative of a low-dimensional strange attractor underlying the modal competition noise.
Dynamic structure factor of vibrating fractals: proteins as a case study.
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-01-01
We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kR(g)≫1, where R(g) is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u(−>)(i)(t)-u(−>)(j)(0)](2)>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u(−>)(i)(t)-u(−>)(i)(0)](2)> ~t(ν), where ν is the anomalous diffusion exponent that depends on the spectral dimension d(s) and fractal dimension d(f). As a result, for wave numbers obeying k(2))(2)>≳1, S(k,t) effectively decays as a stretched exponential S(k,t)≃S(k)e(-(Γ(k)t)(β)) with β≃ν, where the relaxation rate is Γ(k)~(k(B)T/mω(o)(2))(1/β)k(2/β), T is the temperature, and mω(o)(2) the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)~k(-d(f)), with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size
My chaotic trajectory: A brief (personalized) history of solar-system dynamics.
NASA Astrophysics Data System (ADS)
Burns, Joseph A.
2014-05-01
I will use this opportunity to recall my professional career. Like many, I was drawn into the space program during the mid-60s and early 70s when the solar system’s true nature was being revealed. Previously, dynamical astronomy discussed the short-term, predictable motions of point masses; simultaneously, small objects (e.g., satellites, asteroids, dust) were thought boring rather than dynamically rich. Many of today’s most active research subjects were unknown: TNOs, planetary rings, exoplanets and debris disks. The continuing stream of startling findings by spacecraft, ground-based surveys and numerical simulations forced a renaissance in celestial mechanics, incorporating new dynamical paradigms and additional physics (e.g., energy loss, catastrophic events, radiation forces). My interests evolved as the space program expanded outward: dust, asteroids, natural satellites, rings; rotations, orbital evolution, origins. Fortunately for me, in the early days, elementary models with simple solutions were often adequate to gain a first-order explanation of many puzzles. One could be a generalist, always learning new things.My choice of research subjects was influenced greatly by: i) Cornell colleagues involved in space missions who shared results: the surprising diversity of planetary satellites, the unanticipated orbital and rotational dynamics of asteroids, the chaotic histories of solar system bodies, the non-intuitive behavior of dust and planetary rings, irregular satellites. ii) Teaching introductory courses in applied math, dynamics and planetary science encouraged understandable models. iii) The stimulation of new ideas owing to service at Icarus and on space policy forums. iv) Most importantly, excellent students and colleagues who pushed me into new research directions, and who then stimulated and educated me about those topics.If time allows, I will describe some of today’s puzzles for me and point out similarities between the past development in our
Dynamical collapse of trajectories
NASA Astrophysics Data System (ADS)
Biemond, J. J. Benjamin; de Moura, Alessandro P. S.; Grebogi, Celso; van de Wouw, Nathan; Nijmeijer, Henk
2012-04-01
Friction induces unexpected dynamical behaviour. In the paradigmatic pendulum and double-well systems with friction, modelled with differential inclusions, distinct trajectories can collapse onto a single point. Transversal homoclinic orbits display collapse and generate chaotic saddles with forward dynamics that is qualitatively different from the backward dynamics. The space of initial conditions converging to the chaotic saddle is fractal, but the set of points diverging from it is not: friction destroys the complexity of the forward dynamics by generating a unique horseshoe-like topology.
Faybishenko, Boris; Doughty, Christine; Stoops, Thomas M.; Wood, thomas R.; Wheatcraft, Stephen W.
1999-12-31
(1) To determine if and when dynamical chaos theory can be used to investigate infiltration of fluid and contaminant transport in heterogeneous soils and fractured rocks. (2) To introduce a new approach to the multiscale characterization of flow and transport in fractured basalt vadose zones and to develop physically based conceptual models on a hierarchy of scales. The following activities are indicative of the success in meeting the project s objectives: A series of ponded infiltration tests, including (1) small-scale infiltration tests (ponded area 0.5 m2) conducted at the Hell s Half Acre site near Shelley, Idaho, and (2) intermediate-scale infiltration tests (ponded area 56 m2) conducted at the Box Canyon site near Arco, Idaho. Laboratory investigations and modeling of flow in a fractured basalt core. A series of small-scale dripping experiments in fracture models. Evaluation of chaotic behavior of flow in laboratory and field experiments using methods from nonlinear dynamics; Evaluation of the impact these dynamics may have on contaminant transport through heterogeneous fractured rocks and soils, and how it can be used to guide remediation efforts; Development of a conceptual model and mathematical and numerical algorithms for flow and transport that incorporate (1) the spatial variability of heterogeneous porous and fractured media, and (2) the description of the temporal dynamics of flow and transport, both of which may be chaotic. Development of appropriate experimental field and laboratory techniques needed to detect diagnostic parameters for chaotic behavior of flow. This approach is based on the assumption that spatial heterogeneity and flow phenomena are affected by nonlinear dynamics, and in particular, by chaotic processes. The scientific and practical value of this approach is that we can predict the range within which the parameters of flow and transport change with time in order to design and manage the remediation, even when we can not predict
The Role of Chaotic Dynamics in the Cooling of Magmatic Systems in Subduction Related Environment
NASA Astrophysics Data System (ADS)
Petrelli, M.; El Omari, K.; Le Guer, Y.; Perugini, D.
2015-12-01
Understanding the dynamics occurring during the thermo-chemical evolution of igneous bodies is of crucial importance in both petrology and volcanology. This is particularly true in subduction related systems where large amount of magmas start, and sometime end, their differentiation histories at mid and lower crust levels. These magmas play a fundamental role in the evolution of both plutonic and volcanic systems but several key questions are still open about their thermal and chemical evolution: 1) what are the dynamics governing the development of these magmatic systems, 2) what are the timescales of cooling, crystallization and chemical differentiation; 4) how these systems contribute to the evolution of shallower magmatic systems? Recent works shed light on the mechanisms acting during the growing of new magmatic bodies and it is now accepted that large crustal igneous bodies result from the accretion and/or amalgamation of smaller ones. What is lacking now is how fluid dynamics of magma bodies can influence the evolution of these igneous systems. In this contribution we focus on the thermo-chemical evolution of a subduction related magmatic system at pressure conditions corresponding to mid-crustal levels (0.7 GPa, 20-25 km). In order to develop a robust model and address the Non-Newtonian behavior of crystal bearing magmas, we link the numerical formulation of the problem to experimental results and rheological modeling. We define quantitatively the thermo-chemical evolution of the system and address the timing required to reach the maximum packing fraction. We will shows that the development of chaotic dynamics significantly speed up the crystallization process decreasing the time needed to reach the maximum packing fraction. Our results have important implications for both the rheological history of the magmatic body and the refilling of shallower magmatic systems.
Fast and secure encryption-decryption method based on chaotic dynamics
Protopopescu, Vladimir A.; Santoro, Robert T.; Tolliver, Johnny S.
1995-01-01
A method and system for the secure encryption of information. The method comprises the steps of dividing a message of length L into its character components; generating m chaotic iterates from m independent chaotic maps; producing an "initial" value based upon the m chaotic iterates; transforming the "initial" value to create a pseudo-random integer; repeating the steps of generating, producing and transforming until a pseudo-random integer sequence of length L is created; and encrypting the message as ciphertext based upon the pseudo random integer sequence. A system for accomplishing the invention is also provided.
Self: an adaptive pressure arising from self-organization, chaotic dynamics, and neural Darwinism.
Bruzzo, Angela Alessia; Vimal, Ram Lakhan Pandey
2007-12-01
In this article, we establish a model to delineate the emergence of "self" in the brain making recourse to the theory of chaos. Self is considered as the subjective experience of a subject. As essential ingredients of subjective experiences, our model includes wakefulness, re-entry, attention, memory, and proto-experiences. The stability as stated by chaos theory can potentially describe the non-linear function of "self" as sensitive to initial conditions and can characterize it as underlying order from apparently random signals. Self-similarity is discussed as a latent menace of a pathological confusion between "self" and "others". Our test hypothesis is that (1) consciousness might have emerged and evolved from a primordial potential or proto-experience in matter, such as the physical attractions and repulsions experienced by electrons, and (2) "self" arises from chaotic dynamics, self-organization and selective mechanisms during ontogenesis, while emerging post-ontogenically as an adaptive pressure driven by both volume and synaptic-neural transmission and influencing the functional connectivity of neural nets (structure).
A secure image encryption method based on dynamic harmony search (DHS) combined with chaotic map
NASA Astrophysics Data System (ADS)
Mirzaei Talarposhti, Khadijeh; Khaki Jamei, Mehrzad
2016-06-01
In recent years, there has been increasing interest in the security of digital images. This study focuses on the gray scale image encryption using dynamic harmony search (DHS). In this research, first, a chaotic map is used to create cipher images, and then the maximum entropy and minimum correlation coefficient is obtained by applying a harmony search algorithm on them. This process is divided into two steps. In the first step, the diffusion of a plain image using DHS to maximize the entropy as a fitness function will be performed. However, in the second step, a horizontal and vertical permutation will be applied on the best cipher image, which is obtained in the previous step. Additionally, DHS has been used to minimize the correlation coefficient as a fitness function in the second step. The simulation results have shown that by using the proposed method, the maximum entropy and the minimum correlation coefficient, which are approximately 7.9998 and 0.0001, respectively, have been obtained.
Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems
NASA Astrophysics Data System (ADS)
Lucarini, Valerio; Faranda, Davide; Wouters, Jeroen; Kuna, Tobias
2014-02-01
In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.
Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems.
Lucarini, Valerio; Faranda, Davide; Wouters, Jeroen; Kuna, Tobias
2014-01-01
In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.
The fractal dimensions of the spatial distribution of young open clusters in the solar neighbourhood
NASA Astrophysics Data System (ADS)
de La Fuente Marcos, R.; de La Fuente Marcos, C.
2006-06-01
Context: .Fractals are geometric objects with dimensionalities that are not integers. They play a fundamental role in the dynamics of chaotic systems. Observation of fractal structure in both the gas and the star-forming sites in galaxies suggests that the spatial distribution of young open clusters should follow a fractal pattern, too. Aims: .Here we investigate the fractal pattern of the distribution of young open clusters in the Solar Neighbourhood using a volume-limited sample from WEBDA and a multifractal analysis. By counting the number of objects inside spheres of different radii centred on clusters, we study the homogeneity of the distribution. Methods: .The fractal dimension D of the spatial distribution of a volume-limited sample of young open clusters is determined by analysing different moments of the count-in-cells. The spectrum of the Minkowski-Bouligand dimension of the distribution is studied as a function of the parameter q. The sample is corrected for dynamical effects. Results: .The Minkowski-Bouligand dimension varies with q in the range 0.71-1.77, therefore the distribution of young open clusters is fractal. We estimate that the average value of the fractal dimension is < D> = 1.7± 0.2 for the distribution of young open clusters studied. Conclusions: .The spatial distribution of young open clusters in the Solar Neighbourhood exhibits multifractal structure. The fractal dimension is time-dependent, increasing over time. The values found are consistent with the fractal dimension of star-forming sites in other spiral galaxies.
Fractal dynamics of body motion in post-stroke hemiplegic patients during walking
NASA Astrophysics Data System (ADS)
Akay, M.; Sekine, M.; Tamura, T.; Higashi, Y.; Fujimoto, T.
2004-06-01
In this paper, we quantify the complexity of body motion during walking in post-stroke hemiplegic patients. The body motion of patients and healthy elderly subjects was measured by using the accelerometry technique. The complexity of body motion was quantified using the maximum likelihood estimator (MLE-) based fractal analysis methods. Our results suggest that the fractal dimensions of the body motion in post-stroke hemiplegic patients at several Brunnstrom stages were significantly higher than those of healthy elderly subjects (p < 0.05). However, in the hemiplegic patients, the fractal dimensions were more related to Brunnstrom stages.
Origin of periodic and chaotic dynamics due to drops moving in a microfluidic loop device.
Maddala, Jeevan; Vanapalli, Siva A; Rengaswamy, Raghunathan
2014-02-01
Droplets moving in a microfluidic loop device exhibit both periodic and chaotic behaviors based on the inlet droplet spacing. We observe that the periodic behavior is an outcome of carrier phase mass conservation principle, which translates into a droplet spacing quantization rule. This rule implies that the summation of exit spacing is equal to an integral multiple of inlet spacing. This principle also enables identification of periodicity in experimental systems with input scatter. We find that the origin of chaotic behavior is through intermittency, which arises when drops enter and leave the junctions at the same time. We derive an analytical expression to estimate the occurrence of these chaotic regions as a function of system parameters. We provide experimental, simulation, and analytical results to validate the origin of periodic and chaotic behavior.
Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale.
Maslennikov, Oleg V; Nekorkin, Vladimir I
2016-07-01
In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.
NASA Astrophysics Data System (ADS)
Tanaka, Naoki; Okamoto, Hiroshi; Naito, Masayoshi
1995-07-01
We propose a method of distinguishing chaos from random fractal sequences which have been difficult to discriminate from chaos. In the proposed method, time series is predicted both toward the future and toward the past, and the accuracy of the two types of predictions is compared. We show, considering the time reversal symmetry of time series, that if the time series is chaotic and originates from a dissipative dynamical system, the accuracy is better for the future prediction than for the past prediction, whereas the accuracy is the same if the time series is a random fractal sequence. The method is also applicable to distinguishing between chaos and stationary noise.
Synchronization of chaotic systems
Pecora, Louis M.; Carroll, Thomas L.
2015-09-15
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
Synchronization of chaotic systems.
Pecora, Louis M; Carroll, Thomas L
2015-09-01
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
Bhaduri, Anirban; Ghosh, Dipak
2016-01-01
The cardiac dynamics during meditation is explored quantitatively with two chaos-based non-linear techniques viz. multi-fractal detrended fluctuation analysis and visibility network analysis techniques. The data used are the instantaneous heart rate (in beats/minute) of subjects performing Kundalini Yoga and Chi meditation from PhysioNet. The results show consistent differences between the quantitative parameters obtained by both the analysis techniques. This indicates an interesting phenomenon of change in the complexity of the cardiac dynamics during meditation supported with quantitative parameters. The results also produce a preliminary evidence that these techniques can be used as a measure of physiological impact on subjects performing meditation. PMID:26909045
Orbital stability analysis and chaotic dynamics of exoplanets in multi-stellar systems
NASA Astrophysics Data System (ADS)
Satyal, Suman
The advancement in detection technology has substantially increased the discovery rate of exoplanets in the last two decades. The confirmation of thousands of exoplanets orbiting the solar type stars has raised new astrophysical challenges, including the studies of orbital dynamics and long-term stability of such planets. Continuous orbital stability of the planet in stellar habitable zone is considered vital for life to develop. Hence, these studies furthers one self-evident aim of mankind to find an answer to the century old question: Are we alone?. This dissertation investigates the planetary orbits in single and binary star systems. Within binaries, a planet could orbit either one or both stars as S-type or P-type, respectively. I have considered S-type planets in two binaries, gamma Cephei and HD 196885, and compute their orbits by using various numerical techniques to assess their periodic, quasi-periodic or chaotic nature. The Hill stability (HS) function, which measures the orbital perturbation induced by the nearby companion, is calculated for each system and then its efficacy as a new chaos indicator is tested against Maximum Lyapunov Exponents (MLE) and Mean Exponential Growth factor of Nearby Orbits (MEGNO). The dynamics of HD 196885 AB is further explored with an emphasis on the planet's higher orbital inclination relative to the binary plane. I have quantitatively mapped out the chaotic and quasi-periodic regions of the system's phase space, which indicates a likely regime of the planet's inclination. In, addition, the resonant angle is inspected to determine whether alternation between libration and circulation occurs as a consequence of Kozai oscillations, a probable mechanism that can drive the planetary orbit to a large inclination. The studies of planetary system in GJ 832 shows potential of hosting multiple planets in close orbits. The phase space of GJ 832c (inner planet) and the Earth-mass test planet(s) are analyzed for periodic
NASA Astrophysics Data System (ADS)
Martelloni, Gianluca; Bagnoli, Franco
2016-04-01
Richardson's treatise on turbulent diffusion in 1926 [24] and today, the list of system displaying anomalous dynamical behavior is quite extensive. We only report some examples: charge carrier transport in amorphous semiconductors [25], porous systems [26], reptation dynamics in polymeric systems [27, 28], transport on fractal geometries [29], the long-time dynamics of DNA sequences [30]. In this scenario, the fractional calculus is used to generalized the Fokker-Planck linear equation -∂P (x,t)=D ∇2P (x,t), ∂t (3) where P (x,t) is the density of probability in the space x=[x1, x2, x3] and time t, while D >0 is the diffusion coefficient. Such processes are characterized by Eq. (1). An example of Eq. (3) generalization is ∂∂tP (x,t)=D∇ αP β(x,t) ‑ ∞ < α ≤ 2 β > ‑ 1 , (4) where the fractional based-derivatives Laplacian Σ(∂α/∂xα)i, (i = 1, 2, 3), of non-linear term Pβ(x,t) is taken into account [31]. Another generalized form is represented by equation ∂∂tδδP(x,t)=D ∇ αP(x,t) δ > 0 α ≤ 2 , (5) that considers also the fractional time-derivative [32]. These fractional-described processes exhibit a power law patters as expressed by Eq. (2). This general introduction introduces the presented work, whose aim is to develop a theoretical model in order to forecast the triggering and propagation of landslides, using the techniques of fractional calculus. The latter is suitable for modeling the water infiltration (i.e., the pore water pressure diffusion in the soil) and the dynamical processes in the fractal media [33]. Alternatively the fractal representation of temporal and spatial derivative (the fractal order only appears in the denominator of the derivative) is considered and the results are compared to the fractional one. The prediction of landslides and the discovering of the triggering mechanism, is one of the challenging problems in earth science. Landslides can be triggered by different factors but in most cases the trigger is an
NASA Astrophysics Data System (ADS)
Martelloni, Gianluca; Bagnoli, Franco
2016-04-01
Richardson's treatise on turbulent diffusion in 1926 [24] and today, the list of system displaying anomalous dynamical behavior is quite extensive. We only report some examples: charge carrier transport in amorphous semiconductors [25], porous systems [26], reptation dynamics in polymeric systems [27, 28], transport on fractal geometries [29], the long-time dynamics of DNA sequences [30]. In this scenario, the fractional calculus is used to generalized the Fokker-Planck linear equation -∂P (x,t)=D ∇2P (x,t), ∂t (3) where P (x,t) is the density of probability in the space x=[x1, x2, x3] and time t, while D >0 is the diffusion coefficient. Such processes are characterized by Eq. (1). An example of Eq. (3) generalization is ∂∂tP (x,t)=D∇ αP β(x,t) - ∞ < α ≤ 2 β > - 1 , (4) where the fractional based-derivatives Laplacian Σ(∂α/∂xα)i, (i = 1, 2, 3), of non-linear term Pβ(x,t) is taken into account [31]. Another generalized form is represented by equation ∂∂tδδP(x,t)=D ∇ αP(x,t) δ > 0 α ≤ 2 , (5) that considers also the fractional time-derivative [32]. These fractional-described processes exhibit a power law patters as expressed by Eq. (2). This general introduction introduces the presented work, whose aim is to develop a theoretical model in order to forecast the triggering and propagation of landslides, using the techniques of fractional calculus. The latter is suitable for modeling the water infiltration (i.e., the pore water pressure diffusion in the soil) and the dynamical processes in the fractal media [33]. Alternatively the fractal representation of temporal and spatial derivative (the fractal order only appears in the denominator of the derivative) is considered and the results are compared to the fractional one. The prediction of landslides and the discovering of the triggering mechanism, is one of the challenging problems in earth science. Landslides can be triggered by different factors but in most cases the trigger is an
Xavier, J C; Strunz, W T; Beims, M W
2015-08-01
We consider the energy flow between a classical one-dimensional harmonic oscillator and a set of N two-dimensional chaotic oscillators, which represents the finite environment. Using linear response theory we obtain an analytical effective equation for the system harmonic oscillator, which includes a frequency dependent dissipation, a shift, and memory effects. The damping rate is expressed in terms of the environment mean Lyapunov exponent. A good agreement is shown by comparing theoretical and numerical results, even for environments with mixed (regular and chaotic) motion. Resonance between system and environment frequencies is shown to be more efficient to generate dissipation than larger mean Lyapunov exponents or a larger number of bath chaotic oscillators.
Quantum chaotic resonances from short periodic orbits.
Novaes, M; Pedrosa, J M; Wisniacki, D; Carlo, G G; Keating, J P
2009-09-01
We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest-living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long-lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.
Self-Similar Random Process and Chaotic Behavior In Serrated Flow of High Entropy Alloys
NASA Astrophysics Data System (ADS)
Chen, Shuying; Yu, Liping; Ren, Jingli; Xie, Xie; Li, Xueping; Xu, Ying; Zhao, Guangfeng; Li, Peizhen; Yang, Fuqian; Ren, Yang; Liaw, Peter K.
2016-07-01
The statistical and dynamic analyses of the serrated-flow behavior in the nanoindentation of a high-entropy alloy, Al0.5CoCrCuFeNi, at various holding times and temperatures, are performed to reveal the hidden order associated with the seemingly-irregular intermittent flow. Two distinct types of dynamics are identified in the high-entropy alloy, which are based on the chaotic time-series, approximate entropy, fractal dimension, and Hurst exponent. The dynamic plastic behavior at both room temperature and 200 °C exhibits a positive Lyapunov exponent, suggesting that the underlying dynamics is chaotic. The fractal dimension of the indentation depth increases with the increase of temperature, and there is an inflection at the holding time of 10 s at the same temperature. A large fractal dimension suggests the concurrent nucleation of a large number of slip bands. In particular, for the indentation with the holding time of 10 s at room temperature, the slip process evolves as a self-similar random process with a weak negative correlation similar to a random walk.
Self-Similar Random Process and Chaotic Behavior In Serrated Flow of High Entropy Alloys
Chen, Shuying; Yu, Liping; Ren, Jingli; Xie, Xie; Li, Xueping; Xu, Ying; Zhao, Guangfeng; Li, Peizhen; Yang, Fuqian; Ren, Yang; Liaw, Peter K.
2016-01-01
The statistical and dynamic analyses of the serrated-flow behavior in the nanoindentation of a high-entropy alloy, Al0.5CoCrCuFeNi, at various holding times and temperatures, are performed to reveal the hidden order associated with the seemingly-irregular intermittent flow. Two distinct types of dynamics are identified in the high-entropy alloy, which are based on the chaotic time-series, approximate entropy, fractal dimension, and Hurst exponent. The dynamic plastic behavior at both room temperature and 200 °C exhibits a positive Lyapunov exponent, suggesting that the underlying dynamics is chaotic. The fractal dimension of the indentation depth increases with the increase of temperature, and there is an inflection at the holding time of 10 s at the same temperature. A large fractal dimension suggests the concurrent nucleation of a large number of slip bands. In particular, for the indentation with the holding time of 10 s at room temperature, the slip process evolves as a self-similar random process with a weak negative correlation similar to a random walk. PMID:27435922
Self-Similar Random Process and Chaotic Behavior In Serrated Flow of High Entropy Alloys.
Chen, Shuying; Yu, Liping; Ren, Jingli; Xie, Xie; Li, Xueping; Xu, Ying; Zhao, Guangfeng; Li, Peizhen; Yang, Fuqian; Ren, Yang; Liaw, Peter K
2016-01-01
The statistical and dynamic analyses of the serrated-flow behavior in the nanoindentation of a high-entropy alloy, Al0.5CoCrCuFeNi, at various holding times and temperatures, are performed to reveal the hidden order associated with the seemingly-irregular intermittent flow. Two distinct types of dynamics are identified in the high-entropy alloy, which are based on the chaotic time-series, approximate entropy, fractal dimension, and Hurst exponent. The dynamic plastic behavior at both room temperature and 200 °C exhibits a positive Lyapunov exponent, suggesting that the underlying dynamics is chaotic. The fractal dimension of the indentation depth increases with the increase of temperature, and there is an inflection at the holding time of 10 s at the same temperature. A large fractal dimension suggests the concurrent nucleation of a large number of slip bands. In particular, for the indentation with the holding time of 10 s at room temperature, the slip process evolves as a self-similar random process with a weak negative correlation similar to a random walk. PMID:27435922
Meng, Zhiyong; Hashmi, Sara M; Elimelech, Menachem
2013-02-15
The time-evolutions of nanoparticle hydrodynamic radius and aggregate fractal dimension during the aggregation of fullerene (C(60)) nanoparticles (FNPs) were measured via simultaneous multiangle static and dynamic light scattering. The FNP aggregation behavior was determined as a function of monovalent (NaCl) and divalent (CaCl(2)) electrolyte concentration, and the impact of addition of dissolved natural organic matter (humic acid) to the solution was also investigated. In the absence of humic acid, the fractal dimension decreased over time with monovalent and divalent salts, suggesting that aggregates become slightly more open and less compact as they grow. Although the aggregates become slightly more open, the magnitude of the fractal dimension suggests intermediate aggregation between the diffusion- and reaction-limited regimes. We observed different aggregation behavior with monovalent and divalent salts upon the addition of humic acid to the solution. For NaCl-induced aggregation, the introduction of humic acid significantly suppressed the aggregation rate of FNPs at NaCl concentrations lower than 150mM. In this case, the aggregation was intermediate or reaction-limited even at NaCl concentrations as high as 500mM, giving rise to aggregates with a fractal dimension of 2.0. For CaCl(2)-induced aggregation, the introduction of humic acid enhanced the aggregation of FNPs at CaCl(2) concentrations greater than about 5mM due to calcium complexation and bridging effects. Humic acid also had an impact on the FNP aggregate structure in the presence of CaCl(2), resulting in a fractal dimension of 1.6 for the diffusion-limited aggregation regime. Our results with CaCl(2) indicate that in the presence of humic acid, FNP aggregates have a more open and loose structure than in the absence of humic acid. The aggregation results presented in this paper have important implications for the transport, chemical reactivity, and toxicity of engineered nanoparticles in aquatic
Determining Chaotic Instabilities in Mechanical Systems
NASA Technical Reports Server (NTRS)
Zak, M. A.
1986-01-01
Theoretical developments enable suppression of chaotic structual motions. Theory enables prediction, avoidance, and suppression of chaotic vibrations in structures, especially using dynamic feedback stabilization. In new formulation, motion both repeatable and predictable.
Ottermanns, Richard; Szonn, Kerstin; Preuβ, Thomas G; Roβ-Nickoll, Martina
2014-01-01
In this study we present evidence that anthropogenic stressors can reduce the resilience of age-structured populations. Enhancement of disturbance in a model-based Daphnia population lead to a repression of chaotic population dynamics at the same time increasing the degree of synchrony between the population's age classes. Based on the theory of chaos-mediated survival an increased risk of extinction was revealed for this population exposed to high concentrations of a chemical stressor. The Lyapunov coefficient was supposed to be a useful indicator to detect disturbance thresholds leading to alterations in population dynamics. One possible explanation could be a discrete change in attractor orientation due to external disturbance. The statistical analysis of Lyapunov coefficient distribution is proposed as a methodology to test for significant non-linear effects of general disturbance on populations. Although many new questions arose, this study forms a theoretical basis for a dynamical definition of population recovery. PMID:24809537
Fractals in biology and medicine
NASA Technical Reports Server (NTRS)
Havlin, S.; Buldyrev, S. V.; Goldberger, A. L.; Mantegna, R. N.; Ossadnik, S. M.; Peng, C. K.; Simons, M.; Stanley, H. E.
1995-01-01
Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.
Fractals in physiology and medicine
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.; West, Bruce J.
1987-01-01
The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.
Chaotic And Periodic Dynamics Of A Slider-Crank Mechanism With Slider Clearance
NASA Astrophysics Data System (ADS)
Farahanchi, F.; Shaw, S. W.
1994-10-01
The problem of a planar slider-crank mechanism with clearance at the sliding (prismatic) joint is investigated. In this study the influence of the clearance gap size, bearing friction, crank speed and impact parameters on the response of the system are investigated. Three types of responses are observed: chaotic, transient chaos and periodic. It is shown that chaotic motion is prevalent over a range of parameters which corresponds to high crank speeds and/or low values of bearing friction with relatively ideal impacts. Periodic response is generally observed at low crank speeds and also at low values of the coefficient of restitution. Poincaré maps and statistical profiles of the impact locations and severity are used to characterize the motion and to obtain information regarding possible patterns of wear due to repeated impacts. As expected, chaotic motions lead to quite uniform distributions of impacts, while periodic motions lead to highly localized impact locations. It is also shown that the system response is essentially unpredictable over a wide range of parameters, thus casting doubt on the usefulness of such models for accurate prediction purposes.
Dimension of fractal basin boundaries
Park, B.S.
1988-01-01
In many dynamical systems, multiple attractors coexist for certain parameter ranges. The set of initial conditions that asymptotically approach each attractor is its basin of attraction. These basins can be intertwined on arbitrary small scales. Basin boundary can be either smooth or fractal. Dynamical systems that have fractal basin boundary show final state sensitivity of the initial conditions. A measure of this sensitivity (uncertainty exponent {alpha}) is related to the dimension of the basin boundary d = D - {alpha}, where D is the dimension of the phase space and d is the dimension of the basin boundary. At metamorphosis values of the parameter, there might happen a conversion from smooth to fractal basin boundary (smooth-fractal metamorphosis) or a conversion from fractal to another fractal basin boundary characteristically different from the previous fractal one (fractal-fractal metamorphosis). The dimension changes continuously with the parameter except at the metamorphosis values where the dimension of the basin boundary jumps discontinuously. We chose the Henon map and the forced damped pendulum to investigate this. Scaling of the basin volumes near the metamorphosis values of the parameter is also being studied for the Henon map. Observations are explained analytically by using low dimensional model map.
Fractal mechanisms and heart rate dynamics. Long-range correlations and their breakdown with disease
NASA Technical Reports Server (NTRS)
Peng, C. K.; Havlin, S.; Hausdorff, J. M.; Mietus, J. E.; Stanley, H. E.; Goldberger, A. L.
1995-01-01
Under healthy conditions, the normal cardiac (sinus) interbeat interval fluctuates in a complex manner. Quantitative analysis using techniques adapted from statistical physics reveals the presence of long-range power-law correlations extending over thousands of heartbeats. This scale-invariant (fractal) behavior suggests that the regulatory system generating these fluctuations is operating far from equilibrium. In contrast, it is found that for subjects at high risk of sudden death (e.g., congestive heart failure patients), these long-range correlations break down. Application of fractal scaling analysis and related techniques provides new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as motivating development of novel physiologic models of systems that appear to be heterodynamic rather than homeostatic.
Evaluation of bridge instability caused by dynamic scour based on fractal theory
NASA Astrophysics Data System (ADS)
Lin, Tzu-Kang; Wu, Rih-Teng; Chang, Kuo-Chun; Shian Chang, Yu
2013-07-01
Given their special structural characteristics, bridges are prone to suffer from the effects of many hazards, such as earthquakes, wind, or floods. As most of the recent unexpected damage and destruction of bridges has been caused by hydraulic issues, monitoring the scour depth of bridges has become an important topic. Currently, approaches to scour monitoring mainly focus on either installing sensors on the substructure of a bridge or identifying the physical parameters of a bridge, which commonly face problems of system survival or reliability. To solve those bottlenecks, a novel structural health monitoring (SHM) concept was proposed by utilizing the two dominant parameters of fractal theory, including the fractal dimension and the topothesy, to evaluate the instability condition of a bridge structure rapidly. To demonstrate the performance of this method, a series of experiments has been carried out. The function of the two parameters was first determined using data collected from a single bridge column scour test. As the fractal dimension gradually decreased, following the trend of the scour depth, it was treated as an alternative to the fundamental frequency of a bridge structure in the existing methods. Meanwhile, the potential of a positive correlation between the topothesy and the amplitude of vibration data was also investigated. The excellent sensitivity of the fractal parameters related to the scour depth was then demonstrated in a full-bridge experiment. Moreover, with the combination of these two parameters, a safety index to detect the critical scour condition was proposed. The experimental results have demonstrated that the critical scour condition can be predicted by the proposed safety index. The monitoring system developed greatly advances the field of bridge scour health monitoring and offers an alternative choice to traditional scour monitoring technology.
NASA Astrophysics Data System (ADS)
Das, Krishna Pada; Bairagi, Nandadulal; Sen, Prabir
It is generally, but not always, accepted that alternative food plays a stabilizing role in predator-prey interaction. Parasites, on the other hand, have the ability to change both the qualitative and quantitative dynamics of its host population. In recent times, researchers are showing growing interest in formulating models that integrate both the ecological and epidemiological aspects. The present paper deals with the effect of alternative food on a predator-prey system with disease in the predator population. We show that the system, in the absence of alternative food, exhibits different dynamics viz. stable coexistence, limit cycle oscillations, period-doubling bifurcation and chaos when infection rate is gradually increased. However, when predator consumes alternative food coupled with its focal prey, the system returns to regular oscillatory state from chaotic state through period-halving bifurcations. Our study shows that alternative food may have larger impact on the community structure and may increase population persistence.
NASA Astrophysics Data System (ADS)
Bisnovatyi-Kogan, G. S.; Tsupko, O. Yu.
2015-10-01
> In this paper we review a recently developed approximate method for investigation of dynamics of compressible ellipsoidal figures. Collapse and subsequent behaviour are described by a system of ordinary differential equations for time evolution of semi-axes of a uniformly rotating, three-axis, uniform-density ellipsoid. First, we apply this approach to investigate dynamic stability of non-spherical bodies. We solve the equations that describe, in a simplified way, the Newtonian dynamics of a self-gravitating non-rotating spheroidal body. We find that, after loss of stability, a contraction to a singularity occurs only in a pure spherical collapse, and deviations from spherical symmetry prevent the contraction to the singularity through a stabilizing action of nonlinear non-spherical oscillations. The development of instability leads to the formation of a regularly or chaotically oscillating body, in which dynamical motion prevents the formation of the singularity. We find regions of chaotic and regular pulsations by constructing a Poincaré diagram. A real collapse occurs after damping of the oscillations because of energy losses, shock wave formation or viscosity. We use our approach to investigate approximately the first stages of collapse during the large scale structure formation. The theory of this process started from ideas of Ya. B. Zeldovich, concerning the formation of strongly non-spherical structures during nonlinear stages of the development of gravitational instability, known as `Zeldovich's pancakes'. In this paper the collapse of non-collisional dark matter and the formation of pancake structures are investigated approximately. Violent relaxation, mass and angular momentum losses are taken into account phenomenologically. We estimate an emission of very long gravitational waves during the collapse, and discuss the possibility of gravitational lensing and polarization of the cosmic microwave background by these waves.
Dzaharudin, Fatimah; Suslov, Sergey A; Manasseh, Richard; Ooi, Andrew
2013-11-01
Microbubble clustering may occur when bubbles become bound to targeted surfaces or are grouped by acoustic radiation forces in medical diagnostic applications. The ability to identify the formation of such clusters from the ultrasound echoes may be of practical use. Nonlinear numerical simulations were performed on clusters of microbubbles modeled by the modified Keller-Miksis equations. Encapsulated bubbles were considered to mimic practical applications but the aim of the study was to examine the effects of inter-bubble spacing and bubble size on the dynamical behavior of the cluster and to see if chaotic or bifurcation characteristics could be helpful in diagnostics. It was found that as microbubbles were clustered closer together, their oscillation amplitude for a given applied ultrasound power was reduced, and for inter-bubble spacing smaller than about ten bubble radii nonlinear subharmonics and ultraharmonics were eliminated. For clustered microbubbles, as for isolated microbubbles, an increase in the applied acoustic power caused bifurcations and transition to chaos. The bifurcations preceding chaotic behavior were identified by Floquet analysis and confirmed to be of the period-doubling type. It was found that as the number of microbubbles in a cluster increased, regularization occurred at lower ultrasound power and more windows of order appeared.
ERIC Educational Resources Information Center
Dewdney, A. K.
1991-01-01
Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the Lorenz attractor, logistic maps, the Mandelbrot set, and mini-Mandelbrot sets. Provides appropriate computer algorithms, as well as further sources of information. (JJK)
ERIC Educational Resources Information Center
Osler, Thomas J.
1999-01-01
Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file.…
Fractals analysis of cardiac arrhythmias.
Saeed, Mohammed
2005-09-01
Heart rhythms are generated by complex self-regulating systems governed by the laws of chaos. Consequently, heart rhythms have fractal organization, characterized by self-similar dynamics with long-range order operating over multiple time scales. This allows for the self-organization and adaptability of heart rhythms under stress. Breakdown of this fractal organization into excessive order or uncorrelated randomness leads to a less-adaptable system, characteristic of aging and disease. With the tools of nonlinear dynamics, this fractal breakdown can be quantified with potential applications to diagnostic and prognostic clinical assessment. In this paper, I review the methodologies for fractal analysis of cardiac rhythms and the current literature on their applications in the clinical context. A brief overview of the basic mathematics of fractals is also included. Furthermore, I illustrate the usefulness of these powerful tools to clinical medicine by describing a novel noninvasive technique to monitor drug therapy in atrial fibrillation.
Lábos, E; Pasik, P; Hámori, J; Nogradi, E
1990-01-01
Computer simulations were carried out in an attempt to understand the possible operating modes of synaptic triadic arrangements as described in the dorsal lateral geniculate nucleus of the monkey. Small networks of "chaotic" units (piecewise linear internal maps) were used to investigate their performance as ON-gates for the transmission of spikes. "Chaotic" units have advantages over "logic" units because the former are asynchronous, it is possible to simulate temporal summation, and also to adjust subthreshold time-constants. It was demonstrated that ensembles with single delay lines, representing "closely-packed" triads, were hardly capable of realizing reliable and efficient ON-gate operations. Networks with multiple delayed lines, patterned after triads "at a distance" coupled with "closely-packed" triads, were capable of secure ON-gate functions. Such gates were input dependent, becoming reliable only when high frequency bursts were used as the source of activity. Moreover, the ON-gate could be temporarily interrupted by square wave bursts applied to the inhibitory units, a situation resembling electron microscopic observations of interneuron to interneuron synapses in the LGNd.
NASA Technical Reports Server (NTRS)
Shirts, R. B.; Reinhardt, W. P.
1982-01-01
Substantial short time regularity, even in the chaotic regions of phase space, is found for what is seen as a large class of systems. This regularity manifests itself through the behavior of approximate constants of motion calculated by Pade summation of the Birkhoff-Gustavson normal form expansion; it is attributed to remnants of destroyed invariant tori in phase space. The remnant torus-like manifold structures are used to justify Einstein-Brillouin-Keller semiclassical quantization procedures for obtaining quantum energy levels, even in the absence of complete tori. They also provide a theoretical basis for the calculation of rate constants for intramolecular mode-mode energy transfer. These results are illustrated by means of a thorough analysis of the Henon-Heiles oscillator problem. Possible generality of the analysis is demonstrated by brief consideration of classical dynamics for the Barbanis Hamiltonian, Zeeman effect in hydrogen and recent results of Wolf and Hase (1980) for the H-C-C fragment.
``the Human BRAIN & Fractal quantum mechanics''
NASA Astrophysics Data System (ADS)
Rosary-Oyong, Se, Glory
In mtDNA ever retrieved from Iman Tuassoly, et.al:Multifractal analysis of chaos game representation images of mtDNA''.Enhances the price & valuetales of HE. Prof. Dr-Ing. B.J. HABIBIE's N-219, in J. Bacteriology, Nov 1973 sought:'' 219 exist as separate plasmidDNA species in E.coli & Salmonella panama'' related to ``the brain 2 distinct molecular forms of the (Na,K)-ATPase..'' & ``neuron maintains different concentration of ions(charged atoms'' thorough Rabi & Heisenber Hamiltonian. Further, after ``fractal space time are geometric analogue of relativistic quantum mechanics''[Ord], sought L.Marek Crnjac: ``Chaotic fractals at the root of relativistic quantum physics''& from famous Nottale: ``Scale relativity & fractal space-time:''Application to Quantum Physics , Cosmology & Chaotic systems'',1995. Acknowledgements to HE. Mr. H. TUK SETYOHADI, Jl. Sriwijaya Raya 3, South-Jakarta, INDONESIA.
Evidence of deterministic components in the apparent randomness of GRBs: clues of a chaotic dynamic.
Greco, G; Rosa, R; Beskin, G; Karpov, S; Romano, L; Guarnieri, A; Bartolini, C; Bedogni, R
2011-01-01
Prompt γ-ray emissions from gamma-ray bursts (GRBs) exhibit a vast range of extremely complex temporal structures with a typical variability time-scale significantly short - as fast as milliseconds. This work aims to investigate the apparent randomness of the GRB time profiles making extensive use of nonlinear techniques combining the advanced spectral method of the Singular Spectrum Analysis (SSA) with the classical tools provided by the Chaos Theory. Despite their morphological complexity, we detect evidence of a non stochastic short-term variability during the overall burst duration - seemingly consistent with a chaotic behavior. The phase space portrait of such variability shows the existence of a well-defined strange attractor underlying the erratic prompt emission structures. This scenario can shed new light on the ultra-relativistic processes believed to take place in GRB explosions and usually associated with the birth of a fast-spinning magnetar or accretion of matter onto a newly formed black hole.
Visibility graphlet approach to chaotic time series.
Mutua, Stephen; Gu, Changgui; Yang, Huijie
2016-05-01
Many novel methods have been proposed for mapping time series into complex networks. Although some dynamical behaviors can be effectively captured by existing approaches, the preservation and tracking of the temporal behaviors of a chaotic system remains an open problem. In this work, we extended the visibility graphlet approach to investigate both discrete and continuous chaotic time series. We applied visibility graphlets to capture the reconstructed local states, so that each is treated as a node and tracked downstream to create a temporal chain link. Our empirical findings show that the approach accurately captures the dynamical properties of chaotic systems. Networks constructed from periodic dynamic phases all converge to regular networks and to unique network structures for each model in the chaotic zones. Furthermore, our results show that the characterization of chaotic and non-chaotic zones in the Lorenz system corresponds to the maximal Lyapunov exponent, thus providing a simple and straightforward way to analyze chaotic systems.
Faybishenko, B.
1997-10-01
'Understanding subsurface flow and transport processes is critical for effective assessment, decision-making, and remediation activities for contaminated sites. However, for fluid flow and contaminant transport through fractured vadose zones, traditional hydrogeological approaches are often found to be inadequate. In this project, the authors examine flow and transport through a fractured vadose zone as a deterministic chaotic dynamical process, and develop a model of it in these terms. Initially, they examine separately the geometric model of fractured rock and the flow dynamics model needed to describe chaotic behavior. Ultimately they will put the geometry and flow dynamics together to develop a chaotic-dynamical model of flow and transport in a fractured vadose zone. They investigate water flow and contaminant transport on several scales, ranging from small-scale laboratory experiments in fracture replicas and fractured cores, to field experiments conducted in a single exposed fracture at a basalt outcrop, and finally to a ponded infiltration test using a pond of 7 by 8 m. In the field experiments, the authors measure the time-variation of water flux, moisture content, and hydraulic head at various locations, as well as the total inflow rate to the subsurface. Such variations reflect the changes in the geometry and physics of water flow that display chaotic behavior, which the authors try to reconstruct using the data obtained. In the analysis of experimental data, a chaotic model can be used to predict the long-term bounds on fluid flow and transport behavior, known as the attractor of the system, and to examine the limits of short-term predictability within these bounds. This approach is especially well suited to the need for short-term predictions to support remediation decisions and long-term bounding studies.'
NASA Astrophysics Data System (ADS)
Potapov, A.; Ali, M. K.
2001-04-01
We consider the problem of stabilizing unstable equilibria by discrete controls (the controls take discrete values at discrete moments of time). We prove that discrete control typically creates a chaotic attractor in the vicinity of an equilibrium. Artificial neural networks with reinforcement learning are known to be able to learn such a control scheme. We consider examples of such systems, discuss some details of implementing the reinforcement learning to controlling unstable equilibria, and show that the arising dynamics is characterized by positive Lyapunov exponents, and hence is chaotic. This chaos can be observed both in the controlled system and in the activity patterns of the controller.
ERIC Educational Resources Information Center
Jurgens, Hartmut; And Others
1990-01-01
The production and application of images based on fractal geometry are described. Discussed are fractal language groups, fractal image coding, and fractal dialects. Implications for these applications of geometry to mathematics education are suggested. (CW)
NASA Astrophysics Data System (ADS)
Oleshko, Klaudia; de Jesús Correa López, María; Romero, Alejandro; Ramírez, Victor; Pérez, Olga
2016-04-01
The effectiveness of fractal toolbox to capture the scaling or fractal probability distribution, and simply fractal statistics of main hydrocarbon reservoir attributes, was highlighted by Mandelbrot (1995) and confirmed by several researchers (Zhao et al., 2015). Notwithstanding, after more than twenty years, it's still common the opinion that fractals are not useful for the petroleum engineers and especially for Geoengineering (Corbett, 2012). In spite of this negative background, we have successfully applied the fractal and multifractal techniques to our project entitled "Petroleum Reservoir as a Fractal Reactor" (2013 up to now). The distinguishable feature of Fractal Reservoir is the irregular shapes and rough pore/solid distributions (Siler, 2007), observed across a broad range of scales (from SEM to seismic). At the beginning, we have accomplished the detailed analysis of Nelson and Kibler (2003) Catalog of Porosity and Permeability, created for the core plugs of siliciclastic rocks (around ten thousand data were compared). We enriched this Catalog by more than two thousand data extracted from the last ten years publications on PoroPerm (Corbett, 2012) in carbonates deposits, as well as by our own data from one of the PEMEX, Mexico, oil fields. The strong power law scaling behavior was documented for the major part of these data from the geological deposits of contrasting genesis. Based on these results and taking into account the basic principles and models of the Physics of Fractals, introduced by Per Back and Kan Chen (1989), we have developed new software (Muukíl Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoir models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mesh' cells was designed and calibrated in the studied area. The statistically sound power law relations were established
Washburn, Auriel; Coey, Charles A; Romero, Veronica; Malone, MaryLauren; Richardson, Michael J
2015-11-01
The current study investigated whether the influence of available task constraints on power-law scaling might be moderated by a participant's task intention. Participants performed a simple rhythmic movement task with the intention of controlling either movement period or amplitude, either with or without an experimental stimulus designed to constrain period. In the absence of the stimulus, differences in intention did not produce any changes in power-law scaling. When the stimulus was present, however, a shift toward more random fluctuations occurred in the corresponding task dimension, regardless of participants' intentions. More importantly, participants' intentions interacted with available task constraints to produce an even greater shift toward random variation when the task dimension constrained by the stimulus was also the dimension the participant intended to control. Together, the results suggest that intentions serve to more tightly constrain behavior to existing environmental constraints, evidenced by changes in the fractal scaling of task performance.
Jeong, J; Joung, M K; Kim, S Y
1998-03-01
The goal of this study is to quantify and determine the way in which the emotional response to music is reflected in the electrical activities of the brain. When the power spectrum of sequences of musical notes is inversely proportional to the frequency on a log-log plot, we call it 1/f music. According to previous research, most listeners agree that 1/f music is much more pleasing than white (1/f0) or brown (1/f2) music. Based on these studies, we used nonlinear methods to investigate the chaotic dynamics of electroencephalograms (EEGs) elicited by computer-generated 1/f music, white music, and brown music. In this analysis, we used the correlation dimension and the largest Lyapunov exponent as measures of complexity and chaos. We developed a new method that is strikingly faster and more accurate than other algorithms for calculating the nonlinear invariant measures from limited noisy data. At the right temporal lobe, 1/f music elicited lower values of both the correlation dimension and the largest Lyapunov exponent than white or brown music. We observed that brains which feel more pleased show decreased chaotic electrophysiological behavior. By observing that the nonlinear invariant measures for the 1/f distribution of the rhythm with the melody kept constant are lower than those for the 1/f distribution of melody with the rhythm kept constant, we could conclude that the rhythm variations contribute much more to a pleasing response to music than the melody variations do. These results support the assumption that chaos plays an important role in brain function, especially emotion.
Ravishankar, A.S. Ghosal, A.
1999-01-01
The dynamics of a feedback-controlled rigid robot is most commonly described by a set of nonlinear ordinary differential equations. In this paper, the authors analyze these equations, representing the feedback-controlled motion of two- and three-degrees-of-freedom rigid robots with revolute (R) and prismatic (P) joints in the absence of compliance, friction, and potential energy, for the possibility of chaotic motions. The authors first study the unforced or inertial motions of the robots, and show that when the Gaussian or Riemannian curvature of the configuration space of a robot is negative, the robot equations can exhibit chaos. If the curvature is zero or positive, then the robot equations cannot exhibit chaos. The authors show that among the two-degrees-of-freedom robots, the PP and the PR robot have zero Gaussian curvature while the RP and RR robots have negative Gaussian curvatures. For the three-degrees-of-freedom robots, they analyze the two well-known RRP and RRR configurations of the Stanford arm and the PUMA manipulator, respectively, and derive the conditions for negative curvature and possible chaotic motions. The criteria of negative curvature cannot be used for the forced or feedback-controlled motions. For the forced motion, the authors resort to the well-known numerical techniques and compute chaos maps, Poincare maps, and bifurcation diagrams. Numerical results are presented for the two-degrees-of-freedom RP and RR robots, and the authors show that these robot equations can exhibit chaos for low controller gains and for large underestimated models. From the bifurcation diagrams, the route to chaos appears to be through period doubling.
SU-E-J-261: Statistical Analysis and Chaotic Dynamics of Respiratory Signal of Patients in BodyFix
Michalski, D; Huq, M; Bednarz, G; Lalonde, R; Yang, Y; Heron, D
2014-06-01
Purpose: To quantify respiratory signal of patients in BodyFix undergoing 4DCT scan with and without immobilization cover. Methods: 20 pairs of respiratory tracks recorded with RPM system during 4DCT scan were analyzed. Descriptive statistic was applied to selected parameters of exhale-inhale decomposition. Standardized signals were used with the delay method to build orbits in embedded space. Nonlinear behavior was tested with surrogate data. Sample entropy SE, Lempel-Ziv complexity LZC and the largest Lyapunov exponents LLE were compared. Results: Statistical tests show difference between scans for inspiration time and its variability, which is bigger for scans without cover. The same is for variability of the end of exhalation and inhalation. Other parameters fail to show the difference. For both scans respiratory signals show determinism and nonlinear stationarity. Statistical test on surrogate data reveals their nonlinearity. LLEs show signals chaotic nature and its correlation with breathing period and its embedding delay time. SE, LZC and LLE measure respiratory signal complexity. Nonlinear characteristics do not differ between scans. Conclusion: Contrary to expectation cover applied to patients in BodyFix appears to have limited effect on signal parameters. Analysis based on trajectories of delay vectors shows respiratory system nonlinear character and its sensitive dependence on initial conditions. Reproducibility of respiratory signal can be evaluated with measures of signal complexity and its predictability window. Longer respiratory period is conducive for signal reproducibility as shown by these gauges. Statistical independence of the exhale and inhale times is also supported by the magnitude of LLE. The nonlinear parameters seem more appropriate to gauge respiratory signal complexity since its deterministic chaotic nature. It contrasts with measures based on harmonic analysis that are blind for nonlinear features. Dynamics of breathing, so crucial for
The Dynamical Foundation of Fractal Stream Chemistry: The Origin of Extremely Long Retention Times
NASA Astrophysics Data System (ADS)
Scher, H.; Metzler, R.; Margolin, G.; Klafter, J.; Berkowitz, B.
2001-12-01
We present a physical model to explain the behavior of long-term, time series measurements of chloride, a natural passive tracer, in rainfall and runoff in catchments [J. W. Kirchner et al., Nature 403, 524 (2000)]. A spectral analysis of the data shows that the chloride concentrations in rainfall have a white noise spectrum, while in streamflow, the spectrum exhibits a fractal 1/f scaling. The empirically derived distribution of tracer travel times h(τ ) follows a power-law, indicating low-level contaminant delivery to streams for a very long time. Our transport model is based on a continuous time random walk (CTRW) with an event time distribution governed by ψ (t) ~ Aβ t-1-β . The CTRW using this power-law ψ (t) (with 0 < β < 1)is interchangeable with the time-fractional advection-dispersion equation (FADE) and has accounted for the universal phenomenon of anomalous transport in a broad range of disordered and complex systems. Anomalous transport has been generally observed in time-dependent measurements of electron hopping/multiple-trapping in amorphous semiconductors, particle migration in fracture networks, and contaminant transport in heterogeneous porous media. In the current application, the events can be realized as travel times on the segments of a catchment network. The travel time distribution is the first passage time distribution F(τ ; l) at l from a pulse input (at τ =0) at the origin. We show that the empirical h(τ ) is the catchment areal composite of F(τ ; l) and the fractal 1/f spectral response found in many catchments is an example of the larger class of transport phenomena cited above. The physical basis of this ψ (t), which determines F(τ ; l), is the origin of the extremely long chemical retention times in catchments.
The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times
NASA Astrophysics Data System (ADS)
Scher, Harvey; Margolin, Gennady; Metzler, Ralf; Klafter, Joseph; Berkowitz, Brian
2002-03-01
We present a physical model to explain the behavior of long-term, time series measurements of chloride, a natural passive tracer, in rainfall and runoff in catchments [Kirchner et al., Nature, 403(524), 2000]. A spectral analysis of the data shows the chloride concentrations in rainfall to have a white noise spectrum, while in streamflow, the spectrum exhibits a fractal 1/f scaling. The empirically derived distribution of tracer travel times h(t) follows a power-law, indicating low-level contaminant delivery to streams for a very long time. Our transport model is based on a continuous time random walk (CTRW) with an event time distribution governed by ψ(t) ~ Aβt-1-β. The CTRW using this power-law ψ(t) (with 0 < β < 1) is interchangeable with the time-fractional advection-dispersion equation (FADE) and has accounted for the universal phenomenon of anomalous transport in a broad range of disordered and complex systems. In the current application, the events can be realized as transit times on portions of the catchment network. The travel time distribution is the first passage time distribution F(t;l) at a distance l from a pulse input (at t = 0) at the origin. We show that the empirical h(t) is the catchment areal composite of F(t;l) and that the fractal 1/f spectral response found in many catchments is an example of the larger class of transport phenomena cited above. The physical basis of ψ(t), which determines F(t;l), is the origin of the extremely long chemical retention times in catchments.
Dimension of chaotic attractors
Farmer, J.D.; Ott, E.; Yorke, J.A.
1982-09-01
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on probabilistic properties (that is, they depend on the frequency with which a typical trajectory visits different regions of the attractor). Both our example and the previous work that we review support the conclusion that all of the probabilistic dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
2012-01-01
Background The invasion-metastasis cascade of cancer involves a process of parallel progression. A biological interface (module) in which cells is linked with ECM (extracellular matrix) by CAMs (cell adhesion molecules) has been proposed as a tool for tracing cancer spatiotemporal dynamics. Methods A mathematical model was established to simulate cancer cell migration. Human uterine leiomyoma specimens, in vitro cell migration assay, quantitative real-time PCR, western blotting, dynamic viscosity, and an in vivo C57BL6 mouse model were used to verify the predictive findings of our model. Results The return to origin probability (RTOP) and its related CAM expression ratio in tumors, so-called "tumor self-seeding", gradually decreased with increased tumor size, and approached the 3D Pólya random walk constant (0.340537) in a periodic structure. The biphasic pattern of cancer cell migration revealed that cancer cells initially grew together and subsequently began spreading. A higher viscosity of fillers applied to the cancer surface was associated with a significantly greater inhibitory effect on cancer migration, in accordance with the Stokes-Einstein equation. Conclusion The positional probability and cell-CAM-ECM interface (module) in the fractal framework helped us decipher cancer spatiotemporal dynamics; in addition we modeled the methods of cancer control by manipulating the microenvironment plasticity or inhibiting the CAM expression to the Pólya random walk, Pólya constant. PMID:22889191
Evidence of deterministic components in the apparent randomness of GRBs: clues of a chaotic dynamic.
Greco, G; Rosa, R; Beskin, G; Karpov, S; Romano, L; Guarnieri, A; Bartolini, C; Bedogni, R
2011-01-01
Prompt γ-ray emissions from gamma-ray bursts (GRBs) exhibit a vast range of extremely complex temporal structures with a typical variability time-scale significantly short - as fast as milliseconds. This work aims to investigate the apparent randomness of the GRB time profiles making extensive use of nonlinear techniques combining the advanced spectral method of the Singular Spectrum Analysis (SSA) with the classical tools provided by the Chaos Theory. Despite their morphological complexity, we detect evidence of a non stochastic short-term variability during the overall burst duration - seemingly consistent with a chaotic behavior. The phase space portrait of such variability shows the existence of a well-defined strange attractor underlying the erratic prompt emission structures. This scenario can shed new light on the ultra-relativistic processes believed to take place in GRB explosions and usually associated with the birth of a fast-spinning magnetar or accretion of matter onto a newly formed black hole. PMID:22355609
Evidence of Deterministic Components in the Apparent Randomness of GRBs: Clues of a Chaotic Dynamic
Greco, G.; Rosa, R.; Beskin, G.; Karpov, S.; Romano, L.; Guarnieri, A.; Bartolini, C.; Bedogni, R.
2011-01-01
Prompt γ-ray emissions from gamma-ray bursts (GRBs) exhibit a vast range of extremely complex temporal structures with a typical variability time-scale significantly short – as fast as milliseconds. This work aims to investigate the apparent randomness of the GRB time profiles making extensive use of nonlinear techniques combining the advanced spectral method of the Singular Spectrum Analysis (SSA) with the classical tools provided by the Chaos Theory. Despite their morphological complexity, we detect evidence of a non stochastic short-term variability during the overall burst duration – seemingly consistent with a chaotic behavior. The phase space portrait of such variability shows the existence of a well-defined strange attractor underlying the erratic prompt emission structures. This scenario can shed new light on the ultra-relativistic processes believed to take place in GRB explosions and usually associated with the birth of a fast-spinning magnetar or accretion of matter onto a newly formed black hole. PMID:22355609
The transience of virtual fractals.
Taylor, R P
2012-01-01
Artists have a long and fruitful tradition of exploiting electronic media to convert static images into dynamic images that evolve with time. Fractal patterns serve as an example: computers allow the observer to zoom in on virtual images and so experience the endless repetition of patterns in a matter that cannot be matched using static images. This year's featured cover artist, Susan Lowedermilk, instead plans to employ persistence of human vision to bring virtual fractals to life. This will be done by incorporating her prints of fractal patterns into zoetropes and phenakistoscopes.
Exterior dimension of fat fractals
NASA Technical Reports Server (NTRS)
Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.
1985-01-01
Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.
Christov, Ivan C.; Lueptow, Richard M.; Ottino, Julio M.; Sturman, Rob
2014-05-22
We study three-dimensional (3D) chaotic dynamics through an analysis of transport in a granular flow in a half-full spherical tumbler rotated sequentially about two orthogonal axes (a bi-axial “blinking” tumbler). The flow is essentially quasi-two-dimensional in any vertical slice of the sphere during rotation about a single axis, and we provide an explicit exact solution to the model in this case. Hence, the cross-sectional flow can be represented by a twist map, allowing us to express the 3D flow as a linked twist map (LTM). We prove that if the rates of rotation about each axis are equal, then (in the absence of stochasticity) particle trajectories are restricted to two-dimensional (2D) surfaces consisting of a portion of a hemispherical shell closed by a “cap''; if the rotation rates are unequal, then particles can leave the surface they start on and traverse a volume of the tumbler. The period-one structures of the governing LTM are examined in detail: analytical expressions are provided for the location of period-one curves, their extent into the bulk of the granular material, and their dependence on the protocol parameters (rates and durations of rotations). Exploiting the restriction of trajectories to 2D surfaces in the case of equal rotation rates about the axes, a method is proposed for identifying and constructing 3D Kolmogorov--Arnold--Moser (KAM) tubes around the normally elliptic period-one curves. The invariant manifold structure arising from the normally hyperbolic period-one curves is also examined. When the motion is restricted to 2D surfaces, the structure of manifolds of the hyperbolic points in the bulk differs from that corresponding to hyperbolic points in the flowing layer. Each is reminiscent of a template provided by a non-integrable perturbation to a Hamiltonian system, though the governing LTM is not. This highlights the novel 3D chaotic behaviors observed in this model dynamical system.
Christov, Ivan C.; Lueptow, Richard M.; Ottino, Julio M.; Sturman, Rob
2014-05-22
We study three-dimensional (3D) chaotic dynamics through an analysis of transport in a granular flow in a half-full spherical tumbler rotated sequentially about two orthogonal axes (a bi-axial “blinking” tumbler). The flow is essentially quasi-two-dimensional in any vertical slice of the sphere during rotation about a single axis, and we provide an explicit exact solution to the model in this case. Hence, the cross-sectional flow can be represented by a twist map, allowing us to express the 3D flow as a linked twist map (LTM). We prove that if the rates of rotation about each axis are equal, then (inmore » the absence of stochasticity) particle trajectories are restricted to two-dimensional (2D) surfaces consisting of a portion of a hemispherical shell closed by a “cap''; if the rotation rates are unequal, then particles can leave the surface they start on and traverse a volume of the tumbler. The period-one structures of the governing LTM are examined in detail: analytical expressions are provided for the location of period-one curves, their extent into the bulk of the granular material, and their dependence on the protocol parameters (rates and durations of rotations). Exploiting the restriction of trajectories to 2D surfaces in the case of equal rotation rates about the axes, a method is proposed for identifying and constructing 3D Kolmogorov--Arnold--Moser (KAM) tubes around the normally elliptic period-one curves. The invariant manifold structure arising from the normally hyperbolic period-one curves is also examined. When the motion is restricted to 2D surfaces, the structure of manifolds of the hyperbolic points in the bulk differs from that corresponding to hyperbolic points in the flowing layer. Each is reminiscent of a template provided by a non-integrable perturbation to a Hamiltonian system, though the governing LTM is not. This highlights the novel 3D chaotic behaviors observed in this model dynamical system.« less
Fractal dynamics of human gait: a reassessment of the 1996 data of Hausdorff et al.
Delignières, Didier; Torre, Kjerstin
2009-04-01
We propose in this paper a reassessment of the original data of Hausdorff et al. (Hausdorff JM, Purdon PL, Peng C-K, Ladin Z, Wei JY, Goldberger AR. J Appl Physiol 80: 1448-1457, 1996). We confirm, using autoregressive fractionally integrated moving average modeling, the presence of genuine fractal correlations in stride interval series in self-paced conditions. In contrast with the conclusions of the authors, we show that correlations did not disappear in metronomic conditions. The series of stride intervals presented antipersistent correlations, and 1/f fluctuations were evidenced in the asynchronies to the metronome. We show that the super central pattern generator model (West B, Scafetta N. Phys Rev E Stat Nonlin Soft Matter Phys 67: 051917, 2003) allows accounting for the experimentally observed correlations in both self-paced and metronomic conditions, by the simple setting of the coupling strength parameter. We conclude that 1/f fluctuations in gait are not overridden by supraspinal influences when walking is paced by a metronome. The source of 1/f noise is still at work in this condition, but expressed differently under the influence of a continuous coupling process. PMID:19228991
Silk, Daniel; Kirk, Paul D W; Barnes, Chris P; Toni, Tina; Rose, Anna; Moon, Simon; Dallman, Margaret J; Stumpf, Michael P H
2011-01-01
Chaos and oscillations continue to capture the interest of both the scientific and public domains. Yet despite the importance of these qualitative features, most attempts at constructing mathematical models of such phenomena have taken an indirect, quantitative approach, for example, by fitting models to a finite number of data points. Here we develop a qualitative inference framework that allows us to both reverse-engineer and design systems exhibiting these and other dynamical behaviours by directly specifying the desired characteristics of the underlying dynamical attractor. This change in perspective from quantitative to qualitative dynamics, provides fundamental and new insights into the properties of dynamical systems. PMID:21971504
Anomalous Diffusion in Fractal Globules
NASA Astrophysics Data System (ADS)
Tamm, M. V.; Nazarov, L. I.; Gavrilov, A. A.; Chertovich, A. V.
2015-05-01
The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X2(t )⟩˜tαF with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing.
Anomalous diffusion in fractal globules.
Tamm, M V; Nazarov, L I; Gavrilov, A A; Chertovich, A V
2015-05-01
The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X(2)(t)⟩∼t(αF) with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing. PMID:25978267
Anomalous diffusion in fractal globules.
Tamm, M V; Nazarov, L I; Gavrilov, A A; Chertovich, A V
2015-05-01
The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X(2)(t)⟩∼t(αF) with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing.
NASA Technical Reports Server (NTRS)
Huikuri, H. V.; Makikallio, T. H.; Peng, C. K.; Goldberger, A. L.; Hintze, U.; Moller, M.
2000-01-01
BACKGROUND: Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS: Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS: Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.
Semiclassical theory of chaotic conductors.
Heusler, Stefan; Müller, Sebastian; Braun, Petr; Haake, Fritz
2006-02-17
We calculate the Landauer conductance through chaotic ballistic devices in the semiclassical limit, to all orders in the inverse number of scattering channels without and with a magnetic field. Families of pairs of entrance-to-exit trajectories contribute, similarly to the pairs of periodic orbits making up the small-time expansion of the spectral form factor of chaotic dynamics. As a clue to the exact result we find that close self-encounters slightly hinder the escape of trajectories into leads. Our result explains why the energy-averaged conductance of individual chaotic cavities, with disorder or "clean," agrees with predictions of random-matrix theory. PMID:16606030
Characterizing chaotic melodies in automatic music composition.
Coca, Andrés E; Tost, Gerard O; Zhao, Liang
2010-09-01
In this paper, we initially present an algorithm for automatic composition of melodies using chaotic dynamical systems. Afterward, we characterize chaotic music in a comprehensive way as comprising three perspectives: musical discrimination, dynamical influence on musical features, and musical perception. With respect to the first perspective, the coherence between generated chaotic melodies (continuous as well as discrete chaotic melodies) and a set of classical reference melodies is characterized by statistical descriptors and melodic measures. The significant differences among the three types of melodies are determined by discriminant analysis. Regarding the second perspective, the influence of dynamical features of chaotic attractors, e.g., Lyapunov exponent, Hurst coefficient, and correlation dimension, on melodic features is determined by canonical correlation analysis. The last perspective is related to perception of originality, complexity, and degree of melodiousness (Euler's gradus suavitatis) of chaotic and classical melodies by nonparametric statistical tests.
Characterizing chaotic melodies in automatic music composition
NASA Astrophysics Data System (ADS)
Coca, Andrés E.; Tost, Gerard O.; Zhao, Liang
2010-09-01
In this paper, we initially present an algorithm for automatic composition of melodies using chaotic dynamical systems. Afterward, we characterize chaotic music in a comprehensive way as comprising three perspectives: musical discrimination, dynamical influence on musical features, and musical perception. With respect to the first perspective, the coherence between generated chaotic melodies (continuous as well as discrete chaotic melodies) and a set of classical reference melodies is characterized by statistical descriptors and melodic measures. The significant differences among the three types of melodies are determined by discriminant analysis. Regarding the second perspective, the influence of dynamical features of chaotic attractors, e.g., Lyapunov exponent, Hurst coefficient, and correlation dimension, on melodic features is determined by canonical correlation analysis. The last perspective is related to perception of originality, complexity, and degree of melodiousness (Euler's gradus suavitatis) of chaotic and classical melodies by nonparametric statistical tests.
NASA Astrophysics Data System (ADS)
Bhaduri, Susmita; Ghosh, Dipak
2016-08-01
There are numerous existing works on investigating the dynamics of particle production process in ultrarelativistic nuclear collision. In the past, fluctuation of spatial pattern has been analyzed in terms of the scaling behavior of voids. But analysis of the scaling behavior of the void in fractal scenario has not been explored yet. In this work, we have analyzed the fractality of void probability distribution with a completely different and rigorous method called visibility graph analysis, analyzing the void-data produced out of fluctuation of pions in 32S-AgBr interaction at 200 GeV in pseudo-rapidity (η) and azimuthal angle (ϕ) space. The power of scale-freeness of visibility graph denoted by PSVG is a measure of fractality, which can be used as a quantitative parameter for the assessment of the state of chaotic system. As the behavior of particle production process depends on the target excitation, we can dwell down the void probability distribution in the event-wise fluctuation resulted out of the high energy interaction for different degree of target excitation, with respect to the fractal scenario and analyze the scaling behavior of the voids. From the analysis of the PSVG parameter, we have observed that scaling behavior of void probability distribution in multipion production changes with increasing target excitation. Since visibility graph method is a classic method of complex network analysis, has been applied over fractional Brownian motion (fBm) and fractional Gaussian noises (fGn) to measure the fractality and long-range dependence of a time series successfully, we can quantitatively confirm that fractal behavior of the void probability distribution in particle production process depends on the target excitation.
NASA Astrophysics Data System (ADS)
Li, Chien-Ming; Du, Yi-Chun; Wu, Jian-Xing; Lin, Chia-Hung; Ho, Yueh-Ren; Chen, Tainsong
2013-08-01
Lower-extremity peripheral arterial disease (PAD) is caused by narrowing or occlusion of vessels in patients like type 2 diabetes mellitus, the elderly and smokers. Patients with PAD are mostly asymptomatic; typical early symptoms of this limb-threatening disorder are intermittent claudication and leg pain, suggesting the necessity for accurate diagnosis by invasive angiography and ankle-brachial pressure index. This index acts as a gold standard reference for PAD diagnosis and categorizes its severity into normal, low-grade and high-grade, with respective cut-off points of ≥0.9, 0.9-0.5 and <0.5. PAD can be assessed using photoplethysmography as a diagnostic screening tool, displaying changes in pulse transit time and shape, and dissimilarities of these changes between lower limbs. The present report proposed photoplethysmogram with fractional-order chaotic system to assess PAD in 14 diabetics and 11 healthy adults, with analysis of dynamic errors based on various butterfly motion patterns, and color relational analysis as classifier for pattern recognition. The results show that the classification of PAD severity among these testees was achieved with high accuracy and efficiency. This noninvasive methodology potentially provides timing and accessible feedback to patients with asymptomatic PAD and their physicians for further invasive diagnosis or strict management of risk factors to intervene in the disease progression.
NASA Technical Reports Server (NTRS)
Makikallio, T. H.; Hoiber, S.; Kober, L.; Torp-Pedersen, C.; Peng, C. K.; Goldberger, A. L.; Huikuri, H. V.
1999-01-01
A number of new methods have been recently developed to quantify complex heart rate (HR) dynamics based on nonlinear and fractal analysis, but their value in risk stratification has not been evaluated. This study was designed to determine whether selected new dynamic analysis methods of HR variability predict mortality in patients with depressed left ventricular (LV) function after acute myocardial infarction (AMI). Traditional time- and frequency-domain HR variability indexes along with short-term fractal-like correlation properties of RR intervals (exponent alpha) and power-law scaling (exponent beta) were studied in 159 patients with depressed LV function (ejection fraction <35%) after an AMI. By the end of 4-year follow-up, 72 patients (45%) had died and 87 (55%) were still alive. Short-term scaling exponent alpha (1.07 +/- 0.26 vs 0.90 +/- 0.26, p <0.001) and power-law slope beta (-1.35 +/- 0.23 vs -1.44 +/- 0.25, p <0.05) differed between survivors and those who died, but none of the traditional HR variability measures differed between these groups. Among all analyzed variables, reduced scaling exponent alpha (<0.85) was the best univariable predictor of mortality (relative risk 3.17, 95% confidence interval 1.96 to 5.15, p <0.0001), with positive and negative predictive accuracies of 65% and 86%, respectively. In the multivariable Cox proportional hazards analysis, mortality was independently predicted by the reduced exponent alpha (p <0.001) after adjustment for several clinical variables and LV function. A short-term fractal-like scaling exponent was the most powerful HR variability index in predicting mortality in patients with depressed LV function. Reduction in fractal correlation properties implies more random short-term HR dynamics in patients with increased risk of death after AMI.
NASA Astrophysics Data System (ADS)
Kwon, T. H.; Hopkins, A. E.; O'donnell, S. E.
1996-07-01
The dynamic scaling behavior of a growing self-affine fractal interface is examined in a simple paper-towel-wetting experiment. A sheet of plain white paper towel is wetted with red food dye solution, and the evolution of the interface is photographed with a 35-mm camera as a function of time. Each snapshot is scanned and digitized to obtain the interface height h(x,t) as a function of time and position. From these the interface width w(L,t) is determined as a function of time t and system size L. It is found that the interface width scales with system size L as w(L,t)~Lα with α=0.67+/-0.04 and scales with time as w(L,t)~tβ with β=0.24+/-0.02. It is also found that average height of the interface scales with time as
NASA Astrophysics Data System (ADS)
Zhang, Xu
This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li-Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.
Evolving random fractal Cantor superlattices for the infrared using a genetic algorithm.
Bossard, Jeremy A; Lin, Lan; Werner, Douglas H
2016-01-01
Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as 'chaotic', but we propose that apparent 'chaotic' natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too 'perfect' to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the 'chaotic' (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and 'chaotic' superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime.
Evolving random fractal Cantor superlattices for the infrared using a genetic algorithm.
Bossard, Jeremy A; Lin, Lan; Werner, Douglas H
2016-01-01
Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as 'chaotic', but we propose that apparent 'chaotic' natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too 'perfect' to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the 'chaotic' (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and 'chaotic' superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime. PMID:26763335
NASA Astrophysics Data System (ADS)
Selvam, A. M.
2016-09-01
Dynamical systems in nature exhibit self-similar fractal space-time fluctuations on all scales indicating long-range correlations and, therefore, the statistical normal distribution with implicit assumption of independence, fixed mean and standard deviation cannot be used for description and quantification of fractal data sets. The author has developed a general systems theory based on classical statistical physics for fractal fluctuations which predicts the following. (1) The fractal fluctuations signify an underlying eddy continuum, the larger eddies being the integrated mean of enclosed smaller-scale fluctuations. (2) The probability distribution of eddy amplitudes and the variance (square of eddy amplitude) spectrum of fractal fluctuations follow the universal Boltzmann inverse power law expressed as a function of the golden mean. (3) Fractal fluctuations are signatures of quantum-like chaos since the additive amplitudes of eddies when squared represent probability densities analogous to the sub-atomic dynamics of quantum systems such as the photon or electron. (4) The model predicted distribution is very close to statistical normal distribution for moderate events within two standard deviations from the mean but exhibits a fat long tail that are associated with hazardous extreme events. Continuous periodogram power spectral analyses of available GHCN annual total rainfall time series for the period 1900-2008 for Indian and USA stations show that the power spectra and the corresponding probability distributions follow model predicted universal inverse power law form signifying an eddy continuum structure underlying the observed inter-annual variability of rainfall. On a global scale, man-made greenhouse gas related atmospheric warming would result in intensification of natural climate variability, seen immediately in high frequency fluctuations such as QBO and ENSO and even shorter timescales. Model concepts and results of analyses are discussed with reference
Fiamma, Marie-Noëlle; Straus, Christian; Thibault, Sylvain; Wysocki, Marc; Baconnier, Pierre; Similowski, Thomas
2007-05-01
In humans, lung ventilation exhibits breath-to-breath variability and dynamics that are nonlinear, complex, sensitive to initial conditions, unpredictable in the long-term, and chaotic. Hypercapnia, as produced by the inhalation of a CO(2)-enriched gas mixture, stimulates ventilation. Hypocapnia, as produced by mechanical hyperventilation, depresses ventilation in animals and in humans during sleep, but it does not induce apnea in awake humans. This emphasizes the suprapontine influences on ventilatory control. How cortical and subcortical commands interfere thus depend on the prevailing CO(2) levels. However, CO(2) also influences the variability and complexity of ventilation. This study was designed to describe how this occurs and to test the hypothesis that CO(2) chemoreceptors are important determinants of ventilatory dynamics. Spontaneous ventilatory flow was recorded in eight healthy subjects. Breath-by-breath variability was studied through the coefficient of variation of several ventilatory variables. Chaos was assessed with the noise titration method (noise limit) and characterized with numerical indexes [largest Lyapunov exponent (LLE), sensitivity to initial conditions; Kolmogorov-Sinai entropy (KSE), unpredictability; and correlation dimension (CD), irregularity]. In all subjects, under all conditions, a positive noise limit confirmed chaos. Hypercapnia reduced breathing variability, increased LLE (P = 0.0338 vs. normocapnia; P = 0.0018 vs. hypocapnia), increased KSE, and slightly reduced CD. Hypocapnia increased variability, decreased LLE and KSE, and reduced CD. These results suggest that chemoreceptors exert a strong influence on ventilatory variability and complexity. However, complexity persists in the quasi-absence of automatic drive. Ventilatory variability and complexity could be determined by the interaction between the respiratory central pattern generator and suprapontine structures. PMID:17218438
Role of inertial forces on the chaotic dynamics of flexible rotating bodies
NASA Astrophysics Data System (ADS)
Calvo, F.
2013-02-01
The nonlinear dynamics of isolated flexible but rotating many-body atomic systems is theoretically investigated, following the dependence on initial conditions through Lyapunov exponents. The tangent-space equations of motion that rule the time evolution of such small perturbations are rewritten in the rotating reference frame, and the various contributions of the centrifugal, Coriolis, and Euler forces are determined. Evaluating the largest Lyapunov in the rotating frame under various approximations, we show on the example of Lennard-Jones clusters that the dynamics in phase space is qualitatively at variance with the effective dynamics on the centrifugal energy surface. Coupling terms between positions and momenta in phase space, especially arising from the Coriolis force, are essential to recover the measure of chaos in the fixed reference frame.
Manos, Thanos; Robnik, Marko
2013-06-01
We study the kicked rotator in the classically fully chaotic regime using Izrailev's N-dimensional model for various N≤4000, which in the limit N→∞ tends to the quantized kicked rotator. We do treat not only the case K=5, as studied previously, but also many different values of the classical kick parameter 5≤K≤35 and many different values of the quantum parameter kε[5,60]. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable Λ=l(∞)/N to the case of anomalous diffusion in the classical phase space by deriving the localization length l(∞) for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents β(BR). (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by β(loc) in the interval [0,1]. The level repulsion parameters β(BR) and β(loc) are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between β(loc) and the relative localization length Λ, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).
Information encoder/decoder using chaotic systems
Miller, Samuel Lee; Miller, William Michael; McWhorter, Paul Jackson
1997-01-01
The present invention discloses a chaotic system-based information encoder and decoder that operates according to a relationship defining a chaotic system. Encoder input signals modify the dynamics of the chaotic system comprising the encoder. The modifications result in chaotic, encoder output signals that contain the encoder input signals encoded within them. The encoder output signals are then capable of secure transmissions using conventional transmission techniques. A decoder receives the encoder output signals (i.e., decoder input signals) and inverts the dynamics of the encoding system to directly reconstruct the original encoder input signals.
Information encoder/decoder using chaotic systems
Miller, S.L.; Miller, W.M.; McWhorter, P.J.
1997-10-21
The present invention discloses a chaotic system-based information encoder and decoder that operates according to a relationship defining a chaotic system. Encoder input signals modify the dynamics of the chaotic system comprising the encoder. The modifications result in chaotic, encoder output signals that contain the encoder input signals encoded within them. The encoder output signals are then capable of secure transmissions using conventional transmission techniques. A decoder receives the encoder output signals (i.e., decoder input signals) and inverts the dynamics of the encoding system to directly reconstruct the original encoder input signals. 32 figs.
The chaotic dynamics of comets and the problems of the Oort cloud
NASA Technical Reports Server (NTRS)
Sagdeev, Roald Z.; Zaslavskiy, G. M.
1991-01-01
The dynamic properties of comets entering the planetary zone from the Oort cloud are discussed. Even a very slight influence of the large planets can trigger stochastic cometary dynamics. Multiple interactions of comets with the large planets produce diffusion of the parameters of cometary orbits and a mean increase in the semi-major axis of comets. Comets are lifted towards the Oort cloud, where collisions with stars begin to play a substantial role. The transport of comets differs greatly from the customary law of diffusion and noticeably alter cometary distribution.
NASA Astrophysics Data System (ADS)
Papantonopoulos, E.; Uematsu, T.; Yanagida, T.
1987-01-01
We present a chaotic inflationary model, in which nonlinear interactions of dilaton and axion fields in the context of the superconformal theory can dynamically give rise to initial conditions for the inflation of the universe and a flat potential that can produce enough inflation. Our model is free from dangerous thermal effects and large energy density fluctuations. On leave from Physics Department, College of General Education, Tohoku University, Sendai 980, Japan
Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems.
Custódio, M S; Manchein, C; Beims, M W
2012-06-01
The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.
Fractal identification of supercell storms
NASA Astrophysics Data System (ADS)
Féral, Laurent; Sauvageot, Henri
2002-07-01
The most intense and violent form of convective storm is the supercell storm, usually associated with heavy rain, hail, and destructive gusty winds, downbursts, and tornadoes. Identifying a storm cell as a supercell storm is not easy. What is shown here, from radar data, is that when an ordinary, or multicell storm evolves towards the supercellular organization, its fractal dimension is modified. Whereas the fractal dimension of the ordinary convective storms, including multicell thunderstorms, is observed around 1.35, in agreement with previous results, the fractal dimension of supercell storms is found close to 1.07. This low value is due to the unicellular character of supercells. The present paper suggests that the fractal dimension is a parameter that should be considered to analyse the dynamical organization of a convective field and to detect and identify the supercell storms, either isolated or among a population of convective storms.
Regular and Chaotic Quantum Dynamics of Two-Level Atoms in a Selfconsistent Radiation Field
NASA Technical Reports Server (NTRS)
Konkov, L. E.; Prants, S. V.
1996-01-01
Dynamics of two-level atoms interacting with their own radiation field in a single-mode high-quality resonator is considered. The dynamical system consists of two second-order differential equations, one for the atomic SU(2) dynamical-group parameter and another for the field strength. With the help of the maximal Lyapunov exponent for this set, we numerically investigate transitions from regularity to deterministic quantum chaos in such a simple model. Increasing the collective coupling constant b is identical with 8(pi)N(sub 0)(d(exp 2))/hw, we observed for initially unexcited atoms a usual sharp transition to chaos at b(sub c) approx. equal to 1. If we take the dimensionless individual Rabi frequency a = Omega/2w as a control parameter, then a sequence of order-to-chaos transitions has been observed starting with the critical value a(sub c) approx. equal to 0.25 at the same initial conditions.
Chaotic synchronization system and electrocardiogram
NASA Astrophysics Data System (ADS)
Pei, Liuqing; Dai, Xinlai; Li, Baodong
1997-01-01
A mathematical model of chaotic synchronization of the heart-blood flow coupling dynamics is proposed, which is based on a seven dimension nonlinear dynamical system constructed by three subsystems of the sinoatrial node natural pacemaker, the cardiac relaxation oscillator and the dynamics of blood-fluid in heart chambers. The existence and robustness of the self-chaotic synchronization of the system are demonstrated by both methods of theoretical analysis and numerical simulation. The spectrum of Lyapunov exponent, the Lyapunov dimension and the Kolmogorov entropy are estimated when the system was undergoing the state of self-chaotic synchronization evolution. The time waveform of the dynamical variable, which represents the membrane potential of the cardiac integrative cell, shows a shape which is similar to that of the normal electrocardiogram (ECG) of human, thus implies that the model possesses physiological significance functionally.
TS fuzzy realization of chaotic Lü system
NASA Astrophysics Data System (ADS)
Li, Dequan
2006-07-01
The Lü attractor is a new chaotic attractor, which connects the Lorenz attractor and the Chen attractor and represents the transition from one to the other. The Letter presents a hybrid TS fuzzy modeling approach for the newly coined chaotic Lü system. Then the abundant and fundamental dynamical behaviors of the chaotic Lü system are completely and comprehensive investigated based on this novel hybrid TS fuzzy model.
NASA Astrophysics Data System (ADS)
Gu, Huaguang
2013-06-01
The transition from chaotic bursting to chaotic spiking has been simulated and analyzed in theoretical neuronal models. In the present study, we report experimental observations in a neural pacemaker of a transition from chaotic bursting to chaotic spiking within a bifurcation scenario from period-1 bursting to period-1 spiking. This was induced by adjusting extracellular calcium or potassium concentrations. The bifurcation scenario began from period-doubling bifurcations or period-adding sequences of bursting pattern. This chaotic bursting is characterized by alternations between multiple continuous spikes and a long duration of quiescence, whereas chaotic spiking is comprised of fast, continuous spikes without periods of quiescence. Chaotic bursting changed to chaotic spiking as long interspike intervals (ISIs) of quiescence disappeared within bursting patterns, drastically decreasing both ISIs and the magnitude of the chaotic attractors. Deterministic structures of the chaotic bursting and spiking patterns are also identified by a short-term prediction. The experimental observations, which agree with published findings in theoretical neuronal models, demonstrate the existence and reveal the dynamics of a neuronal transition from chaotic bursting to chaotic spiking in the nervous system.
Gu, Huaguang
2013-06-01
The transition from chaotic bursting to chaotic spiking has been simulated and analyzed in theoretical neuronal models. In the present study, we report experimental observations in a neural pacemaker of a transition from chaotic bursting to chaotic spiking within a bifurcation scenario from period-1 bursting to period-1 spiking. This was induced by adjusting extracellular calcium or potassium concentrations. The bifurcation scenario began from period-doubling bifurcations or period-adding sequences of bursting pattern. This chaotic bursting is characterized by alternations between multiple continuous spikes and a long duration of quiescence, whereas chaotic spiking is comprised of fast, continuous spikes without periods of quiescence. Chaotic bursting changed to chaotic spiking as long interspike intervals (ISIs) of quiescence disappeared within bursting patterns, drastically decreasing both ISIs and the magnitude of the chaotic attractors. Deterministic structures of the chaotic bursting and spiking patterns are also identified by a short-term prediction. The experimental observations, which agree with published findings in theoretical neuronal models, demonstrate the existence and reveal the dynamics of a neuronal transition from chaotic bursting to chaotic spiking in the nervous system.
Regular and chaotic quantum dynamics in atom-diatom reactive collisions
Gevorkyan, A. S.; Nyman, G.
2008-05-15
A new microirreversible 3D theory of quantum multichannel scattering in the three-body system is developed. The quantum approach is constructed on the generating trajectory tubes which allow taking into account influence of classical nonintegrability of the dynamical quantum system. When the volume of classical chaos in phase space is larger than the quantum cell in the corresponding quantum system, quantum chaos is generated. The probability of quantum transitions is constructed for this case. The collinear collision of the Li + (FH) {sup {yields}}(LiF) + H system is used for numerical illustration of a system generating quantum (wave) chaos.
NASA Astrophysics Data System (ADS)
Steiros, K.; Bruce, P. J. K.; Buxton, O. R. H.; Vassilicos, J. C.
2015-11-01
Experiments have been performed in an octagonal un-baffled water tank, stirred by three radial turbines with different geometry impellers: (1) regular rectangular blades; (2) single-iteration fractal blades; (3) two-iteration fractal blades. Shaft torque was monitored and the power number calculated for each case. Both impellers with fractal geometry blades exhibited a decrease of turbine power number compared to the regular one (15% decrease for single-iteration and 19% for two iterations). Phase locked PIV in the discharge region of the blades revealed that the vortices emanating from the regular blades are more coherent, have higher kinetic energy, and advect faster towards the tank's walls where they are dissipated, compared to their fractal counterparts. This suggests a strong link between vortex production and behaviour and the energy input for the different impellers. Planar PIV measurements in the bulk of the tank showed an increase of turbulence intensity of over 20% for the fractal geometry blades, suggesting higher mixing efficiency. Experiments with pressure measurements on the different geometry blade surfaces are ongoing to investigate the distribution of forces, and calculate hydrodynamic centres of pressure. The authors would like to acknowledge the financial support given by European Union FP7 Marie Curie MULTISOLVE project (Grant Agreement No. 317269).
Stochastic formation of magnetic vortex structures in asymmetric disks triggered by chaotic dynamics
Im, Mi-Young; Lee, Ki-Suk; Vogel, Andreas; Hong, Jung-Il; Meier, Guido; Fischer, Peter
2014-12-17
The non-trivial spin configuration in a magnetic vortex is a prototype for fundamental studies of nanoscale spin behaviour with potential applications in magnetic information technologies. Arrays of magnetic vortices interfacing with perpendicular thin films have recently been proposed as enabler for skyrmionic structures at room temperature, which has opened exciting perspectives on practical applications of skyrmions. An important milestone for achieving not only such skyrmion materials but also general applications of magnetic vortices is a reliable control of vortex structures. However, controlling magnetic processes is hampered by stochastic behaviour, which is associated with thermal fluctuations in general. Here we showmore » that the dynamics in the initial stages of vortex formation on an ultrafast timescale plays a dominating role for the stochastic behaviour observed at steady state. Our results show that the intrinsic stochastic nature of vortex creation can be controlled by adjusting the interdisk distance in asymmetric disk arrays.« less
Stochastic formation of magnetic vortex structures in asymmetric disks triggered by chaotic dynamics
Im, Mi-Young; Lee, Ki-Suk; Vogel, Andreas; Hong, Jung-Il; Meier, Guido; Fischer, Peter
2014-12-17
The non-trivial spin configuration in a magnetic vortex is a prototype for fundamental studies of nanoscale spin behaviour with potential applications in magnetic information technologies. Arrays of magnetic vortices interfacing with perpendicular thin films have recently been proposed as enabler for skyrmionic structures at room temperature, which has opened exciting perspectives on practical applications of skyrmions. An important milestone for achieving not only such skyrmion materials but also general applications of magnetic vortices is a reliable control of vortex structures. However, controlling magnetic processes is hampered by stochastic behaviour, which is associated with thermal fluctuations in general. Here we show that the dynamics in the initial stages of vortex formation on an ultrafast timescale plays a dominating role for the stochastic behaviour observed at steady state. Our results show that the intrinsic stochastic nature of vortex creation can be controlled by adjusting the interdisk distance in asymmetric disk arrays.
NASA Astrophysics Data System (ADS)
Latka, Miroslaw; Glaubic-Latka, Marta; Latka, Dariusz; West, Bruce J.
2004-04-01
We study the middle cerebral artery blood flow velocity (MCAfv) in humans using transcranial Doppler ultrasonography (TCD). Scaling properties of time series of the axial flow velocity averaged over a cardiac beat interval may be characterized by two exponents. The short time scaling exponent (STSE) determines the statistical properties of fluctuations of blood flow velocities in short-time intervals while the Hurst exponent describes the long-term fractal properties. In many migraineurs the value of the STSE is significantly reduced and may approach that of the Hurst exponent. This change in dynamical properties reflects the significant loss of short-term adaptability and the overall hyperexcitability of the underlying cerebral blood flow control system. We call this effect fractal rigidity.
NASA Technical Reports Server (NTRS)
2003-01-01
[figure removed for brevity, see original site]
Released 4 June 2003
Chaotic terrain on Mars is thought to form when there is a sudden removal of subsurface water or ice, causing the surface material to slump and break into blocks. The chaotic terrain in this THEMIS visible image is confined to a crater just south of Elysium Planitia. It is common to see chaotic terrain in the vicinity of the catastrophic outflow channels on Mars, but the terrain in this image is on the opposite side of the planet from these channels, making it somewhat of an oddity.
Image information: VIS instrument. Latitude -5.9, Longitude 108.1 East (251.9 West). 19 meter/pixel resolution.
Note: this THEMIS visual image has not been radiometrically nor geometrically calibrated for this preliminary release. An empirical correction has been performed to remove instrumental effects. A linear shift has been applied in the cross-track and down-track direction to approximate spacecraft and planetary motion. Fully calibrated and geometrically projected images will be released through the Planetary Data System in accordance with Project policies at a later time.
NASA's Jet Propulsion Laboratory manages the 2001 Mars Odyssey mission for NASA's Office of Space Science, Washington, D.C. The Thermal Emission Imaging System (THEMIS) was developed by Arizona State University, Tempe, in collaboration with Raytheon Santa Barbara Remote Sensing. The THEMIS investigation is led by Dr. Philip Christensen at Arizona State University. Lockheed Martin Astronautics, Denver, is the prime contractor for the Odyssey project, and developed and built the orbiter. Mission operations are conducted jointly from Lockheed Martin and from JPL, a division of the California Institute of Technology in Pasadena.
Dey, Snigdhadip; Goswami, Bedartha; Joshi, Amitabh
2015-02-21
Much research in metapopulation dynamics has concentrated on identifying factors that affect coherence in spatially structured systems. In this regard, conditions for the attainment of out-of-phase dynamics have received considerable attention, due to the stabilizing effect of asynchrony on global dynamics. At low to moderate rates of dispersal, two coupled subpopulations with intrinsically chaotic dynamics tend to go out-of-phase with one another and also become periodic in their dynamics. The onset of out-of-phase dynamics and periodicity typically coincide. Here, we propose a possible mechanism for the onset of out-of-phase dynamics, and also the stabilization of chaos to periodicity, in two coupled subpopulations with intrinsically chaotic dynamics. We suggest that the onset of out-of-phase dynamics is due to the propensity of chaotic subpopulations governed by a steep, single-humped one-dimensional population growth model to repeatedly reach low subpopulation sizes that are close to a value Nt = A (A ≠ equilibrium population size, K) for which Nt( + 1) = K. Subpopulations with very similar low sizes, but on opposite sides of A, will become out-of-phase in the next generation, as they will end up at sizes on opposite sides of K, resulting in positive growth for one subpopulation and negative growth for the other. The key to the stabilization of out-of-phase periodic dynamics in this mechanism is the net effect of dispersal placing upper and lower bounds to subpopulation size in the two coupled subpopulations, once they have become out-of-phase. We tested various components of this proposed mechanism by simulations using the Ricker model, and the results of the simulations are consistent with predictions from the hypothesized mechanism. Similar results were also obtained using the logistic and Hassell models, and with the Ricker model incorporating the possibility of extinction, suggesting that the proposed mechanism could be key to the attainment and
Zurek, W.H.; Pas, J.P. |
1995-08-01
Violation of correspondence principle may occur for very macroscopic byt isolated quantum systems on rather short timescales as illustrated by the case of Hyperion, the chaotically tumbling moon of Saturn, for which quantum and classical predictions are expected to diverge on a timescale of approximately 20 years. Motivated by Hyperion, we review salient features of ``quantum chaos`` and show that decoherence is the essential ingredient of the classical limit, as it enables one to solve the apparent paradox caused by the breakdown of the correspondence principle for classically chaotic systems.
Chaotic algorithms: A numerical exploration of the dynamics of a stiff photoconductor model
Markus, A.S. de
1997-04-01
The photoconducting property of semiconductors leads, in general, to a very complex kinetics for the charge carriers due to the non-equilibrium processes involved. In a semi-conductor with one type of trap, the dynamics of the photo-conducting process are described by a set of ordinary coupled non-linear differential equations given by where n and p are the free electron and hole densities, and m the trapped electron density at time t. So far, there is no known closed solution for these set of non-linear differential equations, and therefore, numerical integration techniques have to be employed, as, for example, the standard procedure of the Runge-Kutta (RK) method. Now then, each one of the mechanisms of generation, recombination, and trapping has its own lifetime, which means that different time constants are to be expected in the time dependent behavior of the photocurrent. Thus, depending on the parameters of the model, the system may become stiff if the time scales between n, m, and p separate considerably. This situation may impose a considerable stress upon a fixed step numerical algorithm as the RK, which may produce then unreliable results, and other methods have to be considered. Therefore, the purpose of this note is to examine, for a critical range of parameters, the results of the numerical integration of the stiff system obtained by standard numerical schemes, such as the single-step fourth-order Runge-Kutta method and the multistep Gear method, the latter being appropriate for a rigid system of equations. 7 refs., 2 figs.
NASA Astrophysics Data System (ADS)
Myasishchev, Denis; Bixler, David
2009-04-01
Chaos theory is a current topic in physics research and is of great scientific and applied interest. Chaotic systems include weather patterns, genetic evolution and free market economics. Modeling chaotic phenomena using electronic circuits is a convenient way to analyze nonlinear systems. We have built various types of circuits and examined the conditions under which chaos occurs. Chua's circuit and analog computing circuits (ones that directly model systems of differential equations) were in the spotlight during the fall semester. An R-C phase space diagram for the Chua's circuit was constructed and the phase transitions were examined. Different analog computing circuits were built and the resulting attractors, attractor phases, and bifurcations were recorded. A mechanical system, the two block train model, is the current focus of study. The goal is to examine attractors produced by a mechanical system, a computer simulation, and a corresponding circuit in order to prove that the same experimental results can be obtained from different sources. This way if a mechanical system is too complicated to build, it can be substituted by a suitable circuit.
NASA Astrophysics Data System (ADS)
Liu, Fu-Sui; Chao, Wen
1989-10-01
This paper attempts to establish the dynamics of a microscopic model for a continuous-time random walk. The waiting-time distribution Q(t) is derived from the time-dependent perturbation theory of quantum mechanics for the walker's motion coupled with the media. The walker's motion includes the hopping of a localized particle and a spin (or dipole) flip. The medium is modeled as a harmonic heat bath. The walker moves among a set of degenerate localized states. The scaling behavior of the effective spectrum at low frequency with index β is modeled by using stochastic theory. It is found that Q(t)=exp(-at(2-β)) for 0<=β<2 and Q(t)~t-α for β=2. The applications of our theory include dispersion diffusion, the transient drift of hopping control light excitation in a-Si:H, and thermoremanent magnetization relaxation in spin glasses.
Effect of muscular fatigue on fractal upper limb coordination dynamics and muscle synergies.
Bueno, Diana R; Lizano, J M; Montano, L
2015-08-01
Rehabilitation exercises cause fatigue because tasks are repetitive. Therefore, inevitable human motion performance changes occur during the therapy. Although traditionally fatigue is considered an event that occurs in the musculoskeletal level, this paper studies whether fatigue can be regarded as context that influences lower-dimensional motor control organization and coordination at neural level. Non Negative Factorization Matrix (NNFM) and Detrended Fluctuations Analysis (DFA) are the tools used to analyze the changes in the coordination of motor function when someone is affected by fatigue. The study establishes that synergies remain fairly stable with the onset of fatigue, but the fatigue affects the dynamical coordination understood as a cognitive process. These results have been validated with 9 healthy subjects for three representative exercises for upper limb: biceps, triceps and deltoid. PMID:26737679
Fractal vector optical fields.
Pan, Yue; Gao, Xu-Zhen; Cai, Meng-Qiang; Zhang, Guan-Lin; Li, Yongnan; Tu, Chenghou; Wang, Hui-Tian
2016-07-15
We introduce the concept of a fractal, which provides an alternative approach for flexibly engineering the optical fields and their focal fields. We propose, design, and create a new family of optical fields-fractal vector optical fields, which build a bridge between the fractal and vector optical fields. The fractal vector optical fields have polarization states exhibiting fractal geometry, and may also involve the phase and/or amplitude simultaneously. The results reveal that the focal fields exhibit self-similarity, and the hierarchy of the fractal has the "weeding" role. The fractal can be used to engineer the focal field. PMID:27420485
Magnetohydrodynamics of fractal media
Tarasov, Vasily E.
2006-05-15
The fractal distribution of charged particles is considered. An example of this distribution is the charged particles that are distributed over the fractal. The fractional integrals are used to describe fractal distribution. These integrals are considered as approximations of integrals on fractals. Typical turbulent media could be of a fractal structure and the corresponding equations should be changed to include the fractal features of the media. The magnetohydrodynamics equations for fractal media are derived from the fractional generalization of integral Maxwell equations and integral hydrodynamics (balance) equations. Possible equilibrium states for these equations are considered.
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
NASA Astrophysics Data System (ADS)
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems. PMID:27586621
Commendatore, Pasquale; Currie, Martin; Kubin, Ingrid
2007-04-01
This paper examines the long-term behavior of a discrete-time Footloose Capital model, where capitalists, who are themselves immobile between regions, move their physical capital between regions in response to economic incentives. The spatial location of industry can exhibit cycles of any periodicity or behave chaotically. Long-term behavior is highly sensitive to transport costs and to the responsiveness of capitalists to profit differentials. The concentration of industry in one region can result from high transport costs or from rapid responses by capitalists. In terms of possible dynamical behaviors, the discrete-time model is much richer than the standard continuous-time Footloose Capital model.
Isotopic Evidence For Chaotic Imprint In The Upper Mantle Heterogeneity
NASA Astrophysics Data System (ADS)
Armienti, P.; Gasperini, D.
2006-12-01
Heterogeneities of the asthenospheric mantle along mid-ocean ridges have been documented as the ultimate effect of complex processes dominated by temperature, pressure and composition of the shallow mantle, in a convective regime that involves mass transfer from the deep mantle, occasionally disturbed by the occurrence of hot spots (e.g. Graham et al., 2001; Agranier et al., 2005; Debaille et al., 2006). Alternatively, upper mantle heterogeneity is seen as the natural result of basically athermal processes that are intrinsic to plate tectonics, such as delamination and recycling of continental crust and of subducted aseismic ridges (Meibom and Anderson, 2003; Anderson, 2006). Here we discuss whether the theory of chaotic dynamical systems applied to isotopic space series along the Mid-Atlantic Ridge (MAR) and the East Pacific Rise (EPR) can delimit the length-scale of upper mantle heterogeneities, then if the model of marble-cake mantle (Allègre and Turcotte, 1986) is consistent with a fractal distribution of such heterogeneity. The correlations between the isotopic (Sr, Nd, Hf, Pb) composition of MORB were parameterized as a function of the ridge length. We found that the distribution of isotopic heterogenity along both the MAR and EPR is self- similar in the range of 7000-9000 km. Self-similarity is the imprint of chaotic mantle processes. The existence of strange attractors in the distribution of isotopic composition of the asthenosphere sampled at ridge crests reveals recursion of the same mantle process(es), endured over long periods of time, up to a stationary state. The occurrence of the same fractal dimension for both the MAR and EPR implies independency of contingent events, suggesting common mantle processes, on a planetary scale. We envisage the cyclic route of "melting, melt extraction and recycling" as the main mantle process which could be able to induce scale invariance. It should have happened for a significant number of times over the Earth
The chaotic set and the cross section for chaotic scattering in three degrees of freedom
NASA Astrophysics Data System (ADS)
Jung, C.; Merlo, O.; Seligman, T. H.; Zapfe, W. P. K.
2010-10-01
This article treats chaotic scattering with three degrees of freedom, where one of them is open and the other two are closed, as a first step towards a more general understanding of chaotic scattering in higher dimensions. Despite the strong restrictions, it breaks the essential simplicity implicit in any two-dimensional time-independent scattering problem. Introducing the third degree of freedom by breaking a continuous symmetry, we first explore the topological structure of the homoclinic/heteroclinic tangle and the structures in the scattering functions. Then we work out the implications of these structures for the doubly differential cross section. The most prominent structures in the cross section are rainbow singularities. They form a fractal pattern that reflects the fractal structure of the chaotic invariant set. This allows us to determine structures in the cross section from the invariant set and, conversely, to obtain information about the topology of the invariant set from the cross section. The latter is a contribution to the inverse scattering problem for chaotic systems.
Symmetric encryption algorithms using chaotic and non-chaotic generators: A review
Radwan, Ahmed G.; AbdElHaleem, Sherif H.; Abd-El-Hafiz, Salwa K.
2015-01-01
This paper summarizes the symmetric image encryption results of 27 different algorithms, which include substitution-only, permutation-only or both phases. The cores of these algorithms are based on several discrete chaotic maps (Arnold’s cat map and a combination of three generalized maps), one continuous chaotic system (Lorenz) and two non-chaotic generators (fractals and chess-based algorithms). Each algorithm has been analyzed by the correlation coefficients between pixels (horizontal, vertical and diagonal), differential attack measures, Mean Square Error (MSE), entropy, sensitivity analyses and the 15 standard tests of the National Institute of Standards and Technology (NIST) SP-800-22 statistical suite. The analyzed algorithms include a set of new image encryption algorithms based on non-chaotic generators, either using substitution only (using fractals) and permutation only (chess-based) or both. Moreover, two different permutation scenarios are presented where the permutation-phase has or does not have a relationship with the input image through an ON/OFF switch. Different encryption-key lengths and complexities are provided from short to long key to persist brute-force attacks. In addition, sensitivities of those different techniques to a one bit change in the input parameters of the substitution key as well as the permutation key are assessed. Finally, a comparative discussion of this work versus many recent research with respect to the used generators, type of encryption, and analyses is presented to highlight the strengths and added contribution of this paper. PMID:26966561
Symmetric encryption algorithms using chaotic and non-chaotic generators: A review.
Radwan, Ahmed G; AbdElHaleem, Sherif H; Abd-El-Hafiz, Salwa K
2016-03-01
This paper summarizes the symmetric image encryption results of 27 different algorithms, which include substitution-only, permutation-only or both phases. The cores of these algorithms are based on several discrete chaotic maps (Arnold's cat map and a combination of three generalized maps), one continuous chaotic system (Lorenz) and two non-chaotic generators (fractals and chess-based algorithms). Each algorithm has been analyzed by the correlation coefficients between pixels (horizontal, vertical and diagonal), differential attack measures, Mean Square Error (MSE), entropy, sensitivity analyses and the 15 standard tests of the National Institute of Standards and Technology (NIST) SP-800-22 statistical suite. The analyzed algorithms include a set of new image encryption algorithms based on non-chaotic generators, either using substitution only (using fractals) and permutation only (chess-based) or both. Moreover, two different permutation scenarios are presented where the permutation-phase has or does not have a relationship with the input image through an ON/OFF switch. Different encryption-key lengths and complexities are provided from short to long key to persist brute-force attacks. In addition, sensitivities of those different techniques to a one bit change in the input parameters of the substitution key as well as the permutation key are assessed. Finally, a comparative discussion of this work versus many recent research with respect to the used generators, type of encryption, and analyses is presented to highlight the strengths and added contribution of this paper.
Symmetric encryption algorithms using chaotic and non-chaotic generators: A review.
Radwan, Ahmed G; AbdElHaleem, Sherif H; Abd-El-Hafiz, Salwa K
2016-03-01
This paper summarizes the symmetric image encryption results of 27 different algorithms, which include substitution-only, permutation-only or both phases. The cores of these algorithms are based on several discrete chaotic maps (Arnold's cat map and a combination of three generalized maps), one continuous chaotic system (Lorenz) and two non-chaotic generators (fractals and chess-based algorithms). Each algorithm has been analyzed by the correlation coefficients between pixels (horizontal, vertical and diagonal), differential attack measures, Mean Square Error (MSE), entropy, sensitivity analyses and the 15 standard tests of the National Institute of Standards and Technology (NIST) SP-800-22 statistical suite. The analyzed algorithms include a set of new image encryption algorithms based on non-chaotic generators, either using substitution only (using fractals) and permutation only (chess-based) or both. Moreover, two different permutation scenarios are presented where the permutation-phase has or does not have a relationship with the input image through an ON/OFF switch. Different encryption-key lengths and complexities are provided from short to long key to persist brute-force attacks. In addition, sensitivities of those different techniques to a one bit change in the input parameters of the substitution key as well as the permutation key are assessed. Finally, a comparative discussion of this work versus many recent research with respect to the used generators, type of encryption, and analyses is presented to highlight the strengths and added contribution of this paper. PMID:26966561
Modeste Nguimdo, Romain; Tchitnga, Robert; Woafo, Paul
2013-12-15
We numerically investigate the possibility of using a coupling to increase the complexity in simplest chaotic two-component electronic circuits operating at high frequency. We subsequently show that complex behaviors generated in such coupled systems, together with the post-processing are suitable for generating bit-streams which pass all the NIST tests for randomness. The electronic circuit is built up by unidirectionally coupling three two-component (one active and one passive) oscillators in a ring configuration through resistances. It turns out that, with such a coupling, high chaotic signals can be obtained. By extracting points at fixed interval of 10 ns (corresponding to a bit rate of 100 Mb/s) on such chaotic signals, each point being simultaneously converted in 16-bits (or 8-bits), we find that the binary sequence constructed by including the 10(or 2) least significant bits pass statistical tests of randomness, meaning that bit-streams with random properties can be achieved with an overall bit rate up to 10×100 Mb/s =1Gbit/s (or 2×100 Mb/s =200 Megabit/s). Moreover, by varying the bias voltages, we also investigate the parameter range for which more complex signals can be obtained. Besides being simple to implement, the two-component electronic circuit setup is very cheap as compared to optical and electro-optical systems.
Nguimdo, Romain Modeste; Tchitnga, Robert; Woafo, Paul
2013-12-01
We numerically investigate the possibility of using a coupling to increase the complexity in simplest chaotic two-component electronic circuits operating at high frequency. We subsequently show that complex behaviors generated in such coupled systems, together with the post-processing are suitable for generating bit-streams which pass all the NIST tests for randomness. The electronic circuit is built up by unidirectionally coupling three two-component (one active and one passive) oscillators in a ring configuration through resistances. It turns out that, with such a coupling, high chaotic signals can be obtained. By extracting points at fixed interval of 10 ns (corresponding to a bit rate of 100 Mb/s) on such chaotic signals, each point being simultaneously converted in 16-bits (or 8-bits), we find that the binary sequence constructed by including the 10(or 2) least significant bits pass statistical tests of randomness, meaning that bit-streams with random properties can be achieved with an overall bit rate up to 10×100 Mb/s = 1 Gbit/s (or 2×100 Mb/s =200 Megabit/s). Moreover, by varying the bias voltages, we also investigate the parameter range for which more complex signals can be obtained. Besides being simple to implement, the two-component electronic circuit setup is very cheap as compared to optical and electro-optical systems.
Fractal universe and quantum gravity.
Calcagni, Gianluca
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.
Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.
Xu, M; Paul, M R
2016-06-01
We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization. PMID:27415256
Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection
NASA Astrophysics Data System (ADS)
Xu, M.; Paul, M. R.
2016-06-01
We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20 ≲Dλ≲50 , and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.
Analysis of rattleback chaotic oscillations.
Hanias, Michael; Stavrinides, Stavros G; Banerjee, Santo
2014-01-01
Rattleback is a canoe-shaped object, already known from ancient times, exhibiting a nontrivial rotational behaviour. Although its shape looks symmetric, its kinematic behaviour seems to be asymmetric. When spun in one direction it normally rotates, but when it is spun in the other direction it stops rotating and oscillates until it finally starts rotating in the other direction. It has already been reported that those oscillations demonstrate chaotic characteristics. In this paper, rattleback's chaotic dynamics are studied by applying Kane's model for different sets of (experimentally decided) parameters, which correspond to three different experimental prototypes made of wax, gypsum, and lead-solder. The emerging chaotic behaviour in all three cases has been studied and evaluated by the related time-series analysis and the calculation of the strange attractors' invariant parameters. PMID:24511290
Analysis of Rattleback Chaotic Oscillations
Stavrinides, Stavros G.; Banerjee, Santo
2014-01-01
Rattleback is a canoe-shaped object, already known from ancient times, exhibiting a nontrivial rotational behaviour. Although its shape looks symmetric, its kinematic behaviour seems to be asymmetric. When spun in one direction it normally rotates, but when it is spun in the other direction it stops rotating and oscillates until it finally starts rotating in the other direction. It has already been reported that those oscillations demonstrate chaotic characteristics. In this paper, rattleback's chaotic dynamics are studied by applying Kane's model for different sets of (experimentally decided) parameters, which correspond to three different experimental prototypes made of wax, gypsum, and lead-solder. The emerging chaotic behaviour in all three cases has been studied and evaluated by the related time-series analysis and the calculation of the strange attractors' invariant parameters. PMID:24511290
Chaotic scattering in an open vase-shaped cavity: Topological, numerical, and experimental results
NASA Astrophysics Data System (ADS)
Novick, Jaison Allen
We present a study of trajectories in a two-dimensional, open, vase-shaped cavity in the absence of forces The classical trajectories freely propagate between elastic collisions. Bound trajectories, regular scattering trajectories, and chaotic scattering trajectories are present in the vase. Most importantly, we find that classical trajectories passing through the vase's mouth escape without return. In our simulations, we propagate bursts of trajectories from point sources located along the vase walls. We record the time for escaping trajectories to pass through the vase's neck. Constructing a plot of escape time versus the initial launch angle for the chaotic trajectories reveals a vastly complicated recursive structure or a fractal. This fractal structure can be understood by a suitable coordinate transform. Reducing the dynamics to two dimensions reveals that the chaotic dynamics are organized by a homoclinic tangle, which is formed by the union of infinitely long, intersecting stable and unstable manifolds. This study is broken down into three major components. We first present a topological theory that extracts the essential topological information from a finite subset of the tangle and encodes this information in a set of symbolic dynamical equations. These equations can be used to predict a topologically forced minimal subset of the recursive structure seen in numerically computed escape time plots. We present three applications of the theory and compare these predictions to our simulations. The second component is a presentation of an experiment in which the vase was constructed from Teflon walls using an ultrasound transducer as a point source. We compare the escaping signal to a classical simulation and find agreement between the two. Finally, we present an approximate solution to the time independent Schrodinger Equation for escaping waves. We choose a set of points at which to evaluate the wave function and interpolate trajectories connecting the source
Thamrin, Cindy; Stern, Georgette; Frey, Urs
2010-06-01
There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment.
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
ERIC Educational Resources Information Center
Barton, Ray
1990-01-01
Presented is an educational game called "The Chaos Game" which produces complicated fractal images. Two basic computer programs are included. The production of fractal images by the Sierpinski gasket and the Chaos Game programs is discussed. (CW)
Fractality à la carte: a general particle aggregation model
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension. PMID:26781204
Fractality à la carte: a general particle aggregation model.
Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V
2016-01-19
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
Fractality à la carte: a general particle aggregation model
NASA Astrophysics Data System (ADS)
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
[Chaos and fractals and their applications in electrocardial signal research].
Jiao, Qing; Guo, Yongxin; Zhang, Zhengguo
2009-06-01
Chaos and fractals are ubiquitous phenomena of nature. A system with fractal structure usually behaves chaos. As a complicated nonlinear dynamics system, heart has fractals structure and behaves as chaos. The deeper inherent mechanism of heart can be opened out when the chaos and fractals theory is utilized in the research of the electrical activity of heart. Generally a time series of a system was used for describing the status of the strange attractor of the system. The indices include Poincare plot, fractals dimension, Lyapunov exponent, entropy, scaling exponent, Hurst index and so on. In this article, the basic concepts and the methods of chaos and fractals were introduced firstly. Then the applications of chaos and fractals theories in the study of electrocardial signal were expounded with example of how they are used for ventricular fibrillation.
Fractality à la carte: a general particle aggregation model.
Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension. PMID:26781204
ERIC Educational Resources Information Center
Fraboni, Michael; Moller, Trisha
2008-01-01
Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…
Das, Kalyan; Srinivas, M N; Srinivas, M A S; Gazi, N H
2012-08-01
We consider a biological economic model based on prey-predator interactions to study the dynamical behaviour of a fishery resource system consisting of one prey and two predators surviving on the same prey. The mathematical model is a set of first order non-linear differential equations in three variables with the population densities of one prey and the two predators. All the possible equilibrium points of the model are identified, where the local and global stabilities are investigated. Biological and bionomical equilibriums of the system are also derived. We have analysed the population intensities of fluctuations i.e., variances around the positive equilibrium due to noise with incorporation of a constant delay leading to chaos, and lastly have investigated the stability and chaotic phenomena with a computer simulation.
Hietschold, V; Appold, S; von Kummer, R; Abolmaali, N
2015-01-01
Objective: To investigate radiochemotherapy (RChT)-induced changes of transfer coefficient (Ktrans) and relative tumour blood volume (rTBV) estimated by dynamic contrast-enhanced CT (DCE-CT) and fractal analysis in head and neck tumours (HNTs). Methods: DCE-CT was performed in 15 patients with inoperable HNTs before RChT, and after 2 and 5 weeks. The dynamics of Ktrans and rTBV as well as lacunarity, slope of log(lacunarity) vs log(box size), and fractal dimension were compared with tumour behaviour during RChT and in the 24-month follow-up. Results: In 11 patients, an increase of Ktrans and/or rTBV after 20 Gy followed by a decrease of both parameters after 50 Gy was noted. Except for one local recurrence, no tumour residue was found during the follow-up. In three patients with partial tumour reduction during RChT, a decrease of Ktrans accompanied by an increase in rTBV between 20 and 50 Gy was detected. In one patient with continuous elevation of both parameters, tumour progressed after RChT. Pre-treatment difference in intratumoral heterogeneity with its decline under RChT for the responders vs non-responders was observed. Conclusion: Initial growth of Ktrans and/or rTBV followed by further reduction of both parameters along with the decline of the slope of log(lacunarity) vs log(box size) was associated with positive radiochemotherapeutic response. Increase of Ktrans and/or rTBV under RChT indicated a poor outcome. Advances in knowledge: The modification of Ktrans and rTBV as measured by DCE-CT may be applied for the assessment of tumour sensitivity to chose RChT regimen and, consequently, to reveal clinical impact allowing individualization of RChT strategy in patients with HNT. PMID:25412001
Faybishenko, B.; Doughty, C.; Geller, J.
1998-07-01
Understanding subsurface flow and transport processes is critical for effective assessment, decision-making, and remediation activities for contaminated sites. However, for fluid flow and contaminant transport through fractured vadose zones, traditional hydrogeological approaches are often found to be inadequate. In this project, the authors examine flow and transport through a fractured vadose zone as a deterministic chaotic dynamical process, and develop a model of it in these terms. Initially, the authors examine separately the geometric model of fractured rock and the flow dynamics model needed to describe chaotic behavior. Ultimately they will put the geometry and flow dynamics together to develop a chaotic-dynamical model of flow and transport in a fractured vadose zone. They investigate water flow and contaminant transport on several scales, ranging from small-scale laboratory experiments in fracture replicas and fractured cores, to field experiments conducted in a single exposed fracture at a basalt outcrop, and finally to a ponded infiltration test using a pond of 7 by 8 m. In the field experiments, they measure the time-variation of water flux, moisture content, and hydraulic head at various locations, as well as the total inflow rate to the subsurface. Such variations reflect the changes in the geometry and physics of water flow that display chaotic behavior, which they try to reconstruct using the data obtained. In the analysis of experimental data, a chaotic model can be used to predict the long-term bounds on fluid flow and transport behavior, known as the attractor of the system, and to examine the limits of short-term predictability within these bounds. This approach is especially well suited to the need for short-term predictions to support remediation decisions and long-term bounding studies. View-graphs from ten presentations made at the annual meeting held December 3--4, 1997 are included in an appendix to this report.
Chaotic Map Construction from Common Nonlinearities and Microcontroller Implementations
NASA Astrophysics Data System (ADS)
Ablay, Günyaz
2016-06-01
This work presents novel discrete-time chaotic systems with some known physical system nonlinearities. Dynamic behaviors of the models are examined with numerical methods and Arduino microcontroller-based experimental studies. Many new chaotic maps are generated in the form of x(k + 1) = rx(k) + f(x(k)) and high-dimensional chaotic systems are obtained by weak coupling or cross-coupling the same or different chaotic maps. An application of the chaotic maps is realized with Arduino for chaotic pulse width modulation to drive electrical machines. It is expected that the new chaotic maps and their microcontroller implementations will facilitate practical chaos-based applications in different fields.
Akrami, Amin; Nazeri, Sina
2016-01-01
An important challenge in brain research is to make out the relation between the features of olfactory stimuli and the electroencephalogram (EEG) signal. Yet, no one has discovered any relation between the structures of olfactory stimuli and the EEG signal. This study investigates the relation between the structures of EEG signal and the olfactory stimulus (odorant). We show that the complexity of the EEG signal is coupled with the molecular complexity of the odorant, where more structurally complex odorant causes less fractal EEG signal. Also, odorant having higher entropy causes the EEG signal to have lower approximate entropy. The method discussed here can be applied and investigated in case of patients with brain diseases as the rehabilitation purpose.
Akrami, Amin; Nazeri, Sina
2016-01-01
An important challenge in brain research is to make out the relation between the features of olfactory stimuli and the electroencephalogram (EEG) signal. Yet, no one has discovered any relation between the structures of olfactory stimuli and the EEG signal. This study investigates the relation between the structures of EEG signal and the olfactory stimulus (odorant). We show that the complexity of the EEG signal is coupled with the molecular complexity of the odorant, where more structurally complex odorant causes less fractal EEG signal. Also, odorant having higher entropy causes the EEG signal to have lower approximate entropy. The method discussed here can be applied and investigated in case of patients with brain diseases as the rehabilitation purpose. PMID:27699169
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
NASA Astrophysics Data System (ADS)
Chernodub, Maxim N.; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics.
Chernodub, Maxim N; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.
Fractal structure of eigenmodes of unstable-cavity lasers.
Karman, G P; Woerdman, J P
1998-12-15
We show that the eigenmodes of unstable-cavity lasers have fractal structure, in contrast with the well-known stable-cavity eigenmodes. As with all fractals, the dynamic range over which self-similarity holds is limited; in this case the range is set by diffraction, i.e., by the Fresnel number of the resonator. We determine the fractal dimension of the mode profiles and show that it is related to the aperture shape. PMID:18091952
Is Hydroclimate Fractal? Another Look
NASA Astrophysics Data System (ADS)
Fleming, S. W.
2012-04-01
Fractal dynamics, defined for our purposes as log-space linear scaling of the power spectrum, are important to water resource scientists for two broad reasons. The first is fundamental: such behaviour is commonly believed to be very widespread in nature, and is therefore a central and important feature of physical systems - including watershed hydrological systems. The second is practical: associated properties, such as runs and clustering, violate the usual assumptions of many standard statistical and time series analysis techniques in applied hydrology and climatology, such as flood frequency analysis and long-term trend analysis. Recent work, however, has indicated that some instrumental climatic and hydroclimatic records, which seem initially suggestive of 1/f ^β scaling, may in fact be insufficiently long to reliably distinguish between fractal dynamics and simpler, low-order linear memory processes which are also known to be common in such systems. With the aim of more carefully assessing the general presence of fractal dynamics in light of historical environmental dataset limitations, we apply here a simple new rule-of-thumb for record length sufficiency, in conjunction with standard Fourier transform-based spectral analysis techniques, to a smorgasbord of long-term time series drawn from watershed hydrology, climatology, and glaciology at various study sites worldwide.
Zhu, Zhiwen; Zhang, Qingxin Xu, Jia
2014-05-07
Stochastic bifurcation and fractal and chaos control of a giant magnetostrictive film–shape memory alloy (GMF–SMA) composite cantilever plate subjected to in-plane harmonic and stochastic excitation were studied. Van der Pol items were improved to interpret the hysteretic phenomena of both GMF and SMA, and the nonlinear dynamic model of a GMF–SMA composite cantilever plate subjected to in-plane harmonic and stochastic excitation was developed. The probability density function of the dynamic response of the system was obtained, and the conditions of stochastic Hopf bifurcation were analyzed. The conditions of noise-induced chaotic response were obtained in the stochastic Melnikov integral method, and the fractal boundary of the safe basin of the system was provided. Finally, the chaos control strategy was proposed in the stochastic dynamic programming method. Numerical simulation shows that stochastic Hopf bifurcation and chaos appear in the parameter variation process. The boundary of the safe basin of the system has fractal characteristics, and its area decreases when the noise intensifies. The system reliability was improved through stochastic optimal control, and the safe basin area of the system increased.
Li, Chunhe; Wang, Erkang; Wang, Jin
2012-05-21
We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.
Mercury's capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics.
Correia, Alexandre C M; Laskar, Jacques
2004-06-24
Mercury is locked into a 3/2 spin-orbit resonance where it rotates three times on its axis for every two orbits around the sun. The stability of this equilibrium state is well established, but our understanding of how this state initially arose remains unsatisfactory. Unless one uses an unrealistic tidal model with constant torques (which cannot account for the observed damping of the libration of the planet) the computed probability of capture into 3/2 resonance is very low (about 7 per cent). This led to the proposal that core-mantle friction may have increased the capture probability, but such a process requires very specific values of the core viscosity. Here we show that the chaotic evolution of Mercury's orbit can drive its eccentricity beyond 0.325 during the planet's history, which very efficiently leads to its capture into the 3/2 resonance. In our numerical integrations of 1,000 orbits of Mercury over 4 Gyr, capture into the 3/2 spin-orbit resonant state was the most probable final outcome of the planet's evolution, occurring 55.4 per cent of the time. PMID:15215857
Exploring Fractals in the Classroom.
ERIC Educational Resources Information Center
Naylor, Michael
1999-01-01
Describes an activity involving six investigations. Introduces students to fractals, allows them to study the properties of some famous fractals, and encourages them to create their own fractal artwork. Contains 14 references. (ASK)
Fractals: To Know, to Do, to Simulate.
ERIC Educational Resources Information Center
Talanquer, Vicente; Irazoque, Glinda
1993-01-01
Discusses the development of fractal theory and suggests fractal aggregates as an attractive alternative for introducing fractal concepts. Describes methods for producing metallic fractals and a computer simulation for drawing fractals. (MVL)
NASA Technical Reports Server (NTRS)
Barnsley, Michael F.; Sloan, Alan D.
1989-01-01
Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Data compression with fractals is an approach to reach high compression ratios for large data streams related to images. The high compression ratios are attained at a cost of large amounts of computation. Both lossless and lossy modes are supported by the technique. The technique is stable in that small errors in codes lead to small errors in image data. Applications to the NASA mission are discussed.
NASA Technical Reports Server (NTRS)
Turcotte, D. L.
1986-01-01
The use of renormalization group techniques on fragmentation problems is examined. The equations which represent fractals and the size-frequency distributions of fragments are presented. Method for calculating the size distributions of asteriods and meteorites are described; the frequency-mass distribution for these interplanetary objects are due to fragmentation. The application of two renormalization group models to fragmentation is analyzed. It is observed that the models yield a fractal behavior for fragmentation; however, different values for the fractal dimension are produced . It is concluded that fragmentation is a scale invariant process and that the fractal dimension is a measure of the fragility of the fragmented material.
Modelling chaotic vibrations using NASTRAN
NASA Technical Reports Server (NTRS)
Sheerer, T. J.
1993-01-01
Due to the unavailability and, later, prohibitive cost of the computational power required, many phenomena in nonlinear dynamic systems have in the past been addressed in terms of linear systems. Linear systems respond to periodic inputs with periodic outputs, and may be characterized in the time domain or in the frequency domain as convenient. Reduction to the frequency domain is frequently desireable to reduce the amount of computation required for solution. Nonlinear systems are only soluble in the time domain, and may exhibit a time history which is extremely sensitive to initial conditions. Such systems are termed chaotic. Dynamic buckling, aeroelasticity, fatigue analysis, control systems and electromechanical actuators are among the areas where chaotic vibrations have been observed. Direct transient analysis over a long time period presents a ready means of simulating the behavior of self-excited or externally excited nonlinear systems for a range of experimental parameters, either to characterize chaotic behavior for development of load spectra, or to define its envelope and preclude its occurrence.
Modelling chaotic vibrations using NASTRAN
NASA Astrophysics Data System (ADS)
Sheerer, T. J.
1993-09-01
Due to the unavailability and, later, prohibitive cost of the computational power required, many phenomena in nonlinear dynamic systems have in the past been addressed in terms of linear systems. Linear systems respond to periodic inputs with periodic outputs, and may be characterized in the time domain or in the frequency domain as convenient. Reduction to the frequency domain is frequently desireable to reduce the amount of computation required for solution. Nonlinear systems are only soluble in the time domain, and may exhibit a time history which is extremely sensitive to initial conditions. Such systems are termed chaotic. Dynamic buckling, aeroelasticity, fatigue analysis, control systems and electromechanical actuators are among the areas where chaotic vibrations have been observed. Direct transient analysis over a long time period presents a ready means of simulating the behavior of self-excited or externally excited nonlinear systems for a range of experimental parameters, either to characterize chaotic behavior for development of load spectra, or to define its envelope and preclude its occurrence.
Surface fractals in liposome aggregation.
Roldán-Vargas, Sándalo; Barnadas-Rodríguez, Ramon; Quesada-Pérez, Manuel; Estelrich, Joan; Callejas-Fernández, José
2009-01-01
In this work, the aggregation of charged liposomes induced by magnesium is investigated. Static and dynamic light scattering, Fourier-transform infrared spectroscopy, and cryotransmission electron microscopy are used as experimental techniques. In particular, multiple intracluster scattering is reduced to a negligible amount using a cross-correlation light scattering scheme. The analysis of the cluster structure, probed by means of static light scattering, reveals an evolution from surface fractals to mass fractals with increasing magnesium concentration. Cryotransmission electron microscopy micrographs of the aggregates are consistent with this interpretation. In addition, a comparative analysis of these results with those previously reported in the presence of calcium suggests that the different hydration energy between lipid vesicles when these divalent cations are present plays a fundamental role in the cluster morphology. This suggestion is also supported by infrared spectroscopy data. The kinetics of the aggregation processes is also analyzed through the time evolution of the mean diffusion coefficient of the aggregates. PMID:19257067
Surface fractals in liposome aggregation.
Roldán-Vargas, Sándalo; Barnadas-Rodríguez, Ramon; Quesada-Pérez, Manuel; Estelrich, Joan; Callejas-Fernández, José
2009-01-01
In this work, the aggregation of charged liposomes induced by magnesium is investigated. Static and dynamic light scattering, Fourier-transform infrared spectroscopy, and cryotransmission electron microscopy are used as experimental techniques. In particular, multiple intracluster scattering is reduced to a negligible amount using a cross-correlation light scattering scheme. The analysis of the cluster structure, probed by means of static light scattering, reveals an evolution from surface fractals to mass fractals with increasing magnesium concentration. Cryotransmission electron microscopy micrographs of the aggregates are consistent with this interpretation. In addition, a comparative analysis of these results with those previously reported in the presence of calcium suggests that the different hydration energy between lipid vesicles when these divalent cations are present plays a fundamental role in the cluster morphology. This suggestion is also supported by infrared spectroscopy data. The kinetics of the aggregation processes is also analyzed through the time evolution of the mean diffusion coefficient of the aggregates.
Dokukin, M. E.; Guz, N. V.; Woodworth, C.D.; Sokolov, I.
2015-01-01
Despite considerable advances in understanding the molecular nature of cancer, many biophysical aspects of malignant development are still unclear. Here we study physical alterations of the surface of human cervical epithelial cells during stepwise in vitro development of cancer (from normal to immortal (premalignant), to malignant). We use atomic force microscopy to demonstrate that development of cancer is associated with emergence of simple fractal geometry on the cell surface. Contrary to the previously expected correlation between cancer and fractals, we find that fractal geometry occurs only at a limited period of development when immortal cells become cancerous; further cancer progression demonstrates deviation from fractal. Because of the connection between fractal behaviour and chaos (or far from equilibrium behaviour), these results suggest that chaotic behaviour coincides with the cancer transformation of the immortalization stage of cancer development, whereas further cancer progression recovers determinism of processes responsible for cell surface formation. PMID:25844044
NASA Astrophysics Data System (ADS)
Dokukin, M. E.; Guz, N. V.; Woodworth, C. D.; Sokolov, I.
2015-03-01
Despite considerable advances in understanding the molecular nature of cancer, many biophysical aspects of malignant development are still unclear. Here we study physical alterations of the surface of human cervical epithelial cells during stepwise in vitro development of cancer (from normal to immortal (premalignant), to malignant). We use atomic force microscopy to demonstrate that development of cancer is associated with emergence of simple fractal geometry on the cell surface. Contrary to the previously expected correlation between cancer and fractals, we find that fractal geometry occurs only at a limited period of development when immortal cells become cancerous; further cancer progression demonstrates deviation from fractal. Because of the connection between fractal behaviour and chaos (or far from equilibrium behaviour), these results suggest that chaotic behaviour coincides with the cancer transformation of the immortalization stage of cancer development, whereas further cancer progression recovers determinism of processes responsible for cell surface formation.
Fractal analysis of DNA sequence data
Berthelsen, C.L.
1993-01-01
DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the [open quote]sandbox method[close quote]. Analysis of 164 human DNA sequences compared to three types of control sequences (random, base-content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than to invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.
Fractal Analysis of DNA Sequence Data
NASA Astrophysics Data System (ADS)
Berthelsen, Cheryl Lynn
DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the "sandbox method." Analysis of 164 human DNA sequences compared to three types of control sequences (random, base -content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than do invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.
Learning feature constraints in a chaotic neural memory
NASA Astrophysics Data System (ADS)
Nara, Shigetoshi; Davis, Peter
1997-01-01
We consider a neural network memory model that has both nonchaotic and chaotic regimes. The chaotic regime occurs for reduced neural connectivity. We show that it is possible to adapt the dynamics in the chaotic regime, by reinforcement learning, to learn multiple constraints on feature subsets. This results in chaotic pattern generation that is biased to generate the feature patterns that have received responses. Depending on the connectivity, there can be additional memory pulling effects, due to the correlations between the constrained neurons in the feature subsets and the other neurons.
Paul, Kush; Cauller, Lawrence J.; Llano, Daniel A.
2016-01-01
Sleep and wakefulness are characterized by distinct states of thalamocortical network oscillations. The complex interplay of ionic conductances within the thalamo-reticular-cortical network give rise to these multiple modes of activity and a rapid transition exists between these modes. To better understand this transition, we constructed a simplified computational model based on physiological recordings and physiologically realistic parameters of a three-neuron network containing a thalamocortical cell, a thalamic reticular neuron, and a corticothalamic cell. The network can assume multiple states of oscillatory activity, resembling sleep, wakefulness, and the transition between these two. We found that during the transition period, but not during other states, thalamic and cortical neurons displayed chaotic dynamics, based on the presence of strange attractors, estimation of positive Lyapunov exponents and the presence of a fractal dimension in the spike trains. These dynamics were quantitatively dependent on certain features of the network, such as the presence of corticothalamic feedback and the strength of inhibition between the thalamic reticular nucleus and thalamocortical neurons. These data suggest that chaotic dynamics facilitate a rapid transition between sleep and wakefulness and produce a series of experimentally testable predictions to further investigate the events occurring during the sleep-wake transition period.
Paul, Kush; Cauller, Lawrence J; Llano, Daniel A
2016-01-01
Sleep and wakefulness are characterized by distinct states of thalamocortical network oscillations. The complex interplay of ionic conductances within the thalamo-reticular-cortical network give rise to these multiple modes of activity and a rapid transition exists between these modes. To better understand this transition, we constructed a simplified computational model based on physiological recordings and physiologically realistic parameters of a three-neuron network containing a thalamocortical cell, a thalamic reticular neuron, and a corticothalamic cell. The network can assume multiple states of oscillatory activity, resembling sleep, wakefulness, and the transition between these two. We found that during the transition period, but not during other states, thalamic and cortical neurons displayed chaotic dynamics, based on the presence of strange attractors, estimation of positive Lyapunov exponents and the presence of a fractal dimension in the spike trains. These dynamics were quantitatively dependent on certain features of the network, such as the presence of corticothalamic feedback and the strength of inhibition between the thalamic reticular nucleus and thalamocortical neurons. These data suggest that chaotic dynamics facilitate a rapid transition between sleep and wakefulness and produce a series of experimentally testable predictions to further investigate the events occurring during the sleep-wake transition period.
Paul, Kush; Cauller, Lawrence J.; Llano, Daniel A.
2016-01-01
Sleep and wakefulness are characterized by distinct states of thalamocortical network oscillations. The complex interplay of ionic conductances within the thalamo-reticular-cortical network give rise to these multiple modes of activity and a rapid transition exists between these modes. To better understand this transition, we constructed a simplified computational model based on physiological recordings and physiologically realistic parameters of a three-neuron network containing a thalamocortical cell, a thalamic reticular neuron, and a corticothalamic cell. The network can assume multiple states of oscillatory activity, resembling sleep, wakefulness, and the transition between these two. We found that during the transition period, but not during other states, thalamic and cortical neurons displayed chaotic dynamics, based on the presence of strange attractors, estimation of positive Lyapunov exponents and the presence of a fractal dimension in the spike trains. These dynamics were quantitatively dependent on certain features of the network, such as the presence of corticothalamic feedback and the strength of inhibition between the thalamic reticular nucleus and thalamocortical neurons. These data suggest that chaotic dynamics facilitate a rapid transition between sleep and wakefulness and produce a series of experimentally testable predictions to further investigate the events occurring during the sleep-wake transition period. PMID:27660609
Paul, Kush; Cauller, Lawrence J; Llano, Daniel A
2016-01-01
Sleep and wakefulness are characterized by distinct states of thalamocortical network oscillations. The complex interplay of ionic conductances within the thalamo-reticular-cortical network give rise to these multiple modes of activity and a rapid transition exists between these modes. To better understand this transition, we constructed a simplified computational model based on physiological recordings and physiologically realistic parameters of a three-neuron network containing a thalamocortical cell, a thalamic reticular neuron, and a corticothalamic cell. The network can assume multiple states of oscillatory activity, resembling sleep, wakefulness, and the transition between these two. We found that during the transition period, but not during other states, thalamic and cortical neurons displayed chaotic dynamics, based on the presence of strange attractors, estimation of positive Lyapunov exponents and the presence of a fractal dimension in the spike trains. These dynamics were quantitatively dependent on certain features of the network, such as the presence of corticothalamic feedback and the strength of inhibition between the thalamic reticular nucleus and thalamocortical neurons. These data suggest that chaotic dynamics facilitate a rapid transition between sleep and wakefulness and produce a series of experimentally testable predictions to further investigate the events occurring during the sleep-wake transition period. PMID:27660609
Chaotic magnetic fields: Particle motion and energization
Dasgupta, Brahmananda; Ram, Abhay K.; Li, Gang; Li, Xiaocan
2014-02-11
Magnetic field line equations correspond to a Hamiltonian dynamical system, so the features of a Hamiltonian systems can easily be adopted for discussing some essential features of magnetic field lines. The integrability of the magnetic field line equations are discussed by various authors and it can be shown that these equations are, in general, not integrable. We demonstrate several examples of realistic chaotic magnetic fields, produced by asymmetric current configurations. Particular examples of chaotic force-free field and non force-free fields are shown. We have studied, for the first time, the motion of a charged particle in chaotic magnetic fields. It is found that the motion of a charged particle in a chaotic magnetic field is not necessarily chaotic. We also showed that charged particles moving in a time-dependent chaotic magnetic field are energized. Such energization processes could play a dominant role in particle energization in several astrophysical environments including solar corona, solar flares and cosmic ray propagation in space.
Chaotic Scattering and Anomalous Transport
NASA Astrophysics Data System (ADS)
Hu, B.; Horton, W.; Petrosky, T.
2002-11-01
The non-relativistic classical electron scattering by a fixed ion in a uniform magnetic field exhibits chaotic scattering feature of fractal dependence of the final pitch angle on the impact parameter. We have constructed a discrete map(B. Hu, W. Horton and T. Petrosky, Phys. Rev. E 65, 056212 (2002).) for the region v>> 3.5 × 10^4 B^1/3, where v is the electron velocity in m/s and B is the magnetic field in Tesla. The map agrees quite well with the numerical integration of the equation of motion. For neutron star atmosphere and white dwarf atmosphere, the Debye length and the average distance between ions are much greater than the electron gyroradius, but the deBroglie wavelength is comparable or smaller than the electron gyroradius, thus quantum effect should be considered. We create ensembles for the initial conditions in different parameter regions, and study the transition between the asymptotic states, the distribution of some quantities, e.g., final pitch angles, trapping times and bouncing numbers. We shall also consider multi-ion scattering and transport problem, and search for possible anomalies in the electric resistivity and thermal conductivity.
Self-organization and fractality in a metabolic processes of the Krebs cycle.
Grytsay, V I; Musatenko, I V
2013-01-01
The metabolic processes of the Krebs cycle is studied with the help of a mathematical model. The autocatalytic processes resulting in both the formation of the self-organization in the Krebs cycle and the appearance of a cyclicity of its dynamics are determined. Some structural-functional connections creating the synchronism of an autoperiodic functioning at the transport in the respiratory chain and the oxidative phosphorylation are investigated. The conditions for breaking the synchronization of processes, increasing the multiplicity of cyclicity, and for the appearance of chaotic modes are analyzed. The phase-parametric diagram of a cascade of bifurcations showing the transition to a chaotic mode by the Feigenbaum scenario is obtained. The fractal nature of the revealed cascade of bifurcations is demonstrated. The strange attractors formed as a result of the folding are obtained. The results obtained give the idea of structural-functional connections, due to which the self-organization appears in the metabolism running in a cell. The constructed mathematical model can be applied to the study of the toxic and allergic effects of drugs and various substances on cell metabolism.
Fractal images induce fractal pupil dilations and constrictions.
Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P
2014-09-01
Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed.
Fractal images induce fractal pupil dilations and constrictions.
Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P
2014-09-01
Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed. PMID:24978815
Senkerik, Roman; Zelinka, Ivan; Pluhacek, Michal; Davendra, Donald; Oplatková Kominkova, Zuzana
2014-01-01
Evolutionary technique differential evolution (DE) is used for the evolutionary tuning of controller parameters for the stabilization of set of different chaotic systems. The novelty of the approach is that the selected controlled discrete dissipative chaotic system is used also as the chaotic pseudorandom number generator to drive the mutation and crossover process in the DE. The idea was to utilize the hidden chaotic dynamics in pseudorandom sequences given by chaotic map to help differential evolution algorithm search for the best controller settings for the very same chaotic system. The optimizations were performed for three different chaotic systems, two types of case studies and developed cost functions.
Pluhacek, Michal; Davendra, Donald; Oplatková Kominkova, Zuzana
2014-01-01
Evolutionary technique differential evolution (DE) is used for the evolutionary tuning of controller parameters for the stabilization of set of different chaotic systems. The novelty of the approach is that the selected controlled discrete dissipative chaotic system is used also as the chaotic pseudorandom number generator to drive the mutation and crossover process in the DE. The idea was to utilize the hidden chaotic dynamics in pseudorandom sequences given by chaotic map to help differential evolution algorithm search for the best controller settings for the very same chaotic system. The optimizations were performed for three different chaotic systems, two types of case studies and developed cost functions. PMID:25243230
Chaos, fractals, and our concept of disease.
Varela, Manuel; Ruiz-Esteban, Raul; Mestre de Juan, Maria Jose
2010-01-01
The classic anatomo-clinic paradigm based on clinical syndromes is fraught with problems. Nevertheless, for multiple reasons, clinicians are reluctant to embrace a more pathophysiological approach, even though this is the prevalent paradigm under "which basic sciences work. In recent decades, nonlinear dynamics ("chaos theory") and fractal geometry have provided powerful new tools to analyze physiological systems. However, these tools are embedded in the pathophysiological perspective and are not easily translated to our classic syndromes. This article comments on the problems raised by the conventional anatomo-clinic paradigm and reviews three areas in which the influence of nonlinear dynamics and fractal geometry can be especially prominent: disease as a loss of complexity, the idea of homeostasis, and fractals in pathology.
Laser light scattering as a probe of fractal colloid aggregates
NASA Technical Reports Server (NTRS)
Weitz, David A.; Lin, M. Y.
1989-01-01
The extensive use of laser light scattering is reviewed, both static and dynamic, in the study of colloid aggregation. Static light scattering enables the study of the fractal structure of the aggregates, while dynamic light scattering enables the study of aggregation kinetics. In addition, both techniques can be combined to demonstrate the universality of the aggregation process. Colloidal aggregates are now well understood and therefore represent an excellent experimental system to use in the study of the physical properties of fractal objects. However, the ultimate size of fractal aggregates is fundamentally limited by gravitational acceleration which will destroy the fractal structure as the size of the aggregates increases. This represents a great opportunity for spaceborne experimentation, where the reduced g will enable the growth of fractal structures of sufficient size for many interesting studies of their physical properties.
CHAOTIC CAPTURE OF NEPTUNE TROJANS
Nesvorny, David; Vokrouhlicky, David
2009-06-15
Neptune Trojans (NTs) are swarms of outer solar system objects that lead/trail planet Neptune during its revolutions around the Sun. Observations indicate that NTs form a thick cloud of objects with a population perhaps {approx}10 times more numerous than that of Jupiter Trojans and orbital inclinations reaching {approx}25 deg. The high inclinations of NTs are indicative of capture instead of in situ formation. Here we study a model in which NTs were captured by Neptune during planetary migration when secondary resonances associated with the mean-motion commensurabilities between Uranus and Neptune swept over Neptune's Lagrangian points. This process, known as chaotic capture, is similar to that previously proposed to explain the origin of Jupiter's Trojans. We show that chaotic capture of planetesimals from an {approx}35 Earth-mass planetesimal disk can produce a population of NTs that is at least comparable in number to that inferred from current observations. The large orbital inclinations of NTs are a natural outcome of chaotic capture. To obtain the {approx}4:1 ratio between high- and low-inclination populations suggested by observations, planetary migration into a dynamically excited planetesimal disk may be required. The required stirring could have been induced by Pluto-sized and larger objects that have formed in the disk.
Chaotic Neural Networks and Beyond
NASA Astrophysics Data System (ADS)
Aihara, Kazuyuki; Yamada, Taiji; Oku, Makito
2013-01-01
A chaotic neuron model which is closely related to deterministic chaos observed experimentally with squid giant axons is explained, and used to construct a chaotic neural network model. Further, such a chaotic neural network is extended to different chaotic models such as a largescale memory relation network, a locally connected network, a vector-valued network, and a quaternionic-valued neuron.
Haire, T J; Ganney, P S; Langton, C M
2001-01-01
Cancellous bone consists of a framework of solid trabeculae interspersed with bone marrow. The structure of the bone tissue framework is highly convoluted and complex, being fractal and statistically self-similar over a limited range of magnifications. To date, the structure of natural cancellous bone tissue has been defined using 2D and 3D imaging, with no facility to modify and control the structure. The potential of four computer-generated paradigms has been reviewed based upon knowledge of other fractal structures and chaotic systems, namely Diffusion Limited Aggregation (DLA), Percolation and Epidemics, Cellular Automata, and a regular Grid with randomly relocated nodes. The resulting structures were compared for their ability to create realistic structures of cancellous bone rather than reflecting growth and form processes. Although the creation of realistic computer-generated cancellous bone structures is difficult, it should not be impossible. Future work considering the combination of fractal and chaotic paradigms is underway. PMID:11328644
Landini, G
2011-01-01
Fractal geometry, developed by B. Mandelbrot, has provided new key concepts necessary to the understanding and quantification of some aspects of pattern and shape randomness, irregularity, complexity and self-similarity. In the field of microscopy, fractals have profound implications in relation to the effects of magnification and scaling on morphology and to the methodological approaches necessary to measure self-similar structures. In this article are reviewed the fundamental concepts on which fractal geometry is based, their relevance to the microscopy field as well as a number of technical details that can help improving the robustness of morphological analyses when applied to microscopy problems.
Fractal Physiology and the Fractional Calculus: A Perspective
West, Bruce J.
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a
Chaotic motif sampler: detecting motifs from biological sequences by using chaotic neurodynamics
NASA Astrophysics Data System (ADS)
Matsuura, Takafumi; Ikeguchi, Tohru
Identification of a region in biological sequences, motif extraction problem (MEP) is solved in bioinformatics. However, the MEP is an NP-hard problem. Therefore, it is almost impossible to obtain an optimal solution within a reasonable time frame. To find near optimal solutions for NP-hard combinatorial optimization problems such as traveling salesman problems, quadratic assignment problems, and vehicle routing problems, chaotic search, which is one of the deterministic approaches, has been proposed and exhibits better performance than stochastic approaches. In this paper, we propose a new alignment method that employs chaotic dynamics to solve the MEPs. It is called the Chaotic Motif Sampler. We show that the performance of the Chaotic Motif Sampler is considerably better than that of the conventional methods such as the Gibbs Site Sampler and the Neighborhood Optimization for Multiple Alignment Discovery.
Chaotic Stochasticity: A Ubiquitous Source of Unpredictability in Epidemics
NASA Astrophysics Data System (ADS)
Rand, D. A.; Wilson, H. B.
1991-11-01
We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.
NASA Astrophysics Data System (ADS)
Martin, R. F., Jr.; Holland, D. L.; Svetich, J.
2014-12-01
We consider dynamical signatures of ion motion that discriminate between a current sheet magnetic field reversal and a magnetic neutral line field. These two related dynamical systems have been studied previously as chaotic scattering systems with application to the Earth's magnetotail. Both systems exhibit chaotic scattering over a wide range of parameter values. The structure and properties of their respective phase spaces have been used to elucidate potential dynamical signatures that affect spacecraft measured ion distributions. In this work we consider the problem of discrimination between these two magnetic structures using charged particle dynamics. For example we show that signatures based on the well known energy resonance in the current sheet field provide good discrimination since the resonance is not present in the neutral line case. While both fields can lead to fractal exit region structuring, their characteristics are different and also may provide some field discrimination. Application to magnetotail field and particle parameters will be presented
Downing, D.J.; Fedorov, V.; Lawkins, W.F.; Morris, M.D.; Ostrouchov, G.
1996-05-01
Large data series with more than several million multivariate observations, representing tens of megabytes or even gigabytes of data, are difficult or impossible to analyze with traditional software. The shear amount of data quickly overwhelms both the available computing resources and the ability of the investigator to confidently identify meaningful patterns and trends which may be present. The purpose of this research is to give meaningful definition to `large data set analysis` and to describe and illustrate a technique for identifying unusual events in large data series. The technique presented here is based on the theory of nonlinear dynamical systems.
ERIC Educational Resources Information Center
Clark, Garry
1999-01-01
Reports on a mathematical investigation of fractals and highlights the thinking involved, problem solving strategies used, generalizing skills required, the role of technology, and the role of mathematics. (ASK)
Fractal and Multifractal Analysis of Human Gait
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.
2003-09-01
We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.
Cascade Chaotic System With Applications.
Zhou, Yicong; Hua, Zhongyun; Pun, Chi-Man; Chen, C L Philip
2015-09-01
Chaotic maps are widely used in different applications. Motivated by the cascade structure in electronic circuits, this paper introduces a general chaotic framework called the cascade chaotic system (CCS). Using two 1-D chaotic maps as seed maps, CCS is able to generate a huge number of new chaotic maps. Examples and evaluations show the CCS's robustness. Compared with corresponding seed maps, newly generated chaotic maps are more unpredictable and have better chaotic performance, more parameters, and complex chaotic properties. To investigate applications of CCS, we introduce a pseudo-random number generator (PRNG) and a data encryption system using a chaotic map generated by CCS. Simulation and analysis demonstrate that the proposed PRNG has high quality of randomness and that the data encryption system is able to protect different types of data with a high-security level.
Hausdorff, Jeffrey M
2007-08-01
Until recently, quantitative studies of walking have typically focused on properties of a typical or average stride, ignoring the stride-to-stride fluctuations and considering these fluctuations to be noise. Work over the past two decades has demonstrated, however, that the alleged noise actually conveys important information. The magnitude of the stride-to-stride fluctuations and their changes over time during a walk - gait dynamics - may be useful in understanding the physiology of gait, in quantifying age-related and pathologic alterations in the locomotor control system, and in augmenting objective measurement of mobility and functional status. Indeed, alterations in gait dynamics may help to determine disease severity, medication utility, and fall risk, and to objectively document improvements in response to therapeutic interventions, above and beyond what can be gleaned from measures based on the average, typical stride. This review discusses support for the idea that gait dynamics has meaning and may be useful in providing insight into the neural control of locomotion and for enhancing functional assessment of aging, chronic disease, and their impact on mobility.
Hausdorff, Jeffrey M
2007-01-01
Until recently, quantitative studies of walking have typically focused on properties of a typical or average stride, ignoring the stride-to-stride fluctuations and considering these fluctuations to be noise. Work over the past two decades has demonstrated, however, that the alleged noise actually conveys important information. The magnitude of the stride-to-stride fluctuations and their changes over time during a walk – gait dynamics – may be useful in understanding the physiology of gait, in quantifying age-related and pathologic alterations in the locomotor control system, and in augmenting objective measurement of mobility and functional status Indeed, alterations in gait dynamics may help to determine disease severity, medication utility, and fall risk, and to objectively document improvements in response to therapeutic interventions, above and beyond what can be gleaned from measures based on the average, typical stride. This review discusses support for the idea that gait dynamics has meaning and may be useful in providing insight into the neural control of locomtion and for enhancing functional assessment of aging, chronic disease, and their impact on mobility. PMID:17618701
Synchronization regimes in conjugate coupled chaotic oscillators.
Karnatak, Rajat; Ramaswamy, Ram; Prasad, Awadhesh
2009-09-01
Nonlinear oscillators that are mutually coupled via dissimilar (or conjugate) variables display distinct regimes of synchronous behavior. In identical chaotic oscillators diffusively coupled in this manner, complete synchronization occurs only by chaos suppression when the coupled subsystems drive each other into a regime of periodic dynamics. Furthermore, the coupling does not vanish but acts as an "internal" drive. When the oscillators are mismatched, phase synchronization occurs, while in a master slave configuration, generalized synchrony results. These effects are demonstrated in a system of coupled chaotic Rossler oscillators.
Chaotic neurodynamics for autonomous agents.
Harter, Derek; Kozma, Robert
2005-05-01
Mesoscopic level neurodynamics study the collective dynamical behavior of neural populations. Such models are becoming increasingly important in understanding large-scale brain processes. Brains exhibit aperiodic oscillations with a much more rich dynamical behavior than fixed-point and limit-cycle approximation allow. Here we present a discretized model inspired by Freeman's K-set mesoscopic level population model. We show that this version is capable of replicating the important principles of aperiodic/chaotic neurodynamics while being fast enough for use in real-time autonomous agent applications. This simplification of the K model provides many advantages not only in terms of efficiency but in simplicity and its ability to be analyzed in terms of its dynamical properties. We study the discrete version using a multilayer, highly recurrent model of the neural architecture of perceptual brain areas. We use this architecture to develop example action selection mechanisms in an autonomous agent. PMID:15940987
Fractal structure of the time distribution of microfracturing in rocks
NASA Astrophysics Data System (ADS)
Feng, Xia-Ting; Seto, Masahiro
1999-01-01
Using acoustic emission data obtained from laboratory double torsion tests, we have analysed the fractal nature of a series of 29 granite microfracturing processes in time. The data represent a wide variety of timescales, stress environments (increasing load with a constant displacement rate, relaxation, creep), soaking conditions [air, water, dodecyl trimethyl ammonium bromide (DTAB), polyethelene oxide (PEO)], and material anisotropy. We find that the time distribution of rock microfracturing displays fractal and multifractal properties. In some cases, it has a single fractal or a multifractal structure. In other cases, it changes from a single fractal structure into a multifractal structure as the system evolves dynamically. We suggest that the heterogeneity of the rock, the distribution of joints or weak planes, the stress level, and the nature of the microfracturing mechanism lead to these multifractal properties. Whatever the fractal structure of the system, a lower fractal dimension is generally produced at near-failure of the rock due to an increased clustering. This result concerning the fractal-dimension decrease is consistent with the conclusion drawn from the spatial distribution of rock microfracturing. Therefore, from the vantage point of observation of the time distribution of rock microfracturing, the decrease of the fractal dimension has a potential use as a rock failure predictor.
Chaotic Behavior of a Brownian Particle in a Periodic Potential
NASA Astrophysics Data System (ADS)
Fang, Jian-Shu; Liu, Wing-Ki; Zhan, Li-Xin
2005-07-01
The classical deterministic dynamics of a Brownian particle with a time-dependent periodic perturbation in a spatially periodic potential is investigated. We have constructed a perturbed chaotic solution near the heteroclinic orbit of the nonlinear dynamics system by using the Constant-Variation method. Theoretical analysis and numerical result show that the motion of the Brownian particle is a kind of chaotic motion. The corresponding chaotic region in parameter space is obtained analytically and numerically. The project supported by the Natural Science Foundation of Hunan Educational Bureau of China under Grant No. 04C063
Cognitive radio resource allocation based on coupled chaotic genetic algorithm
NASA Astrophysics Data System (ADS)
Zu, Yun-Xiao; Zhou, Jie; Zeng, Chang-Chang
2010-11-01
A coupled chaotic genetic algorithm for cognitive radio resource allocation which is based on genetic algorithm and coupled Logistic map is proposed. A fitness function for cognitive radio resource allocation is provided. Simulations are conducted for cognitive radio resource allocation by using the coupled chaotic genetic algorithm, simple genetic algorithm and dynamic allocation algorithm respectively. The simulation results show that, compared with simple genetic and dynamic allocation algorithm, coupled chaotic genetic algorithm reduces the total transmission power and bit error rate in cognitive radio system, and has faster convergence speed.
Controlled transitions between cupolets of chaotic systems
NASA Astrophysics Data System (ADS)
Morena, Matthew A.; Short, Kevin M.; Cooke, Erica E.
2014-03-01
We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.
Controlled transitions between cupolets of chaotic systems.
Morena, Matthew A; Short, Kevin M; Cooke, Erica E
2014-03-01
We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.
Controlled transitions between cupolets of chaotic systems
Morena, Matthew A. Short, Kevin M.; Cooke, Erica E.
2014-03-15
We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.
NASA Astrophysics Data System (ADS)
Chadee, X. T.
2007-05-01
The fractal dimension, Lyapunov-exponent spectrum, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily wind speeds over the Caribbean region. It can be shown that this dimension is greater than 8. However, the number of data points may be too small to obtain a reliable estimate of the Grassberger-Procaccia (1983a) correlation dimension because of the limitations discussed by Ruelle (1990). These results lead us to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia (1983b) are sensitive to the selection of the time delay and are thus unreliable. A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. (1991). Using this method, it is found that the error-doubling time is 2-3 days for time series over the Caribbean region. This is comparable to the predictability time found by Waelbrock (1995) for a single station in Mexico. The predictability time over land is slightly less than that over ocean which tends to have higher climatic signal-to-noise ratio. This analysis impacts on the selection of prediction tools (deterministic chaotic linear and non-linear maps or linear stochastic modeling) for wind speeds in the short term for wind energy farm resource planning and management. We conclude that short term wind predictions in the Caribbean region, for a few days ahead, may be best done with a stochastic model instead of a deterministic chaotic model. References Grassberger, P., and I. Procaccia. 1983a. Measuring the strangeness of attractors. Physica D 9: 189-208. Grassberger, P., and I. Procaccia. 1983b. Estimating the Kolmogorov entropy from a chaotic signal. Phys. Rev. A. 28
A Chaotic System with Different Families of Hidden Attractors
NASA Astrophysics Data System (ADS)
Pham, Viet-Thanh; Volos, Christos; Jafari, Sajad; Vaidyanathan, Sundarapandian; Kapitaniak, Tomasz; Wang, Xiong
The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.
Inhomogeneous stationary and oscillatory regimes in coupled chaotic oscillators.
Liu, Weiqing; Volkov, Evgeny; Xiao, Jinghua; Zou, Wei; Zhan, Meng; Yang, Junzhong
2012-09-01
The dynamics of linearly coupled identical Lorenz and Pikovsky-Rabinovich oscillators are explored numerically and theoretically. We concentrate on the study of inhomogeneous stable steady states ("oscillation death (OD)" phenomenon) and accompanying periodic and chaotic regimes that emerge at an appropriate choice of the coupling matrix. The parameters, for which OD occurs, are determined by stability analysis of the chosen steady state. Three model-specific types of transitions to and from OD are observed: (1) a sharp transition to OD from a nonsymmetric chaotic attractor containing random intervals of synchronous chaos; (2) transition to OD from the symmetry-breaking chaotic regime created by negative coupling; (3) supercritical bifurcation of OD into inhomogeneous limit cycles and further evolution of the system to inhomogeneous chaotic regimes that coexist with complete synchronous chaos. These results may fill a gap in the understanding of the mechanism of OD in coupled chaotic systems.
A New Simple Chaotic Circuit Based on Memristor
NASA Astrophysics Data System (ADS)
Wu, Renping; Wang, Chunhua
In this paper, a new memristor is proposed, and then an emulator built from off-the-shelf solid state components imitating the behavior of the proposed memristor is presented. Multisim simulation and breadboard experiment are done on the emulator, exhibiting a pinched hysteresis loop in the voltage-current plane when the emulator is driven by a periodic excitation voltage. In addition, a new simple chaotic circuit is designed by using the proposed memristor and other circuit elements. It is exciting that this circuit with only a linear negative resistor, a capacitor, an inductor and a memristor can generate a chaotic attractor. The dynamical behaviors of the proposed chaotic system are analyzed by Lyapunov exponents, phase portraits and bifurcation diagrams. Finally, an electronic circuit is designed to implement the chaotic system. For the sake of simple circuit topology, the proposed chaotic circuit can be easily manufactured at low cost.
Exact folded-band chaotic oscillator
NASA Astrophysics Data System (ADS)
Corron, Ned J.; Blakely, Jonathan N.
2012-06-01
An exactly solvable chaotic oscillator with folded-band dynamics is shown. The oscillator is a hybrid dynamical system containing a linear ordinary differential equation and a nonlinear switching condition. Bounded oscillations are provably chaotic, and successive waveform maxima yield a one-dimensional piecewise-linear return map with segments of both positive and negative slopes. Continuous-time dynamics exhibit a folded-band topology similar to Rössler's oscillator. An exact solution is written as a linear convolution of a fixed basis pulse and a discrete binary sequence, from which an equivalent symbolic dynamics is obtained. The folded-band topology is shown to be dependent on the symbol grammar.
Entropy computing via integration over fractal measures.
Słomczynski, Wojciech; Kwapien, Jarosław; Zyczkowski, Karol
2000-03-01
We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.
Fractal Patterns and Chaos Games
ERIC Educational Resources Information Center
Devaney, Robert L.
2004-01-01
Teachers incorporate the chaos game and the concept of a fractal into various areas of the algebra and geometry curriculum. The chaos game approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations.
Building Fractal Models with Manipulatives.
ERIC Educational Resources Information Center
Coes, Loring
1993-01-01
Uses manipulative materials to build and examine geometric models that simulate the self-similarity properties of fractals. Examples are discussed in two dimensions, three dimensions, and the fractal dimension. Discusses how models can be misleading. (Contains 10 references.) (MDH)
Shadowing Lemma and Chaotic Orbit Determination
NASA Astrophysics Data System (ADS)
Milani Comparetti, Andrea; Spoto, Federica
2015-08-01
Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. We test both the convergence of the orbit determination procedure and the behavior of the uncertainties as a function of the maximum number n of map iterations observed; this by using a simple discrete model, namely the standard map. Two problems appear: first, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula containing the Lyapounov time and the relative roundoff error. Second, the uncertainty of the results is sharply increased if a dynamical parameter (contained in the standard map formula) is added to the initial conditions as parameter to be estimated. In particular the uncertainty of the dynamical parameter, and of at least one of the initial conditions, decreases like n^a with a<0 but not large (of the order of unity). If only the initial conditions are estimated, their uncertainty decreases exponentially with n, thus it becomes very small. All these phenomena occur when the chosen initial conditions belong to a chaotic orbit (as shown by one of the well known Lyapounov indicators). If they belong to a non-chaotic orbit the computational horizon is much larger, if it exists at all, and the decrease of the uncertainty appears to be polynomial in all parameters, like n^a with a approximately 1/2; the difference between the case with and without dynamical parameter estimated disappears. These phenomena, which we can investigate in a simple model, have significant implications in practical problems of orbit determination involving chatic phenomena, such as the chaotic rotation state of a celestial body and a chaotic orbit of a planet-crossing asteroid undergoing many close approaches.
On chaotic conductivity in the magnetotail
NASA Technical Reports Server (NTRS)
Holland, Daniel L.; Chen, James
1992-01-01
The concept of chaotic conductivity and the acceleration of particles due to a constant dawn dusk electric field are studied in a magnetotail-like magnetic field. A test particle simulation is used including the full nonlinear dynamics. It is found that the acceleration process can be understood without invoking chaos and that the cross tail current is determined by the particle dynamics and distributions. It is concluded that in general there is no simple relationship between the electric field and the current.
Chaotic Orbits for Systems of Nonlocal Equations
NASA Astrophysics Data System (ADS)
Dipierro, Serena; Patrizi, Stefania; Valdinoci, Enrico
2016-07-01
We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Results are presented of a preliminary investigation of the fractal nature of the plan-view shapes of lava flows in Hawaii (based on field measurements and aerial photographs), as well as in Idaho and the Galapagos Islands (using aerial photographs only). The shapes of the lava flow margins are found to be fractals: lava flow shape is scale-invariant. This observation suggests that nonlinear forces are operating in them because nonlinear systems frequently produce fractals. A'a and pahoehoe flows can be distinguished by their fractal dimensions (D). The majority of the a'a flows measured have D between 1.05 and 1.09, whereas the pahoehoe flows generally have higher D (1.14-1.23). The analysis is extended to other planetary bodies by measuring flows from orbital images of Venus, Mars, and the moon. All are fractal and have D consistent with the range of terrestrial a'a and have D consistent with the range of terrestrial a'a and pahoehoe values.
Origin of coherent structures in a discrete chaotic medium
Rabinovich, M.I.; Torres, J.J.; Varona, P.; Huerta, R.; Varona, P.; Huerta, R.; Weidman, P.
1999-08-01
Using as an example a large lattice of locally interacting Hindmarsh-Rose chaotic neurons, we disclose the origin of ordered structures in a discrete nonequilibrium medium with fast and slow chaotic oscillations. The origin of the ordering mechanism is related to the appearance of a periodic average dynamics in the group of chaotic neurons whose individual slow activity is significantly synchronized by the group mean field. Introducing the concept of a {open_quotes}coarse grain{close_quotes} as a cluster of neuron elements with periodic averaged behavior allows consideration of the dynamics of a medium composed of these clusters. A study of this medium reveals spatially ordered patterns in the periodic and slow dynamics of the coarse grains that are controlled by the average intensity of the fast chaotic pulsation. {copyright} {ital 1999} {ital The American Physical Society}
He, Chiquan; Zhao, Kuiyi
2003-04-01
By using the principles and methods of fractal geometry theory, the relationship between above ground biomass and plant length or sheath height of Carex lasiocarpa population was studied. The results showed that there was a good static fractal relationship between them, and the resulted fractal dimension was an efficient description of the accumulation of above ground biomass in each organ. The dynamic fractal relationship showed that during the whole growing season, the increase of above ground biomass had a self-similarity, being a fractal growth process, and the pattern of its increase was the fractal dimension D. Based on these results, a fractal growth model of Carex lasiocarpa population was established, which regarded the bigger grass as the result of the amplification of seedling growth.
Chaotic keyed hash function based on feedforward feedback nonlinear digital filter
NASA Astrophysics Data System (ADS)
Zhang, Jiashu; Wang, Xiaomin; Zhang, Wenfang
2007-03-01
In this Letter, we firstly construct an n-dimensional chaotic dynamic system named feedforward feedback nonlinear filter (FFNF), and then propose a novel chaotic keyed hash algorithm using FFNF. In hashing process, the original message is modulated into FFNF's chaotic trajectory by chaotic shift keying (CSK) mode, and the final hash value is obtained by the coarse-graining quantization of chaotic trajectory. To expedite the avalanche effect of hash algorithm, a cipher block chaining (CBC) mode is introduced. Theoretic analysis and numerical simulations show that the proposed hash algorithm satisfies the requirement of keyed hash function, and it is easy to implement by the filter structure.
Advective coalescence in chaotic flows.
Nishikawa, T; Toroczkai, Z; Grebogi, C
2001-07-16
We investigate the reaction kinetics of small spherical particles with inertia, obeying coalescence type of reaction, B+B-->B, and being advected by hydrodynamical flows with time-periodic forcing. In contrast to passive tracers, the particle dynamics is governed by the strongly nonlinear Maxey-Riley equations, which typically create chaos in the spatial component of the particle dynamics, appearing as filamental structures in the distribution of the reactants. Defining a stochastic description supported on the natural measure of the attractor, we show that, in the limit of slow reaction, the reaction kinetics assumes a universal behavior exhibiting a t(-1) decay in the amount of reagents, which become distributed on a subset of dimension D2, where D2 is the correlation dimension of the chaotic flow. PMID:11461595
Fractal Globules: A New Approach to Artificial Molecular Machines
Avetisov, Vladik A.; Ivanov, Viktor A.; Meshkov, Dmitry A.; Nechaev, Sergei K.
2014-01-01
The over-damped relaxation of elastic networks constructed by contact maps of hierarchically folded fractal (crumpled) polymer globules was investigated in detail. It was found that the relaxation dynamics of an anisotropic fractal globule is very similar to the behavior of biological molecular machines like motor proteins. When it is perturbed, the system quickly relaxes to a low-dimensional manifold, M, with a large basin of attraction and then slowly approaches equilibrium, not escaping M. Taking these properties into account, it is suggested that fractal globules, even those made by synthetic polymers, are artificial molecular machines that can transform perturbations into directed quasimechanical motion along a defined path. PMID:25418305
Emergence of fractals in aggregation with stochastic self-replication.
Hassan, Md Kamrul; Hassan, Md Zahedul; Islam, Nabila
2013-10-01
We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verify it numerically using the idea of data collapse. Furthermore, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given, and the relationship between this fractal dimension and a conserved quantity is pointed out.
Fractal globules: a new approach to artificial molecular machines.
Avetisov, Vladik A; Ivanov, Viktor A; Meshkov, Dmitry A; Nechaev, Sergei K
2014-11-18
The over-damped relaxation of elastic networks constructed by contact maps of hierarchically folded fractal (crumpled) polymer globules was investigated in detail. It was found that the relaxation dynamics of an anisotropic fractal globule is very similar to the behavior of biological molecular machines like motor proteins. When it is perturbed, the system quickly relaxes to a low-dimensional manifold, M, with a large basin of attraction and then slowly approaches equilibrium, not escaping M. Taking these properties into account, it is suggested that fractal globules, even those made by synthetic polymers, are artificial molecular machines that can transform perturbations into directed quasimechanical motion along a defined path. PMID:25418305
Fractal globules: a new approach to artificial molecular machines.
Avetisov, Vladik A; Ivanov, Viktor A; Meshkov, Dmitry A; Nechaev, Sergei K
2014-11-18
The over-damped relaxation of elastic networks constructed by contact maps of hierarchically folded fractal (crumpled) polymer globules was investigated in detail. It was found that the relaxation dynamics of an anisotropic fractal globule is very similar to the behavior of biological molecular machines like motor proteins. When it is perturbed, the system quickly relaxes to a low-dimensional manifold, M, with a large basin of attraction and then slowly approaches equilibrium, not escaping M. Taking these properties into account, it is suggested that fractal globules, even those made by synthetic polymers, are artificial molecular machines that can transform perturbations into directed quasimechanical motion along a defined path.
NASA Astrophysics Data System (ADS)
Cheng, Qiuming
2016-04-01
Singularity theory states that extreme geo-processes result in anomalous amounts of energy release or material accumulation within a narrow spatial-temporal interval. The products (e.g. mass density and energy density) caused by extreme geo-processes depict singularity without the ordinary derivative and antiderivative (integration) properties. Based on the definition of fractal density, the density measured in fractal dimensional space, in the current paper the author is proposing several operations including fractal derivative and fractal integral to analyze singularity of fractal density. While the ordinary derivative including fractional derivatives as a fundamental tool measuring the sensitivity of change of function (quantity as dependent variable) with change of another quantity as independent variable, the changes are measured in the ordinary space with additive property, fractal derivative (antiderivative) measures the ratio of changes of two quantities measured in fractal space-fractal dimensional space. For example, if the limit of ratio of increment of quantity (Δf) over the associated increment of time (Δtα) measured in α - dimensional space approaches to a finite value, then the limit is referred a α-dimensional fractal derivative of function fand denoted as f' = lim Δf--= df- α Δt→0 Δtα dtα According to the definition of the fractal derivative the ordinary derivative becomes the special case if the space becomes non-fractal space with α value as an integer. In the rest of the paper we demonstrate that fractal density concept and fractal derivative can be applied in describing singularity property of products caused by extreme or avalanche events. The extreme earth-thermal processes such as hydrothermal mineralization occurred in the earth crust, heat flow over ocean ridges, igneous activities or juvenile crust grows, originated from cascade earth dynamics (mantle convection, plate tectonics, and continent crust grow etc.) were analyzed
Chaotic advection, diffusion, and reactions in open flows
Tel, Tamas; Karolyi, Gyoergy; Pentek, Aron; Scheuring, Istvan; Toroczkai, Zoltan; Grebogi, Celso; Kadtke, James
2000-03-01
We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.
Kopelman, R
1988-09-23
Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal "memories." The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of "fractal-like kinetics" are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds.
Evolving random fractal Cantor superlattices for the infrared using a genetic algorithm
Bossard, Jeremy A.; Lin, Lan; Werner, Douglas H.
2016-01-01
Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as ‘chaotic’, but we propose that apparent ‘chaotic’ natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too ‘perfect’ to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the ‘chaotic’ (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and ‘chaotic’ superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime. PMID:26763335
NASA Astrophysics Data System (ADS)
Semenova, N.; Zakharova, A.; Schöll, E.; Anishchenko, V.
2015-11-01
We analyze nonlocally coupled networks of identical chaotic oscillators with either time-discrete or time-continuous dynamics (Henon map, Lozi map, Lorenz system). We hypothesize that chimera states, in which spatial domains of coherent (synchronous) and incoherent (desynchronized) dynamics coexist, can be obtained only in networks of oscillators with nonhyperbolic chaotic attractors and cannot be found in networks of systems with hyperbolic chaotic attractors. This hypothesis is supported by analytical results and numerical simulations for hyperbolic and nonhyperbolic cases.
Fractal dimensions of sinkholes
NASA Astrophysics Data System (ADS)
Reams, Max W.
1992-05-01
Sinkhole perimeters are probably fractals ( D=1.209-1.558) for sinkholes with areas larger than 10,000 m 2, based on area-perimeter plots of digitized data from karst surfaces developed on six geologic units in the United States. The sites in Florida, Kentucky, Indiana and Missouri were studied using maps with a scale of 1:24, 000. Size-number distributions of sinkhole perimeters and areas may also be fractal, although data for small sinkholes is needed for verification. Studies based on small-scale maps are needed to evaluate the number and roughness of small sinkhole populations.
NASA Astrophysics Data System (ADS)
Meir, Yigal; Aharony, Amnon
1989-05-01
We investigate the problem of flow in porous media near the percolation threshold by studying the generelized model of Viscous Fingering (VF) on fractal structures. We obtain analytic expressions for the fractal dimensions of the resulting structures, which are in excellent agreement with existing experimental results, and exact relations for the exponent Dt, which describes the scaling of the time it takes the fluid to cross the sample, with the sample size, in terms of geometrical exponents for various experimental situations. Lastly, we discuss the relation between the continuous viscous fingers model and stochastic processes such as dielectric breakdown model (DBM) and diffusion limited aggregation (DLA).
Enqvist, Kari; Koivisto, Tomi; Rigopoulos, Gerasimos E-mail: T.S.Koivisto@astro.uio.no
2012-05-01
We consider inflation within the context of what is arguably the simplest non-metric extension of Einstein gravity. There non-metricity is described by a single graviscalar field with a non-minimal kinetic coupling to the inflaton field Ψ, parameterized by a single parameter γ. There is a simple equivalent description in terms of a massless field and an inflaton with a modified potential. We discuss the implications of non-metricity for chaotic inflation and find that it significantly alters the inflaton dynamics for field values Ψ∼>M{sub P}/γ, dramatically changing the qualitative behaviour in this regime. In the equivalent single-field description this is described as a cuspy potential that forms of barrier beyond which the inflation becomes a ghost field. This imposes an upper bound on the possible number of e-folds. For the simplest chaotic inflation models, the spectral index and the tensor-to-scalar ratio receive small corrections dependent on the non-metricity parameter. We also argue that significant post-inflationary non-metricity may be generated.
Synchronization of mobile chaotic oscillator networks
NASA Astrophysics Data System (ADS)
Fujiwara, Naoya; Kurths, Jürgen; Díaz-Guilera, Albert
2016-09-01
We study synchronization of systems in which agents holding chaotic oscillators move in a two-dimensional plane and interact with nearby ones forming a time dependent network. Due to the uncertainty in observing other agents' states, we assume that the interaction contains a certain amount of noise that turns out to be relevant for chaotic dynamics. We find that a synchronization transition takes place by changing a control parameter. But this transition depends on the relative dynamic scale of motion and interaction. When the topology change is slow, we observe an intermittent switching between laminar and burst states close to the transition due to small noise. This novel type of synchronization transition and intermittency can happen even when complete synchronization is linearly stable in the absence of noise. We show that the linear stability of the synchronized state is not a sufficient condition for its stability due to strong fluctuations of the transverse Lyapunov exponent associated with a slow network topology change. Since this effect can be observed within the linearized dynamics, we can expect such an effect in the temporal networks with noisy chaotic oscillators, irrespective of the details of the oscillator dynamics. When the topology change is fast, a linearized approximation describes well the dynamics towards synchrony. These results imply that the fluctuations of the finite-time transverse Lyapunov exponent should also be taken into account to estimate synchronization of the mobile contact networks.
Ciszak, M.; Marino, F.; Ortolan, A.; Canton, T. Dal
2009-08-15
Phase space and attractor dimensions in a gravitational wave detector output can be estimated in order to identify chaotic (deterministic) signals in the presence of additive Gaussian noise. These quantities are evaluated, respectively, by means of conditional probabilities and the Grassberger-Procaccia algorithm, both methods relying on embedding in a suitable space of dimension d. By testing with different embedding dimensions, a deterministic--though erratic--signal can be detected by comparing the corresponding conditional probabilities via Kolmogorov-Smirnoff test and checking whether the correlation (fractal) dimension differs from d. Results of the two approaches are eventually compared, both for chaotic and periodic trajectories.
Viscosity and thermal fields associated with strongly chaotic non-Newtonian thermal convection
NASA Technical Reports Server (NTRS)
Malevsky, A. V.; Yuen, D. A.; Weyer, L. M.
1992-01-01
The thermomechanical structure is investigated in strongly chaotic non-Newtonian thermal convection for both base-heated and internally-heated systems. Temperature can build up in stagnant regions and a non-Newtonian mantle can tolerate less internal heating. Viscosity fields of the strongly chaotic regime show a granular structure. The horizontal spectra of viscosity fluctuations obey a power-law and yield a fractal dimension of 1.6 to 1.8 for the isoviscosity lines, providing evidence for 2D turbulence. Long-wavelength viscosity variations are smoothed out by the turbulent non-Newtonian flows.
Chaotic cross-waves in tanks of finite sizes
NASA Astrophysics Data System (ADS)
Krasnopolskaya, Tatyana; Pechuk, Evgeniy; Spektor, Viacheslav
2014-11-01
The phenomenon of chaotic cross-waves generation in fluid free surface in two finite size containers is studied. The waves may be excited by harmonic axisymmetric deformations of the inner shell in the volume between two cylinders and in a rectangular tank when one wall is a flap wavemaker. The existence of chaotic attractors was established for the dynamical system presenting cross-waves and forced waves interaction at fluid free-surface in a volume between two cylinders of finite length. In the case of one cross-wave in a rectangular tank no chaotic regimes were found.
Random symmetry breaking and freezing in chaotic networks.
Peleg, Y; Kinzel, W; Kanter, I
2012-09-01
Parameter space of a driven damped oscillator in a double well potential presents either a chaotic trajectory with sign oscillating amplitude or a nonchaotic trajectory with a fixed sign amplitude. A network of such delay coupled damped oscillators is shown to present chaotic dynamics while the sign amplitude of each damped oscillator is randomly frozen. This phenomenon of random broken global symmetry of the network simultaneous with random freezing of each degree of freedom is accompanied by the existence of exponentially many randomly frozen chaotic attractors with the size of the network. Results are exemplified by a network of modified Duffing oscillators with infinite range pseudoinverse delayed interactions. PMID:23031002
Classical Liquids in Fractal Dimension.
Heinen, Marco; Schnyder, Simon K; Brady, John F; Löwen, Hartmut
2015-08-28
We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a noninteger dimension dl. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here, we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semianalytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.
Fractal analysis of narwhal space use patterns.
Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R
2004-01-01
Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice. PMID:16351924
Fractal analysis of narwhal space use patterns.
Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R
2004-01-01
Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice.
Hausdorff, Jeffrey M.
2009-01-01
Parkinson’s disease (PD) is a common, debilitating neurodegenerative disease. Gait disturbances are a frequent cause of disability and impairment for patients with PD. This article provides a brief introduction to PD and describes the gait changes typically seen in patients with this disease. A major focus of this report is an update on the study of the fractal properties of gait in PD, the relationship between this feature of gait and stride length and gait variability, and the effects of different experimental conditions on these three gait properties. Implications of these findings are also briefly described. This update highlights the idea that while stride length, gait variability, and fractal scaling of gait are all impaired in PD, distinct mechanisms likely contribute to and are responsible for the regulation of these disparate gait properties. PMID:19566273
Dimension of a fractal streamer structure
NASA Astrophysics Data System (ADS)
Lehtinen, Nikolai G.; Østgaard, Nikolai
2015-04-01
Streamer corona plays an important role in formation of leader steps in lightning. In order to understand its dynamics, the streamer front velocity is calculated in a 1D model with curvature. We concentrate on the role of photoionization mechanism in the propagation of the streamer ionization front, the other important mechanisms being electron drift and electron diffusion. The results indicate, in particular, that the effect of photoionization on the streamer velocity for both positive and negative streamers is mostly determined by the photoionization length, with a weaker dependence on the amount of photoionization, and that the velocity is decreased for positive curvature, i.e., convex fronts. These results are used in a fractal model in which the front propagation velocity is simulated as the cluster growth probability [Niemeyer et al, 1984, doi:10.1103/PhysRevLett.52.1033]. Monte Carlo simulations of the cluster growth for various ratios of background electric field E to the breakdown field Eb show that the emerging transverse size of the streamers is of the order of the photoionization length, and at the larger scale the streamer structure is a fractal similar to the one obtained in a diffusion-limited aggregation (DLA) system. In the absence of electron attachment (Eb = 0), the fractal dimension is the same (D ˜ 1.67) as in the DLA model, and is reduced, i.e., the fractal has less branching, for Eb > 0.
ERIC Educational Resources Information Center
Camp, Dane R.
1991-01-01
After introducing the two-dimensional Koch curve, which is generated by simple recursions on an equilateral triangle, the process is extended to three dimensions with simple recursions on a regular tetrahedron. Included, for both fractal sequences, are iterative formulae, illustrations of the first several iterations, and a sample PASCAL program.…
ERIC Educational Resources Information Center
Marks, Tim K.
1992-01-01
Presents a three-lesson unit that uses fractal geometry to measure the coastline of Massachusetts. Two lessons provide hands-on activities utilizing compass and grid methods to perform the measurements and the third lesson analyzes and explains the results of the activities. (MDH)
Moon, Francis C.
1999-07-20
The technical research was directed at problems involving the dynamics of fluid flow and elastic structures. Such problems occur in heat-exchange systems in energy generating plants. Fluid excited vibrations of structures can result in unwanted impact forces which can lead to metal fatigue failures. Mathematical theories based on linear models have been used for several decades. In this research the authors explored the phenomena associated with nonlinear effects using experimental models, mathematical models and numerical computation. A number of nonlinear effects were observed experimentally including chaotic dynamics, multi-fractal Poincare maps, quasi-periodic vibrations, subcritical Hopf bifurcations, helical waves in a tube row and spatial localization.
Shadowing Lemma and chaotic orbit determination
NASA Astrophysics Data System (ADS)
Spoto, Federica; Milani, Andrea
2016-03-01
Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. The Shadowing Lemma (Anosov 1967; Bowen in J Differ Equ 18:333-356, 1975) can be seen as a way to connect the orbit obtained using the observations with a real trajectory. An orbit is a shadowing of the trajectory if it stays close to the real trajectory for some amount of time. In a simple discrete model, the standard map, we tackle the problem of chaotic orbit determination when observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing orbit is computed by the least squares orbit determination. We test both the convergence of the orbit determination iterative procedure and the behaviour of the uncertainties as a function of the maximum number of map iterations observed. When the initial conditions belong to a chaotic orbit, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula. Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is added to the initial conditions as parameter to be estimated. The Shadowing Lemma does not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena have significant implications, which remain to be studied, in practical problems of orbit determination involving chaos, such as the chaotic rotation state of a celestial body and a chaotic orbit of a planet-crossing asteroid undergoing many close approaches.
Chaotic exploration and learning of locomotion behaviors.
Shim, Yoonsik; Husbands, Phil
2012-08-01
We present a general and fully dynamic neural system, which exploits intrinsic chaotic dynamics, for the real-time goal-directed exploration and learning of the possible locomotion patterns of an articulated robot of an arbitrary morphology in an unknown environment. The controller is modeled as a network of neural oscillators that are initially coupled only through physical embodiment, and goal-directed exploration of coordinated motor patterns is achieved by chaotic search using adaptive bifurcation. The phase space of the indirectly coupled neural-body-environment system contains multiple transient or permanent self-organized dynamics, each of which is a candidate for a locomotion behavior. The adaptive bifurcation enables the system orbit to wander through various phase-coordinated states, using its intrinsic chaotic dynamics as a driving force, and stabilizes on to one of the states matching the given goal criteria. In order to improve the sustainability of useful transient patterns, sensory homeostasis has been introduced, which results in an increased diversity of motor outputs, thus achieving multiscale exploration. A rhythmic pattern discovered by this process is memorized and sustained by changing the wiring between initially disconnected oscillators using an adaptive synchronization method. Our results show that the novel neurorobotic system is able to create and learn multiple locomotion behaviors for a wide range of body configurations and physical environments and can readapt in realtime after sustaining damage. PMID:22509965
Reconstruction of time-delay systems from chaotic time series.
Bezruchko, B P; Karavaev, A S; Ponomarenko, V I; Prokhorov, M D
2001-11-01
We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutions and for modeling experimental systems with delay-induced dynamics from their chaotic time series.
Spread spectrum communication system with chaotic frequency modulation
NASA Astrophysics Data System (ADS)
Volkovskii, A. R.; Tsimring, L. Sh.; Rulkov, N. F.; Langmore, I.
2005-09-01
A new spread spectrum communication system utilizing chaotic frequency modulation of sinusoidal signals is discussed. A single phase lock loop (PLL) system in the receiver is used both to synchronize the local chaotic oscillator and to recover the information signal. We study the dynamics of the synchronization process, stability of the PLL system, and evaluate the bit-error-rate performance of this chaos-based communication system.
Statistics of chaotic resonances in an optical microcavity
NASA Astrophysics Data System (ADS)
Wang, Li; Lippolis, Domenico; Li, Ze-Yang; Jiang, Xue-Feng; Gong, Qihuang; Xiao, Yun-Feng
2016-04-01
Distributions of eigenmodes are widely concerned in both bounded and open systems. In the realm of chaos, counting resonances can characterize the underlying dynamics (regular vs chaotic), and is often instrumental to identify classical-to-quantum correspondence. Here, we study, both theoretically and experimentally, the statistics of chaotic resonances in an optical microcavity with a mixed phase space of both regular and chaotic dynamics. Information on the number of chaotic modes is extracted by counting regular modes, which couple to the former via dynamical tunneling. The experimental data are in agreement with a known semiclassical prediction for the dependence of the number of chaotic resonances on the number of open channels, while they deviate significantly from a purely random-matrix-theory-based treatment, in general. We ascribe this result to the ballistic decay of the rays, which occurs within Ehrenfest time, and importantly, within the time scale of transient chaos. The present approach may provide a general tool for the statistical analysis of chaotic resonances in open systems.
Statistics of chaotic resonances in an optical microcavity.
Wang, Li; Lippolis, Domenico; Li, Ze-Yang; Jiang, Xue-Feng; Gong, Qihuang; Xiao, Yun-Feng
2016-04-01
Distributions of eigenmodes are widely concerned in both bounded and open systems. In the realm of chaos, counting resonances can characterize the underlying dynamics (regular vs chaotic), and is often instrumental to identify classical-to-quantum correspondence. Here, we study, both theoretically and experimentally, the statistics of chaotic resonances in an optical microcavity with a mixed phase space of both regular and chaotic dynamics. Information on the number of chaotic modes is extracted by counting regular modes, which couple to the former via dynamical tunneling. The experimental data are in agreement with a known semiclassical prediction for the dependence of the number of chaotic resonances on the number of open channels, while they deviate significantly from a purely random-matrix-theory-based treatment, in general. We ascribe this result to the ballistic decay of the rays, which occurs within Ehrenfest time, and importantly, within the time scale of transient chaos. The present approach may provide a general tool for the statistical analysis of chaotic resonances in open systems. PMID:27176237
Chaotic neural network applied to two-dimensional motion control.
Yoshida, Hiroyuki; Kurata, Shuhei; Li, Yongtao; Nara, Shigetoshi
2010-03-01
Chaotic dynamics generated in a chaotic neural network model are applied to 2-dimensional (2-D) motion control. The change of position of a moving object in each control time step is determined by a motion function which is calculated from the firing activity of the chaotic neural network. Prototype attractors which correspond to simple motions of the object toward four directions in 2-D space are embedded in the neural network model by designing synaptic connection strengths. Chaotic dynamics introduced by changing system parameters sample intermediate points in the high-dimensional state space between the embedded attractors, resulting in motion in various directions. By means of adaptive switching of the system parameters between a chaotic regime and an attractor regime, the object is able to reach a target in a 2-D maze. In computer experiments, the success rate of this method over many trials not only shows better performance than that of stochastic random pattern generators but also shows that chaotic dynamics can be useful for realizing robust, adaptive and complex control function with simple rules.
Statistics of chaotic resonances in an optical microcavity.
Wang, Li; Lippolis, Domenico; Li, Ze-Yang; Jiang, Xue-Feng; Gong, Qihuang; Xiao, Yun-Feng
2016-04-01
Distributions of eigenmodes are widely concerned in both bounded and open systems. In the realm of chaos, counting resonances can characterize the underlying dynamics (regular vs chaotic), and is often instrumental to identify classical-to-quantum correspondence. Here, we study, both theoretically and experimentally, the statistics of chaotic resonances in an optical microcavity with a mixed phase space of both regular and chaotic dynamics. Information on the number of chaotic modes is extracted by counting regular modes, which couple to the former via dynamical tunneling. The experimental data are in agreement with a known semiclassical prediction for the dependence of the number of chaotic resonances on the number of open channels, while they deviate significantly from a purely random-matrix-theory-based treatment, in general. We ascribe this result to the ballistic decay of the rays, which occurs within Ehrenfest time, and importantly, within the time scale of transient chaos. The present approach may provide a general tool for the statistical analysis of chaotic resonances in open systems.
Nanoflow over a fractal surface
NASA Astrophysics Data System (ADS)
Papanikolaou, Michail; Frank, Michael; Drikakis, Dimitris
2016-08-01
This paper investigates the effects of surface roughness on nanoflows using molecular dynamics simulations. A fractal model is employed to model wall roughness, and simulations are performed for liquid argon confined by two solid walls. It is shown that the surface roughness reduces the velocity in the proximity of the walls with the reduction being accentuated when increasing the roughness depth and wettability of the solid wall. It also makes the flow three-dimensional and anisotropic. In flows over idealized smooth surfaces, the liquid forms parallel, well-spaced layers, with a significant gap between the first layer and the solid wall. Rough walls distort the orderly distribution of fluid layers resulting in an incoherent formation of irregularly shaped fluid structures around and within the wall cavities.
Lagrangian coherent structures, transport and chaotic mixing in simple kinematic ocean models
NASA Astrophysics Data System (ADS)
Budyansky, M. V.; Uleysky, M. Yu.; Prants, S. V.
2007-02-01
Methods of dynamical system's theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean—a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle's scattering and transport. A numerical analysis reveals a non-attracting chaotic invariant set Λ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle's coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time. The clusters appear due to existence of regions of stability in the phase space which is the physical space in the advection problem.
Cryptography using multiple one-dimensional chaotic maps
NASA Astrophysics Data System (ADS)
Pareek, N. K.; Patidar, Vinod; Sud, K. K.
2005-10-01
Recently, Pareek et al. [Phys. Lett. A 309 (2003) 75] have developed a symmetric key block cipher algorithm using a one-dimensional chaotic map. In this paper, we propose a symmetric key block cipher algorithm in which multiple one-dimensional chaotic maps are used instead of a one-dimensional chaotic map. However, we also use an external secret key of variable length (maximum 128-bits) as used by Pareek et al. In the present cryptosystem, plaintext is divided into groups of variable length (i.e. number of blocks in each group is different) and these are encrypted sequentially by using randomly chosen chaotic map from a set of chaotic maps. For block-by-block encryption of variable length group, number of iterations and initial condition for the chaotic maps depend on the randomly chosen session key and encryption of previous block of plaintext, respectively. The whole process of encryption/decryption is governed by two dynamic tables, which are updated time to time during the encryption/decryption process. Simulation results show that the proposed cryptosystem requires less time to encrypt the plaintext as compared to the existing chaotic cryptosystems and further produces the ciphertext having flat distribution of same size as the plaintext.
[Dimensional fractal of post-paddy wheat root architecture].
Chen, Xin-xin; Ding, Qi-shuo; Li, Yi-nian; Xue, Jin-lin; Lu, Ming-zhou; Qiu, Wei
2015-06-01
To evaluate whether crop rooting system was directionally dependent, a field digitizer was used to measure post-paddy wheat root architectures. The acquired data was transferred to Pro-E, in which virtual root architecture was reconstructed and projected to a series of planes each separated in 10° apart. Fractal dimension and fractal abundance of root projections in all the 18 planes were calculated, revealing a distinctive architectural distribution of wheat root in each direction. This strongly proved that post-paddy wheat root architecture was directionally dependent. From seedling to turning green stage, fractal dimension of the 18 projections fluctuated significantly, illustrating a dynamical root developing process in the period. At the jointing stage, however, fractal indices of wheat root architecture resumed its regularity in each dimension. This wheat root architecture recovered its dimensional distinctness. The proposed method was applicable for precision modeling field state root distribution in soil.
Quantum Chaotic Attractor in a Dissipative System
NASA Astrophysics Data System (ADS)
Liu, W. Vincent; Schieve, William C.
1997-04-01
A dissipative quantum system is treated here by coupling it with a heat bath of harmonic oscillators. Through quantum Langevin equations and Ehrenfest's theorem, we establish explicitly the quantum Duffing equations with a double-well potential chosen. A quantum noise term appears the only driving force in dynamics. Numerical studies show that the chaotic attractor exists in this system while chaos is certainly forbidden in the classical counterpart.
Complexity and synchronization in stochastic chaotic systems
NASA Astrophysics Data System (ADS)
Son Dang, Thai; Palit, Sanjay Kumar; Mukherjee, Sayan; Hoang, Thang Manh; Banerjee, Santo
2016-02-01
We investigate the complexity of a hyperchaotic dynamical system perturbed by noise and various nonlinear speech and music signals. The complexity is measured by the weighted recurrence entropy of the hyperchaotic and stochastic systems. The synchronization phenomenon between two stochastic systems with complex coupling is also investigated. These criteria are tested on chaotic and perturbed systems by mean conditional recurrence and normalized synchronization error. Numerical results including surface plots, normalized synchronization errors, complexity variations etc show the effectiveness of the proposed analysis.
Universal characteristics of fractal fluctuations in prime number distribution
NASA Astrophysics Data System (ADS)
Selvam, A. M.
2014-11-01
The frequency of occurrence of prime numbers at unit number spacing intervals exhibits self-similar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations and population dynamics. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualizes the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations or an inter-connected network with associated long-range correlations. The model predictions are as follows: (1) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (2) Fractals signify quantum-like chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (3) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organization of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.
Huang, F.; Peng, R. D.; Liu, Y. H.; Chen, Z. Y.; Ye, M. F.; Wang, L.
2012-09-15
Fractal dust grains of different shapes are observed in a radially confined magnetized radio frequency plasma. The fractal dimensions of the dust structures in two-dimensional (2D) horizontal dust layers are calculated, and their evolution in the dust growth process is investigated. It is found that as the dust grains grow the fractal dimension of the dust structure decreases. In addition, the fractal dimension of the center region is larger than that of the entire region in the 2D dust layer. In the initial growth stage, the small dust particulates at a high number density in a 2D layer tend to fill space as a normal surface with fractal dimension D = 2. The mechanism of the formation of fractal dust grains is discussed.
Fractals in geology and geophysics
NASA Technical Reports Server (NTRS)
Turcotte, Donald L.
1989-01-01
The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.
Topological analysis of chaotic orbits: Revisiting Hyperion
NASA Technical Reports Server (NTRS)
Boyd, Patricia T.; Mindlin, Gabriel B.; Gilmore, Robert; Solari, Hernan G.
1994-01-01
There is emerging interest in the possibility of chaotic evolution in astrophysical systems. To mention just one example, recent well-sampled ground-based observations of the Saturian satellite Hyperion strongly suggest that it is exhibiting chaotic behavior. We present a general technique, the method of close returns, for the analysis of data from astronomical objects believed to be exhibiting chaotic motion. The method is based on the extraction of pieces of the evolution that exhibit nearly periodic behavior-episodes during which the object stays near in phase space to some unstable periodic orbit. Such orbits generally act as skeletal features, tracing the topological organization of the manifold on which the chaotic dynamics takes place. This method does not require data sets as lengthy as other nonlinear analysis techniques do and is therefore well suited to many astronomical observing programs. Well sampled data covering between twenty and forty characteristic periods of the system have been found to be sufficient for the application of this technique. Additional strengths of this method are its robustness in the presence of noise and the ability for a user to clearly distinguish between periodic, random, and chaotic behavior by inspection of the resulting two-dimensional image. As an example of its power, we analyze close returns in a numerically generated data set, based on a model for Hyperion extensively studied in the literature, corresponding to nightly observations of the satellite. We show that with a small data set, embedded unstable periodic orbits can be extracted and that these orbits can be responsible for nearly periodic behavior lasting a substantial fraction of the observing run.
Engineering synchronization of chaotic oscillators
Padmanaban, E.; Dana, Syamal K.
2011-04-19
We propose a controller based coupling design for engineering synchronization in chaotic oscillators for unidirectional as well as bi-directional mode. In the synchronization regimes, it is possible to amplify/ attenuate a chaotic attractor with respect to other chaotic attractors. Numerical examples are presented for a Lorenz system, a Roessler oscillator, and a Sprott system. Physical implementation of the scheme is done in electronic circuit to design the controller for verification of the theory.
Langevin Equation on Fractal Curves
NASA Astrophysics Data System (ADS)
Satin, Seema; Gangal, A. D.
2016-07-01
We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the Fα-Calculus.
Eliazar, Iddo; Klafter, Joseph
2008-06-01
We explore six classes of fractal probability laws defined on the positive half-line: Weibull, Frechét, Lévy, hyper Pareto, hyper beta, and hyper shot noise. Each of these classes admits a unique statistical power-law structure, and is uniquely associated with a certain operation of renormalization. All six classes turn out to be one-dimensional projections of underlying Poisson processes which, in turn, are the unique fixed points of Poissonian renormalizations. The first three classes correspond to linear Poissonian renormalizations and are intimately related to extreme value theory (Weibull, Frechét) and to the central limit theorem (Lévy). The other three classes correspond to nonlinear Poissonian renormalizations. Pareto's law--commonly perceived as the "universal fractal probability distribution"--is merely a special case of the hyper Pareto class.
Fractal multifiber microchannel plates
NASA Technical Reports Server (NTRS)
Cook, Lee M.; Feller, W. B.; Kenter, Almus T.; Chappell, Jon H.
1992-01-01
The construction and performance of microchannel plates (MCPs) made using fractal tiling mehtods are reviewed. MCPs with 40 mm active areas having near-perfect channel ordering were produced. These plates demonstrated electrical performance characteristics equivalent to conventionally constructed MCPs. These apparently are the first MCPs which have a sufficiently high degree of order to permit single channel addressability. Potential applications for these devices and the prospects for further development are discussed.
NASA Technical Reports Server (NTRS)
Touma, Jihad; Wisdom, Jack
1993-01-01
The discovery (by Laskar, 1989, 1990) that the evolution of the solar system is chaotic, made in a numerical integration of the averaged secular approximation of the equations of motions for the planets, was confirmed by Sussman and Wisdom (1992) by direct numerical integration of the whole solar system. This paper presents results of direct integrations of the rotation of Mars in the chaotically evolved planetary system, made using the same model as that used by Sussman and Wisdom. The numerical integration shows that the obliquity of Mars undergoes large chaotic variations, which occur as the system evolves in the chaotic zone associated with a secular spin-orbit resonance.
Chaotic signal reconstruction with application to noise radar system
NASA Astrophysics Data System (ADS)
Liu, Lidong; Hu, Jinfeng; He, Zishu; Han, Chunlin; Li, Huiyong; Li, Jun
2011-12-01
Chaotic signals are potentially attractive in engineering applications, most of which require an accurate estimation of the actual chaotic signal from a noisy background. In this article, we present an improved symbolic dynamics-based method (ISDM) for accurate estimating the initial condition of chaotic signal corrupted by noise. Then, a new method, called piecewise estimation method (PEM), for chaotic signal reconstruction based on ISDM is proposed. The reconstruction performance using PEM is much better than that using the existing initial condition estimation methods. Next, PEM is applied in a noncoherent reception noise radar scheme and an improved noncoherent reception scheme is given. The simulation results show that the improved noncoherent scheme has better correlation performance and range resolution especially at low signal-to-noise ratios (SNRs).
Investigating Fractal Geometry Using LOGO.
ERIC Educational Resources Information Center
Thomas, David A.
1989-01-01
Discusses dimensionality in Euclidean geometry. Presents methods to produce fractals using LOGO. Uses the idea of self-similarity. Included are program listings and suggested extension activities. (MVL)
NASA Astrophysics Data System (ADS)
Burdzy, Krzysztof; Hołyst, Robert; Pruski, Łukasz
2013-05-01
We investigate a process of random walks of a point particle on a two-dimensional square lattice of size n×n with periodic boundary conditions. A fraction p⩽20% of the lattice is occupied by holes (p represents macroporosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M, can be pushed aside. The system approaches equilibrium in (nlnn)2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F(M) is given by Mγ where γ=-1.18 for p=0.1, decreasing to γ=-1.28 for p=0.01. Irrespective of the initial distribution of holes on the lattice, the final equilibrium distribution of holes forms a fractal with fractal dimension changing from a=1.56 for p=0.20 to a=1.42 for p=0.001 (for n=4,000). The trace of a random walker forms a distribution with expected fractal dimension 2.
Darwinian Evolution and Fractals
NASA Astrophysics Data System (ADS)
Carr, Paul H.
2009-05-01
Did nature's beauty emerge by chance or was it intelligently designed? Richard Dawkins asserts that evolution is blind aimless chance. Michael Behe believes, on the contrary, that the first cell was intelligently designed. The scientific evidence is that nature's creativity arises from the interplay between chance AND design (laws). Darwin's ``Origin of the Species,'' published 150 years ago in 1859, characterized evolution as the interplay between variations (symbolized by dice) and the natural selection law (design). This is evident in recent discoveries in DNA, Madelbrot's Fractal Geometry of Nature, and the success of the genetic design algorithm. Algorithms for generating fractals have the same interplay between randomness and law as evolution. Fractal statistics, which are not completely random, characterize such phenomena such as fluctuations in the stock market, the Nile River, rainfall, and tree rings. As chaos theorist Joseph Ford put it: God plays dice, but the dice are loaded. Thus Darwin, in discovering the evolutionary interplay between variations and natural selection, was throwing God's dice!
The Use of Fractals for the Study of the Psychology of Perception:
NASA Astrophysics Data System (ADS)
Mitina, Olga V.; Abraham, Frederick David
The present article deals with perception of time (subjective assessment of temporal intervals), complexity and aesthetic attractiveness of visual objects. The experimental research for construction of functional relations between objective parameters of fractals' complexity (fractal dimension and Lyapunov exponent) and subjective perception of their complexity was conducted. As stimulus material we used the program based on Sprott's algorithms for the generation of fractals and the calculation of their mathematical characteristics. For the research 20 fractals were selected which had different fractal dimensions that varied from 0.52 to 2.36, and the Lyapunov exponent from 0.01 to 0.22. We conducted two experiments: (1) A total of 20 fractals were shown to 93 participants. The fractals were displayed on the screen of a computer for randomly chosen time intervals ranging from 5 to 20 s. For each fractal displayed, the participant responded with a rating of the complexity and attractiveness of the fractal using ten-point scale with an estimate of the duration of the presentation of the stimulus. Each participant also answered the questions of some personality tests (Cattell and others). The main purpose of this experiment was the analysis of the correlation between personal characteristics and subjective perception of complexity, attractiveness, and duration of fractal's presentation. (2) The same 20 fractals were shown to 47 participants as they were forming on the screen of the computer for a fixed interval. Participants also estimated subjective complexity and attractiveness of fractals. The hypothesis on the applicability of the Weber-Fechner law for the perception of time, complexity and subjective attractiveness was confirmed for measures of dynamical properties of fractal images.
Reengineering through natural structures: the fractal factory
NASA Astrophysics Data System (ADS)
Sihn, Wilfried
1995-08-01
Many branches of European industry have had to recognize that their lead in the world market has been caught up with, particularly through Asian competition. In many cases a deficit of up to 30% in costs and productivity already exists. The reasons are rigid, Tayloristic company structures. The companies are not in a position to react flexibly to constantly changing environmental conditions. This article illustrates the methods of the `fractal company' which are necessary to solve the structure crisis. The fractal company distinguishes itself through its dynamics and its vitality, as well as its independent reaction to the changing circumstances. The developed methods, procedures, and framework conditions such as company structuring, human networking, hierarchy formation, and models for renumeration and working time are explained. They are based on practical examples from IPA's work with the automobile industry, their suppliers, and the engineering industry.
Differential Diagnosis: Shape and Function, Fractal Tools in the Pathology Lab.
Bianciardi, Giorgio
2015-10-01
Fractal analysis is a useful objective tool in describing complexity of shapes and signals providing information for understanding pathological changes. We present fractal approaches and software used in our pathology laboratory to analyze shapes of tumors in tissues and cells, to evaluate the microvessel network complexity in hereditary diseases or the complexity of the surface of blood cells in atherosclerosis-linked condition, as well to analyze function in vasculopathic subjects by chaotic analysis of electrocardiographic signals, in order to perform differential diagnosis. The fractal parameters appear to converge towards distinct values in pathological conditions compared to healthy, approaching the characteristics values of a percolation process or the diffusion-limited aggregation process, respectively: a bifurcation that allows to support the diagnostic process of the pathologist in his daily work. These methods, presented here as a kind of a cookbook ready for the pathologist, are low cost and not time consuming. PMID:26375935
Differential Diagnosis: Shape and Function, Fractal Tools in the Pathology Lab.
Bianciardi, Giorgio
2015-10-01
Fractal analysis is a useful objective tool in describing complexity of shapes and signals providing information for understanding pathological changes. We present fractal approaches and software used in our pathology laboratory to analyze shapes of tumors in tissues and cells, to evaluate the microvessel network complexity in hereditary diseases or the complexity of the surface of blood cells in atherosclerosis-linked condition, as well to analyze function in vasculopathic subjects by chaotic analysis of electrocardiographic signals, in order to perform differential diagnosis. The fractal parameters appear to converge towards distinct values in pathological conditions compared to healthy, approaching the characteristics values of a percolation process or the diffusion-limited aggregation process, respectively: a bifurcation that allows to support the diagnostic process of the pathologist in his daily work. These methods, presented here as a kind of a cookbook ready for the pathologist, are low cost and not time consuming.
Is the chaotic clock ticking correctly?
NASA Astrophysics Data System (ADS)
Knežević, Z.; Jovanović, B.
1997-12-01
The authors performed an extensive analysis of the dynamics of all presently known members of the Veritas asteroid family, in order to check the estimate of its age given originally by Milani and Farinella (1994), who used the so-called "chaotic chronology" method for this purpose. The authors started from a much larger sample of family members and carried out many more numerical integrations of the entire family and, in particular, of the chaotic bodies. They show that the dynamics in the region of the phase space occupied by the family is more complex than previously believed, and that there are bodies with motions ranging from remarkable stability to strong chaos. Seven strongly chaotic bodies exhibit a qualitatively similar behavior, in agreement with the predictions of Milani et al. (1996), but also a number of previously unknown features. Although, in general, the authors' results do not contradict the conclusions of Milani and Farinella and fit reasonably well with their estimate of the family age, there are also some interesting results of this analysis, which open new questions and require a more thorough and dedicated investigation.
Covariant Lyapunov analysis of chaotic Kolmogorov flows.
Inubushi, Masanobu; Kobayashi, Miki U; Takehiro, Shin-ichi; Yamada, Michio
2012-01-01
Hyperbolicity is an important concept in dynamical system theory; however, we know little about the hyperbolicity of concrete physical systems including fluid motions governed by the Navier-Stokes equations. Here, we study numerically the hyperbolicity of the Navier-Stokes equation on a two-dimensional torus (Kolmogorov flows) using the method of covariant Lyapunov vectors developed by Ginelli et al. [Phys. Rev. Lett. 99, 130601 (2007)]. We calculate the angle between the local stable and unstable manifolds along an orbit of chaotic solution to evaluate the hyperbolicity. We find that the attractor of chaotic Kolmogorov flows is hyperbolic at small Reynolds numbers, but that smaller angles between the local stable and unstable manifolds are observed at larger Reynolds numbers, and the attractor appears to be nonhyperbolic at a certain Reynolds numbers. Also, we observed some relations between these hyperbolic properties and physical properties such as time correlation of the vorticity and the energy dissipation rate.