Sporadically Fractal Basin Boundaries of Chaotic Systems
Hunt, B.R.; Ott, E.; Rosa, E. Jr.
1999-05-01
We demonstrate a new type of basin boundary for typical chaotic dynamical systems. For the case of a two dimensional map, this boundary has the character of the graph of a function that is smooth and differentiable except on a set of fractal dimensions less than one. In spite of the basin boundary being smooth {open_quotes}almost everywhere,{close_quotes} its fractal dimension exceeds one (implying degradation of one{close_quote}s ability to predict the attractor an orbit approaches in the presence of small initial condition uncertainty). We call such a boundary {ital sporadically fractal}. {copyright} {ital 1999} {ital The American Physical Society}
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.
The chaotic atom model via a fractal approximation of motion
NASA Astrophysics Data System (ADS)
Agop, M.; Nica, P.; Gurlui, S.; Focsa, C.; Magop, D.; Borsos, Z.
2011-10-01
A new model of the atom is built based on a complete and detailed nonlinear dynamics analysis (complete time series, Poincaré sections, complete phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis), through the correlation of the chaotic-stochastic model with a fractal one. Some specific mechanisms that ensure the atom functionality are proposed: gun, chaotic gun and multi-gun effects for the excited states (the classical analogue of quantum absorption) and the fractalization of the trajectories for the stationary states (a natural way of introducing the quantification).
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.
Fractal boundaries in open hydrodynamical flows: Signatures of chaotic saddles
NASA Astrophysics Data System (ADS)
Péntek, Áron; Toroczkai, Zoltán; Tél, Tamás; Grebogi, Celso; Yorke, James A.
1995-05-01
We introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently. It is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors. The fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set. We point out that these boundaries coincide with streak lines passing through appropriately chosen points. As an illustrative numerical experiment, we consider a model of the von Kármán vortex street, a time periodic two-dimensional flow of a viscous fluid around a cylinder.
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.
Is a chaotic multi-fractal approach for rainfall possible?
NASA Astrophysics Data System (ADS)
Sivakumar, Bellie
2001-04-01
Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono-fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi-fractal approaches. The assumption behind multi-fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi-fractals. However, it has been observed recently that multi-fractal approaches might provide positive evidence of a multi-fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi-fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi-fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi-fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi-fractal approach for
Experiments in chaotic dynamics
NASA Astrophysics Data System (ADS)
Moon, F. C.
Mathematical tools for the description of chaotic phenomena in physical systems are described and demonstrated, summarizing in part the principles presented in the author's book-length treatise on chaotic vibrations (Moon, 1987). Consideration is given to phase-plane and pseudo-phase-plane techniques, bifurcation diagrams, FFTs, autocorrelation functions, single and double Poincare maps, reduction to one-dimensional maps, Liapunov exponents, fractal dimensions, invariant distributions, chaos diagrams, and basin-boundary diagrams. The results obtained by application of these methods to data from typical mechanical and electronic oscillation experiments are presented graphically and discussed in detail.
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).
Dynamical fractional chaotic inflation
NASA Astrophysics Data System (ADS)
Harigaya, Keisuke; Ibe, Masahiro; Schmitz, Kai; Yanagida, Tsutomu T.
2014-12-01
Chaotic inflation based on a simple monomial scalar potential, V (ϕ )∝ϕp, is an attractive large-field model of inflation capable of generating a sizable tensor-to-scalar ratio r . Therefore, assuming that future cosmic microwave background observations will confirm the large r value reported by BICEP2, it is important to determine what kind of dynamical mechanism could possibly endow the inflaton field with such a simple effective potential. In this paper, we answer this question in the context of field theory, i.e. in the framework of dynamical chaotic inflation, where strongly interacting supersymmetric gauge dynamics around the scale of grand unification dynamically generate a fractional power-law potential via the quantum effect of dimensional transmutation. In constructing explicit models, we significantly extend our previous work, as we now consider a large variety of possible underlying gauge dynamics and relax our conditions on the field content of the model. This allows us to realize almost arbitrary rational values for the power p in the inflaton potential. The present paper may hence be regarded as a first step toward a more complete theory of dynamical chaotic inflation.
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.
Fractal scattering dynamics of the three-dimensional HOCl molecule
NASA Astrophysics Data System (ADS)
Lin, Yi-Der; Barr, Alex M.; Reichl, L. E.; Jung, Christof
2013-01-01
We compare the 2D and 3D classical fractal scattering dynamics of Cl and HO for energies just above dissociation of the HOCl molecule, using a realistic potential energy surface for the HOCl molecule and techniques developed to analyze 3D chaotic scattering processes. For parameter regimes where the HO dimer initially has small vibrational energy, only small intervals of initial conditions show fractal scattering behavior and the scattering process is well described by a 2D model. For parameter regimes where the HO dimer initially has large vibrational energy, the scattering process is fully 3D and is dominated by fractal behavior.
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.
Bialek, J.M.
1988-01-01
Chaotic behavior may be observed in deterministic Hamiltonian Systems with as few as three dimensions, i.e, X, P, and t. The amount of chaotic behavior depends on the relative influence of the integrable and non-integrable parts of the Hamiltonian. The Standard Map is such a system and the amount of chaotic behavior may be varied by adjusting a single parameter. The global phase space portrait is a complicated mixture of quiescent and chaotic regions. First a new calculational method, characterized by a Fractual Diagram, is presented. This allows the quantitative prediction of the boundaries between regular and chaotic regions in phase space. Where these barriers are located gives qualitative insight into diffusion in phase space. The method is illustrated with the Standard Map but may be applied to any Hamiltonian System. The second phenomenon is the Universal Behavior predicted to occur for all area preserving maps. As a parameter is varied causing the mapping to become more chaotic a pattern is observed in the location and stability of the fixed points of the maps. The fixed points undergo an infinite sequence of period doubling bifurcations in a finite range of the parameter. The relative locations of the fixed point bifurcation and the parameter intervals between bifurcations both asymptotically approach constants which are Universal in that the same constants keep appearing in different problems.
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.
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.
Nonlinear dynamics in a microfluidic loop device: Chaos and Fractals
NASA Astrophysics Data System (ADS)
Maddala, Jeevan; Rengaswamy, Raghunathan
2012-11-01
Discrete decision making and resistive interactions between droplets in a microfluidic loop device induces fascinating nonlinear dynamics such as multi-stability and period doubling. Droplets entering the device at fixed time intervals can exit at different periods or chaotically. One of the periodic behaviors that is observed in a loop is the three-period behavior; this is consistent with the notion that three period behavior implies chaos. Switching between these different dynamical regimes is achieved by changing the inlet droplet feeding frequency. Chaotic behavior is observed between islands of periodic behavior. We show through simulations and experimental observations that the transitions between periods are indeed chaotic. Network model is used to study the dynamic behavior for different inlet feeding frequencies resulting in the development of a bifurcation map. The bifurcation map shows that the three period dynamics is preceded by chaos. A Lyapunov exponent is used to further validate these results. The exit droplet spacing shows several fascinating patterns when the model is simulated for a large number of droplets in the chaotic regime. One such chaotic regime produces a fractal that has a boundary of cardioid. The correlation dimension for a fractal pattern produced by this particular loop system is calculated to be 0.7.
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).
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.
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.
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. PMID:25019866
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.
Fractal boundaries in magnetotail particle dynamics
NASA Astrophysics Data System (ADS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-07-01
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 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.
Dynamic visual cryptography based on chaotic oscillations
NASA Astrophysics Data System (ADS)
Petrauskiene, Vilma; Palivonaite, Rita; Aleksa, Algiment; Ragulskis, Minvydas
2014-01-01
Dynamic visual cryptography scheme based on chaotic oscillations is proposed in this paper. Special computational algorithms are required for hiding the secret image in the cover moiré grating, but the decryption of the secret is completely visual. The secret image is leaked in the form of time-averaged geometric moiré fringes when the cover image is oscillated by a chaotic law. The relationship among the standard deviation of the stochastic time variable, the pitch of the moiré grating and the pixel size ensuring visual decryption of the secret is derived. The parameters of these chaotic oscillations must be carefully preselected before the secret image is leaked from the cover image. Several computational experiments are used to illustrate the functionality and the applicability of the proposed image hiding technique.
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…
Chaotic dynamics of a magnetic nanoparticle.
Bragard, J; Pleiner, H; Suarez, O J; Vargas, P; Gallas, J A C; Laroze, D
2011-09-01
We study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a magnetic field with a constant longitudinal and a time-dependent transverse component using the Landau-Lifshitz-Gilbert equation. We characterize the dynamical behavior of the system through calculation of the Lyapunov exponents, Poincaré sections, bifurcation diagrams, and Fourier power spectra. In particular we explore the positivity of the largest Lyapunov exponent as a function of the magnitude and frequency of the applied magnetic field and its direction with respect to the main anisotropy axis of the magnetic particle. We find that the system presents multiple transitions between regular and chaotic behaviors. We show that the dynamical phases display a very complicated structure of intricately intermingled chaotic and regular phases. PMID:22060537
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
Urey Prize Lecture - Chaotic dynamics in the solar system
NASA Astrophysics Data System (ADS)
Wisdom, J.
1987-11-01
Newton's equations have chaotic solutions as well as regular solutions. There are several physical situations in the solar system where chaotic solutions of Newton's equations play an important role. There are examples of both chaotic rotation and chaotic orbital evolution. Hyperion is currently tumbling chaotically. Many of the other irregularly shaped satellites in the solar system have had chaotic rotations in the past. This episode of chaotic tumbling could have had a significant effect on the orbital histories of these satellites. Chaotic orbital evolution seems to be an essential ingredient in the explanation of the Kirkwood gaps in the distribution of asteroids. Chaotic trajectories at the 3/1 commensurability have the correct properties to provide a dynamical route for the transport of meteoritic material from the asteroid belt to Earth.
Chaotic spectra: How to extract dynamic information
Taylor, H.S.; Gomez Llorente, J.M.; Zakrzewski, J.; Kulander, K.C.
1988-10-01
Nonlinear dynamics is applied to chaotic unassignable atomic and molecular spectra with the aim of extracting detailed information about regular dynamic motions that exist over short intervals of time. It is shown how this motion can be extracted from high resolution spectra by doing low resolution studies or by Fourier transforming limited regions of the spectrum. These motions mimic those of periodic orbits (PO) and are inserts into the dominant chaotic motion. Considering these inserts and the PO as a dynamically decoupled region of space, resonant scattering theory and stabilization methods enable us to compute ladders of resonant states which interact with the chaotic quasi-continuum computed in principle from basis sets placed off the PO. The interaction of the resonances with the quasicontinuum explains the low resolution spectra seen in such experiments. It also allows one to associate low resolution features with a particular PO. The motion on the PO thereby supplies the molecular movements whose quantization causes the low resolution spectra. Characteristic properties of the periodic orbit based resonances are discussed. The method is illustrated on the photoabsorption spectrum of the hydrogen atom in a strong magnetic field and on the photodissociation spectrum of H/sub 3//sup +/. Other molecular systems which are currently under investigation using this formalism are also mentioned. 53 refs., 10 figs., 2 tabs.
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.
Topological analysis of chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Gilmore, Robert
1998-10-01
Topological methods have recently been developed for the analysis of dissipative dynamical systems that operate in the chaotic regime. They were originally developed for three-dimensional dissipative dynamical systems, but they are applicable to all ``low-dimensional'' dynamical systems. These are systems for which the flow rapidly relaxes to a three-dimensional subspace of phase space. Equivalently, the associated attractor has Lyapunov dimension dL<3. Topological methods supplement methods previously developed to determine the values of metric and dynamical invariants. However, topological methods possess three additional features: they describe how to model the dynamics; they allow validation of the models so developed; and the topological invariants are robust under changes in control-parameter values. The topological-analysis procedure depends on identifying the stretching and squeezing mechanisms that act to create a strange attractor and organize all the unstable periodic orbits in this attractor in a unique way. The stretching and squeezing mechanisms are represented by a caricature, a branched manifold, which is also called a template or a knot holder. This turns out to be a version of the dynamical system in the limit of infinite dissipation. This topological structure is identified by a set of integer invariants. One of the truly remarkable results of the topological-analysis procedure is that these integer invariants can be extracted from a chaotic time series. Furthermore, self-consistency checks can be used to confirm the integer values. These integers can be used to determine whether or not two dynamical systems are equivalent; in particular, they can determine whether a model developed from time-series data is an accurate representation of a physical system. Conversely, these integers can be used to provide a model for the dynamical mechanisms that generate chaotic data. In fact, the author has constructed a doubly discrete classification of strange
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.
Chaotic Dynamics of an Elastically Bouncing Dumbbell
NASA Astrophysics Data System (ADS)
Rees, Colin; Franklin, Scott
2009-03-01
The dynamics of an elastically bouncing dumbbell is analogous to those of an ball bouncing on a sinusoidally oscillating surface with one important exception: the dumbbell's angular velocity, analogous to the surface's oscillation frequency, changes with each bounce, making the subsequent motion significantly more complicated. We investigate this dynamical system over a range of aspect ratios and initial energy, finding periodic, quasi-periodic and chaotic motions. As the initial energy is increased, the dumbbell can flip over and tumble. We find for large aspect ratios, however, narrow bands of energies well above this minimum where tumbling suddenly ceases. Because energy is conserved, the dynamics of a bounce are uniquely determined by the angle and angular velocity. The Lyapunov exponents of paths in this two dimensional phase space can be calculated, with the hope of identifying periodic islands within the chaotic sea. Finally, for certain parameters, the angle at each collision moves from its initial value in a subdiffusive manner, and we determine the characteristic exponents.
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)
Altmann, Eduardo G.; Portela, Jefferson S. E.; Tél, Tamás
2015-02-01
We investigate chaotic dynamical systems for which the intensity of trajectories might grow unlimited in time. We show that i) the intensity grows exponentially in time and is distributed spatially according to a fractal measure with an information dimension smaller than that of the phase space, ii) such exploding cases can be described by an operator formalism similar to the one applied to chaotic systems with absorption (decaying intensities), but iii) the invariant quantities characterizing explosion and absorption are typically not directly related to each other, e.g., the decay rate and fractal dimensions of absorbing maps typically differ from the ones computed in the corresponding inverse (exploding) maps. We illustrate our general results through numerical simulation in the cardioid billiard mimicking a lasing optical cavity, and through analytical calculations in the baker map.
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.
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.
Chaotic Dynamics in Partial Differential Equations.
NASA Astrophysics Data System (ADS)
Li, Yanguang
The existence of chaotic behavior, for a certain damped and driven perturbation of the nonlinear Schroedinger equation under even periodic boundary conditions, is established. More specifically, the existence of a symmetric pair of homoclinic orbits is established for the perturbed NLS equation through two main arguments: Argument 1 is a combination of Melnikov analysis and a geometric singular perturbation theory for the pde. The geometric singular perturbation theory involves the theory of persistence of invariant manifolds for the pde and the theory of Hadamard-Fenichel fiber coordinatization for those invariant manifolds. Argument 2 is a purely geometric argument. Finally, an argument is sketched which, we believe, provides a core of an existence proof for Smale "horseshoes" and a symbolic dynamics in a neighborhood of the persistent homoclinic orbits.
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.
Fractal templates in the escape dynamics of trapped ultracold atoms
Mitchell, Kevin A.; Steck, Daniel A.
2007-09-15
We consider the dynamic escape of a small packet of ultracold atoms launched from within an optical dipole trap. Based on a theoretical analysis of the underlying nonlinear dynamics, we predict that fractal behavior can be seen in experimental escape data. These data can be collected by measuring the time-dependent escape rate for packets launched over a range of angles. This fractal pattern is particularly well resolved below the Bose-Einstein transition temperature - a direct result of the extreme phase-space localization of the condensate. We predict that several self-similar layers of this novel fractal should be measurable, and we explain how this fractal pattern can be predicted and analyzed with recently developed techniques in symbolic dynamics.
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.
Chaotic dynamics of weakly nonlinear systems
Vavriv, D.M.
1996-06-01
A review is given on the recent results in studying chaotic phenomena in weakly nonlinear systems. We are concerned with the class of chaotic states that can arise in physical systems with any degree of nonlinearity however small. The conditions for, and the mechanisms of, the transitions to chaos are discussed. These findings are illustrated by the results of the stability analysis of practical microwave and optical devices. {copyright} {ital 1996 American Institute of Physics.}
Chaotic dynamics in a simple dynamical green ocean plankton model
NASA Astrophysics Data System (ADS)
Cropp, Roger; Moroz, Irene M.; Norbury, John
2014-11-01
The exchange of important greenhouse gases between the ocean and atmosphere is influenced by the dynamics of near-surface plankton ecosystems. Marine plankton ecosystems are modified by climate change creating a feedback mechanism that could have significant implications for predicting future climates. The collapse or extinction of a plankton population may push the climate system across a tipping point. Dynamic green ocean models (DGOMs) are currently being developed for inclusion into climate models to predict the future state of the climate. The appropriate complexity of the DGOMs used to represent plankton processes is an ongoing issue, with models tending to become more complex, with more complicated dynamics, and an increasing propensity for chaos. We consider a relatively simple (four-population) DGOM of phytoplankton, zooplankton, bacteria and zooflagellates where the interacting plankton populations are connected by a single limiting nutrient. Chaotic solutions are possible in this 4-dimensional model for plankton population dynamics, as well as in a reduced 3-dimensional model, as we vary two of the key mortality parameters. Our results show that chaos is robust to the variation of parameters as well as to the presence of environmental noise, where the attractor of the more complex system is more robust than the attractor of its simplified equivalent. We find robust chaotic dynamics in low trophic order ecological models, suggesting that chaotic dynamics might be ubiquitous in the more complex models, but this is rarely observed in DGOM simulations. The physical equations of DGOMs are well understood and are constrained by conservation principles, but the ecological equations are not well understood, and generally have no explicitly conserved quantities. This work, in the context of the paucity of the empirical and theoretical bases upon which DGOMs are constructed, raises the interesting question of whether DGOMs better represent reality if they include
Design of Microlasers and Beam Splitters using chaotic ray dynamics
NASA Astrophysics Data System (ADS)
Luna-Acosta, German A.; Mendez-Bermudez, J. Antonio; Bewdix, Oliver
2005-03-01
. We consider chaotic waveguides formed by single or multiple 2D chaotic cavities connected to leads. The cavities are chaotic in the sense that the ray/particle dynamics within them is chaotic. Specifically the phase space is mixed, with chaotic regions surrounding stable islands where motion is regular. Stable islands are inaccessible to the incoming rays/particles. In contrast, incoming plane waves can dynamically tunnel into them at a certain set of discrete values of frequency/energy. The support of the corresponding quasi-bound state is along the trajectories of periodic orbits trapped within the cavity. We take advantage of this difference in the ray/wave behavior to demonstrate how chaotic waveguides, electromagnetic or electronic, can be used to design beam splitters and microlasers[1]. We also present some preliminary experimental results in a microwave realization of a chaotic waveguide. [1] J. A. M'edez-Berm'udez, G. A. Luna-Acosta, P. Seba, and K. N. Pichugin, Phys. Rev. B 67, 161104(R) (2003).
Dynamical features of reaction-diffusion fronts in fractals.
Méndez, Vicenç; Campos, Daniel; Fort, Joaquim
2004-01-01
The speed of front propagation in fractals is studied by using (i) the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and (ii) the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration. PMID:14995742
Chaotic dynamics of loosely supported tubes in crossflow
Cai, Y.; Chen, S.S.
1991-07-01
By means of the unsteady-flow theory and a bilinear mathematical model, a theoretical study was conducted of the chaotic dynamics associated with the fluidelastic instability of loosely supported tubes. Calculations were performed for the RMS of tube displacement, bifurcation diagram, phase portrait, power spectral density, and Poincare map. Analytical results show the existence of chaotic, quasiperiodic, and periodic regions when flow velocity exceeds a threshold value. 38 refs., 15 figs., 2 tabs.
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
Chaotic dynamics in flow through unsaturated fractured media
NASA Astrophysics Data System (ADS)
Faybishenko, Boris
Predictions of flow and transport within fractured rock in the vadose zone cannot be made without first characterizing the physics of unstable flow phenomena in unsaturated fractures. This paper introduces a new approach for studying complex flow processes in heterogeneous fractured media, using the methods of nonlinear dynamics and chaos--in particular reconstructing the system dynamics and calculating chaotic diagnostic parameters from time-series data. To demonstrate the application of chaotic analysis, this author analyzed the time-series pressure fluctuations from two water-air flow experiments conducted by Persoff and Pruess [Water Resour. Res. 31 (1995) 1175] in replicas of rough-walled rock fractures under controlled boundary conditions. This analysis showed that chaotic flow in fractures creates relaxational oscillations of liquid, gas, and capillary pressures. These pressure oscillations were used to calculate the diagnostic parameters of deterministic chaos, including correlation time, global embedding dimension, local embedding dimension, Lyapunov dimension, Lyapunov exponents, and correlation dimension. The results of the Persoff-Pruess experiments were then compared with the chaotic analysis of laboratory dripping-water experiments in fracture models and field-infiltration experiments in fractured basalt. This comparison allowed us to conjecture that intrinsic fracture flow and dripping, as well as extrinsic water dripping (from a fracture) subjected to a capillary-barrier effect, are deterministic-chaotic processes with a certain random component. The unsaturated fractured rock is a dynamic system that exhibits chaotic behavior because the flow processes are nonlinear, dissipative, and sensitive to initial conditions, with chaotic fluctuations generated by intrinsic properties of the system, not random external factors. Identifying a system as deterministically chaotic is important for developing appropriate short- and long-term prediction models
Chaotic dynamics in a two-dimensional optical lattice.
Horsley, Eric; Koppell, Stewart; Reichl, L E
2014-01-01
The classical nonlinear dynamics of a dilute gas of rubidium atoms in an optical lattice is studied for a range of polarizations of the laser beams forming the lattice. The dynamics ranges from integrable to chaotic, and mechanisms leading to the onset of chaos in the lattice are described. PMID:24580307
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.
Quantifying chaotic dynamics from integrate-and-fire processes
NASA Astrophysics Data System (ADS)
Pavlov, A. N.; Pavlova, O. N.; Mohammad, Y. K.; Kurths, J.
2015-01-01
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.
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
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.
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
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.
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.
Quantized chaotic dynamics and non-commutative KS entropy
Klimek, S.; Lesniewski, A.
1996-06-01
We study the quantization of two examples of classically chaotic dynamics, the Anosov dynamics of {open_quote}{open_quote}cat maps{close_quote}{close_quote} on a two dimensional torus, and the dynamics of baker{close_quote}s maps. Each of these dynamics is implemented as a discrete group of automorphisms of a von Neumann algebra of functions on a quantized torus. We compute the non-commutative generalization of the Kolmogorov-Sinai entropy, namely the Connes-Sto/rmer entropy, of the generator of this group, and find that its value is equal to the classical value. This can be interpreted as a sign of persistence of chaotic behavior in a dynamical system under quantization. Copyright {copyright} 1996 Academic Press, Inc.
Coevolutionary extremal dynamics on gasket fractal
NASA Astrophysics Data System (ADS)
Lee, Kyoung Eun; Sung, Joo Yup; Cha, Moon-Yong; Maeng, Seong Eun; Bang, Yu Sik; Lee, Jae Woo
2009-11-01
We considered a Bak-Sneppen model on a Sierpinski gasket fractal. We calculated the avalanche size distribution and the distribution of distances between subsequent minimal sites. To observe the temporal correlations of the avalanche, we estimated the return time distribution, the first-return time, and the all-return time distribution. The avalanche size distribution follows the power law, P(s)∼s, with the exponent τ=1.004(7). The distribution of jumping sites also follows the power law, P(r)∼r, with the critical exponent π=4.12(4). We observe the periodic oscillation of the distribution of the jumping distances which originated from the jumps of the level when the minimal site crosses the stage of the fractal. The first-return time distribution shows the power law, P(t)∼t, with the critical exponent τ=1.418(7). The all-return time distribution is also characterized by the power law, P(t)∼t, with the exponent τ=0.522(4). The exponents of the return time satisfy the scaling relation τ+τ=2 for τ⩽2.
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.
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. PMID:25273196
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.
Long-Range Correlations in Stride Intervals May Emerge from Non-Chaotic Walking Dynamics
Ahn, Jooeun; Hogan, Neville
2013-01-01
Stride intervals of normal human walking exhibit long-range temporal correlations. Similar to the fractal-like behaviors observed in brain and heart activity, long-range correlations in walking have commonly been interpreted to result from chaotic dynamics and be a signature of health. Several mathematical models have reproduced this behavior by assuming a dominant role of neural central pattern generators (CPGs) and/or nonlinear biomechanics to evoke chaos. In this study, we show that a simple walking model without a CPG or biomechanics capable of chaos can reproduce long-range correlations. Stride intervals of the model revealed long-range correlations observed in human walking when the model had moderate orbital stability, which enabled the current stride to affect a future stride even after many steps. This provides a clear counterexample to the common hypothesis that a CPG and/or chaotic dynamics is required to explain the long-range correlations in healthy human walking. Instead, our results suggest that the long-range correlation may result from a combination of noise that is ubiquitous in biological systems and orbital stability that is essential in general rhythmic movements. PMID:24086274
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 around astrophysical objects with nonisotropic stresses
Dubeibe, F. L.; Pachon, Leonardo A.; Sanabria-Gomez, Jose D.
2007-01-15
The existence of chaotic behavior for the geodesics of the test particles orbiting compact objects is a subject of much current research. Some years ago, Gueron and Letelier [Phys. Rev. E 66, 046611 (2002)] reported the existence of chaotic behavior for the geodesics of the test particles orbiting compact objects like black holes induced by specific values of the quadrupolar deformation of the source using as models the Erez--Rosen solution and the Kerr black hole deformed by an internal multipole term. In this work, we are interested in the study of the dynamic behavior of geodesics around astrophysical objects with intrinsic quadrupolar deformation or nonisotropic stresses, which induces nonvanishing quadrupolar deformation for the nonrotating limit. For our purpose, we use the Tomimatsu-Sato spacetime [Phys. Rev. Lett. 29 1344 (1972)] and its arbitrary deformed generalization obtained as the particular vacuum case of the five parametric solution of Manko et al. [Phys. Rev. D 62, 044048 (2000)] characterizing the geodesic dynamics throughout the Poincare sections method. We found only regular motion for the geodesics in the Tomimatsu-Sato {delta}=2 solution. Additionally, using the deformed generalization of Tomimatsu-Sato {delta}=2 solution given by Manko et al. we found chaotic motion for oblate deformation instead of prolate deformation, which is in contrast to the results by Gueron and Letelier. It opens the possibility that the particles forming the accretion disk around a large variety of different astrophysical bodies (nonprolate, e.g., neutron stars) could exhibit chaotic dynamics. We also conjecture that the existence of an arbitrary deformation parameter is necessary for the existence of chaotic dynamics.
Hypotheses on the functional roles of chaotic transitory dynamics
NASA Astrophysics Data System (ADS)
Tsuda, Ichiro
2009-03-01
In contrast to the conventional static view of the brain, recent experimental data show that an alternative view is necessary for an appropriate interpretation of its function. Some selected problems concerning the cortical transitory dynamics are discussed. For the first time, we propose five scenarios for the appearance of chaotic itinerancy, which provides typical transitory dynamics. Second, we describe the transitory behaviors that have been observed in human and animal brains. Finally, we propose nine hypotheses on the functional roles of such dynamics, focusing on the dynamics embedded in data and the dynamical interpretation of brain activity within the framework of cerebral hermeneutics.
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.
Fractal and complex network analyses of protein molecular dynamics
NASA Astrophysics Data System (ADS)
Zhou, Yuan-Wu; Liu, Jin-Long; Yu, Zu-Guo; Zhao, Zhi-Qin; Anh, Vo
2014-12-01
Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h(q) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h(2) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H≈0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h(2) of MF-DFA on the time series, exponent λ of the exponential degree distribution and fractal dimension dB of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between
Synchronization in complex dynamical networks coupled with complex chaotic system
NASA Astrophysics Data System (ADS)
Wei, Qiang; Xie, Cheng-Jun; Wang, Bo
2015-11-01
This paper investigates synchronization in complex dynamical networks with time delay and perturbation. The node of complex dynamical networks is composed of complex chaotic system. A complex feedback controller is designed to realize different component of complex state variable synchronize up to different scaling complex function when complex dynamical networks realize synchronization. The synchronization scaling function is changed from real field to complex field. Synchronization in complex dynamical networks with constant delay and time-varying coupling delay are investigated, respectively. Numerical simulations show the effectiveness of the proposed method.
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
NASA Astrophysics Data System (ADS)
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(c2) 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 q0 and frequency ωp 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.
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.
Chaotic dynamics in accelerator physics. Progress report
Cary, J.R.
1992-11-30
Substantial progress was in several areas of accelerator dynamics. For developing understanding of longitudinal adiabatic dynamics, and for creating efficiency enhancements of recirculating free-electron lasers, was substantially completed. A computer code for analyzing the critical KAM tori that bound the dynamic aperture in circular machines was developed. Studies of modes that arise due to the interaction of coating beams with a narrow-spectrum impedance have begun. During this research educational and research ties with the accelerator community at large have been strengthened.
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
The geometry of chaotic dynamics — a complex network perspective
NASA Astrophysics Data System (ADS)
Donner, R. V.; Heitzig, J.; Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J.
2011-12-01
Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ɛ-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ɛ-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ɛ-recurrence networks exhibit an important link between dynamical systems and graph theory.
Chaotic electron dynamics in gyrotron resonators
Kominis, Y.; Dumbrajs, O.; Avramides, K.A.; Hizanidis, K.; Vomvoridis, J.L.
2005-04-15
Phase space analysis of electron dynamics is used in combination with the canonical perturbation method and the KAM (Kolmogorov-Arnold-Moser) theory in order to study the dependence of the efficient gyrotron operation on the rf field profile and frequency mismatch. Knowledge of the boundaries of the electron motion provided through robust (slightly distorted) KAM surfaces is useful for optimizing depressed collectors and thereby for enhancement of overall efficiency of gyrotron operation.
Chaotic dynamics in circulation with Tohoku University vibrating flow pump.
Nitta, S; Yambe, T; Kobayashi, S; Hashimoto, H; Yoshizawa, M; Mastuki, H; Tabayashi, K; Takeda, H
1999-01-01
For the development of a totally implantable ventricular assist system (VAS), we have been developing the vibrating flow pump (VFP), which can generate oscillated blood flow with a relative high frequency (10-50 Hz) for a totally implantable system. In this study, the effects of left ventricular assistance with this unique oscillated blood flow were analyzed by the use of nonlinear mathematics for evaluation as the whole circulatory regulatory system, not as the decomposed parts of the system. Left heart bypasses using the VFP from the left atrium to the descending aorta were performed in chronic animal experiments using healthy adult goats. The ECG, arterial blood pressure, VFP pump flow, and the flow of the descending aorta were recorded in the data recorder during awake conditions and analyzed in a personal computer system through an A-D convertor. By the use of nonlinear mathematics, time series data were embedded into the phase space, the Lyapunov numerical method, fractal dimension analysis, and power spectrum analysis were performed to evaluate nonlinear dynamics. During left ventricular assistance with the VFP, Mayer wave fluctuations were decreased in the power spectrum, the fractal dimension of the hemodynamics was significantly decreased, and peripheral vascular resistance was significantly decreased. These results suggest that nonlinear dynamics, which mediate the cardiovascular dynamics, may be affected during left ventricular (LV) bypass with oscillated flow. The decreased power of the Mayer wave in the spectrum caused the limit cycle attractor of the hemodynamics and decreased peripheral resistance. Decreased sympathetic discharges may be the origin of the decreased Mayer wave and fractal dimension. These nonlinear dynamic analyses may be useful to design optimal VAS control. PMID:9950190
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…
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 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. PMID:24387569
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.
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.
NASA Astrophysics Data System (ADS)
Kato, Tomohiro; Hasegawa, Mikio
Chaotic dynamics has been shown to be effective in improving the performance of combinatorial optimization algorithms. In this paper, the performance of chaotic dynamics in the asymmetric traveling salesman problem (ATSP) is investigated by introducing three types of heuristic solution update methods. Numerical simulation has been carried out to compare its performance with simulated annealing and tabu search; thus, the effectiveness of the approach using chaotic dynamics for driving heuristic methods has been shown. The chaotic method is also evaluated in the case of a combinatorial optimization problem in the real world, which can be solved by the same heuristic operation as that for the ATSP. We apply the chaotic method to the DNA fragment assembly problem, which involves building a DNA sequence from several hundred fragments obtained by the genome sequencer. Our simulation results show that the proposed algorithm using chaotic dynamics in a block shift operation exhibits the best performance for the DNA fragment assembly problem.
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.
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
Lightning and the Heart: Fractal Behavior in Cardiac Function
BASSINGTHWAIGHTE, JAMES B.; van BEEK, J. H. G. M.
2010-01-01
Physical systems, from galactic clusters to diffusing molecules, often show fractal behavior. Likewise, living systems might often be well described by fractal algorithms. Such fractal descriptions in space and time imply that there is order in chaos, or put the other way around, chaotic dynamical systems in biology are more constrained and orderly than seen at first glance. The vascular network, the syncytium of cells, the processes of diffusion and transmembrane transport might be fractal features of the heart. These fractal features provide a basis which enables one to understand certain aspects of more global behavior such as atrial or ventricular fibrillation and perfusion heterogeneity. The heart might be regarded as a prototypical organ from these points of view. A particular example of the use of fractal geometry is in explaining myocardial flow heterogeneity via delivery of blood through an asymmetrical fractal branching network. PMID:21938081
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
Fractal dynamics of electric discharges in a thundercloud.
Iudin, D I; Trakhtengerts, V Y; Hayakawa, M
2003-07-01
We have investigated the fractal dynamics of intracloud microdischarges responsible for the formation of a so-called drainage system of electric charge transport inside a cloud volume. Microdischarges are related to the nonlinear stage of multiflow instability development, which leads to the generation of a small-scale intracloud electric structure. The latter is modeled by using a two-dimensional lattice of finite-state automata. The results of numerical simulations show that the developed drainage system belongs to the percolation-cluster family. We then point out the parameter region relevant to the proposed model, in which the thundercloud exhibits behavior corresponding to a regime of self-organized criticality. The initial development and statistical properties of dynamic conductive clusters are investigated, and a kinetic equation is introduced, which permits us to find state probabilities of electric cells and to estimate macroscopic parameters of the system. PMID:12935264
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. PMID:16893802
Detection of Ordered and Chaotic Motion Using the Dynamical Spectra
NASA Astrophysics Data System (ADS)
Voglis, N.; Contopoulos, G.; Efthymiopoulos, C.
1999-01-01
Two simple and efficient numerical methods to explore the phase space structure are presented, based on the properties of the "dynamical spectra". 1) We calculate a "spectral distance" D of the dynamical spectra for two different initial deviation vectors. D → 0 in the case of chaotic orbits, while D → const ≠ 0 in the case of ordered orbits. This method is by orders of magnitude faster than the method of the Lyapunov Characteristic Number (LCN). 2) We define a sensitive indicator called ROTOR (ROtational TOri Recongnizer) for 2D maps. The ROTOR remains zero in time on a rotational torus, while it tends to infinity at a rate ∝ N = number of iterations, in any case other than a rotational torus. We use this method to locate the last KAM torus of an island of stability, as well as the most important cantori causing stickiness near it.
Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics
NASA Astrophysics Data System (ADS)
Panchuk, Anastasiia; Sushko, Iryna; Avrutin, Viktor
In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.
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.
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.
Onset of Chaotic Dynamics in Vortex Sheet Roll-Up
NASA Astrophysics Data System (ADS)
Krasny, Robert; Nitsche, Monika
1997-11-01
Vortex sheet roll-up in planar and axisymmetric geometry is studied numerically. Starting from flat initial data, the sheet rolls up into either a vortex pair or a vortex ring, depending on the geometry. The spiral roll-up proceeds smoothly at early times, but at late times the sheet develops small-scale irregular features. The outer turn becomes folded and sheds a wake behind the vortex ring, and spiral turns in the vortex core become non-uniformly spaced for both cases. These features are attributed to the onset of chaotic dynamics, specifically a heteroclinic tangle in the case of the vortex ring, and a resonance band for both cases. A Poincaré section is presented to support this conjecture. Two factors account for the onset of chaos: the presence of hyperbolic and elliptic points in the instantaneous streamline pattern, and (2) self-sustained small amplitude oscillations in the core vorticity distribution.
Chaotic Dynamics of Driven Flux Drops: A Superconducting ``Dripping Faucet''
NASA Astrophysics Data System (ADS)
Field, Stuart B.; Stan, Gheorghe
2008-02-01
When a current is applied to a type-I superconducting strip containing a narrow channel across its width, magnetic flux spots nucleate at the edge and are then driven along the channel by the current. These flux “drops” are reminiscent of water drops dripping from a faucet, a model system for studying low-dimensional chaos. We use a novel high-bandwidth Hall probe to detect in real time the motion of individual flux spots moving along the channel. Analyzing the time series consisting of the intervals between successive flux drops, we find distinct regions of chaotic behavior characterized by positive Lyapunov exponents, indicating that there is a close analogy between the dynamics of the superconducting and water drop systems.
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.
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. PMID:26871079
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.
Chaotic Soliton Dynamics in Photoexcited trans-Polyacetylene.
Bernasconi, Leonardo
2015-03-01
We study the photogeneration of topological solitons in trans-polyacetylene and their time evolution using ab initio excited-state dynamics. The system is excited to the optically allowed 1(1)Bu state, and the atoms are then propagated classically using quantum mechanical forces computed using hybrid time-dependent density functional theory (TD-DFT). A soliton/antisoliton pair nucleates spontaneously and creates two independent solitons moving at constant velocity, similar to simulations based on uncorrelated lattice models like the Su-Schrieffer-Heeger (SSH) Hamiltonian [Su, W. P.; Schrieffer, J. R.; Heeger, A. J. Phys. Rev. Lett. 1979, 42, 1698]. At T = 0, the solitons coalesce into bound pairs with a two-soliton functional form, whereas chaotic dynamics, in the form of 2-bounce resonances, is observed at soliton/antisoliton collisions at T ≠ 0. This behavior is related to the onset of a strong correlation regime at short intersoliton distance, which is not accounted for by SSH simulations. PMID:26262671
Blended particle filters for large-dimensional chaotic dynamical systems.
Majda, Andrew J; Qi, Di; Sapsis, Themistoklis P
2014-05-27
A major challenge in contemporary data science is the development of statistically accurate particle filters to capture non-Gaussian features in large-dimensional chaotic dynamical systems. Blended particle filters that capture non-Gaussian features in an adaptively evolving low-dimensional subspace through particles interacting with evolving Gaussian statistics on the remaining portion of phase space are introduced here. These blended particle filters are constructed in this paper through a mathematical formalism involving conditional Gaussian mixtures combined with statistically nonlinear forecast models compatible with this structure developed recently with high skill for uncertainty quantification. Stringent test cases for filtering involving the 40-dimensional Lorenz 96 model with a 5-dimensional adaptive subspace for nonlinear blended filtering in various turbulent regimes with at least nine positive Lyapunov exponents are used here. These cases demonstrate the high skill of the blended particle filter algorithms in capturing both highly non-Gaussian dynamical features as well as crucial nonlinear statistics for accurate filtering in extreme filtering regimes with sparse infrequent high-quality observations. The formalism developed here is also useful for multiscale filtering of turbulent systems and a simple application is sketched below. PMID:24825886
Chaotic magnetization dynamics in single-crystal thin-film structures
Shutyi, A. M. Sementsov, D. I.
2009-01-15
The nonlinear dynamics of homogeneously precessing magnetization in perpendicularly magnetized single-crystal films has been investigated in a wide range of ac field frequencies on the basis of a numerical solution to the Landau-Lifshitz equation and construction of the spectrum of Lyapunov exponents. The conditions for implementing and specific features of chaotic dynamic modes are revealed for films of three basic crystallographic orientations: (100), (110), and (111). It is shown that chaotic precession modes can be controlled using external magnetic fields. Time analogs of the Poincare section of chaotic mode trajectories are considered.
Chaotic magnetization dynamics in single-crystal thin-film structures
NASA Astrophysics Data System (ADS)
Shutyi, A. M.; Sementsov, D. I.
2009-01-01
The nonlinear dynamics of homogeneously precessing magnetization in perpendicularly magnetized single-crystal films has been investigated in a wide range of ac field frequencies on the basis of a numerical solution to the Landau-Lifshitz equation and construction of the spectrum of Lyapunov exponents. The conditions for implementing and specific features of chaotic dynamic modes are revealed for films of three basic crystallographic orientations: (100), (110), and (111). It is shown that chaotic precession modes can be controlled using external magnetic fields. Time analogs of the Poincaré section of chaotic mode trajectories are considered.
Universal behavior in the parametric evolution of chaotic saddles
Lai, Y.; Zyczkowski, K.; Grebogi, 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. {copyright} {ital 1999} {ital The American Physical Society}
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).
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.
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.
Celso Grebogi
2000-02-29
This is the final report on a research project that explored (a) controlling complex dynamical systems; (b) using controlled chaotic signals for communication (c) methods of controlling chaos via targeting; (d) deterministic modeling; and miscellaneous work on the interface between chaotic and stable periodic behavior as system parameters vary, bifurcations of non-smooth systems that describe impact oscillators; phenomena that occur in quasiperiodically forced systems, and the fractal and topological properties of chaotic inveriant sets, in particular those arising in fluid flow.
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
A fast chaotic cryptographic scheme with dynamic look-up table
NASA Astrophysics Data System (ADS)
Wong, K. W.
2002-06-01
We propose a fast chaotic cryptographic scheme based on iterating a logistic map. In particular, no random numbers need to be generated and the look-up table used in the cryptographic process is updated dynamically. Simulation results show that the proposed method leads to a substantial reduction in the encryption and decryption time. As a result, chaotic cryptography becomes more practical in the secure transmission of large multi-media files over public data communication network.
NASA Astrophysics Data System (ADS)
Igeta, Hideki; Hasegawa, Mikio
Chaotic dynamics have been effectively applied to improve various heuristic algorithms for combinatorial optimization problems in many studies. Currently, the most used chaotic optimization scheme is to drive heuristic solution search algorithms applicable to large-scale problems by chaotic neurodynamics including the tabu effect of the tabu search. Alternatively, meta-heuristic algorithms are used for combinatorial optimization by combining a neighboring solution search algorithm, such as tabu, gradient, or other search method, with a global search algorithm, such as genetic algorithms (GA), ant colony optimization (ACO), or others. In these hybrid approaches, the ACO has effectively optimized the solution of many benchmark problems in the quadratic assignment problem library. In this paper, we propose a novel hybrid method that combines the effective chaotic search algorithm that has better performance than the tabu search and global search algorithms such as ACO and GA. Our results show that the proposed chaotic hybrid algorithm has better performance than the conventional chaotic search and conventional hybrid algorithms. In addition, we show that chaotic search algorithm combined with ACO has better performance than when combined with GA.
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.
Self-affine fractal variability of human heartbeat interval dynamics in health and disease.
Meyer, M; Stiedl, O
2003-10-01
The complexity of the cardiac rhythm is demonstrated to exhibit self-affine multifractal variability. The dynamics of heartbeat interval time series was analyzed by application of the multifractal formalism based on the Cramèr theory of large deviations. The continuous multifractal large deviation spectrum uncovers the nonlinear fractal properties in the dynamics of heart rate and presents a useful diagnostic framework for discrimination and classification of patients with cardiac disease, e.g., congestive heart failure. The characteristic multifractal pattern in heart transplant recipients or chronic heart disease highlights the importance of neuroautonomic control mechanisms regulating the fractal dynamics of the cardiac rhythm. PMID:12942331
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
Experimental measurement of chaotic attractors in solid mechanics(a)).
Moon, Francis C.
1991-07-01
In this paper a review is given of experimental techniques in chaotic dynamics of solid mechanical systems based on modern ideas of nonlinear dynamics. These methods include Poincare maps, double Poincare sections, symbol dynamics, and fractal dimension. The physical problems discussed include nonlinear elastic beams, forced motion of a string, flow-induced vibration of a rod, forced motions of a magnetic pendulum, and rigid body dynamics of a magnet and high-temperature superconductor. PMID:12779894
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
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.
Regular and chaotic dynamics of magnetization precession in ferrite-garnet films
NASA Astrophysics Data System (ADS)
Shutyĭ, Anatoliy M.; Sementsov, Dmitriy I.
2009-03-01
By numerically solving equations of motion and constructing the spectrum of Lyapunov exponents, nonlinear dynamics of uniformly precessing magnetization in (110) thin film structures with perpendicular magnetic bias is investigated over a wide frequency range of the alternating field. Bifurcational changes in magnetization precession and the states of dynamical bistability are discovered. Conditions for the realization of high-amplitude regular and chaotic dynamic regimes are revealed. The possibility of controlling those precession regimes by using external magnetic fields is shown. The features of time analogs of the Poincaré section of trajectories in the chaotic regimes are studied.
Delayed feedback control method for dynamical systems with chaotic saddles
NASA Astrophysics Data System (ADS)
Kobayashi, Miki U.; Aihara, Kazuyuki
2012-08-01
We consider systems whose orbits diverge after chaotic transient for a finite time, and propose a controlmethod for preventing the divergence. These systems generally possess not chaotic attractors but some chaotic saddles. Our aim of control, i.e., the prevention of divergence, is achieved through the stabilization of unstable periodic orbits embedded in the chaotic saddle by making use of the delayed feedback controlmethod. The key concept of our control strategy is the application of the Proper Interior Maximum (PIM) triple method and the method to detect unstable periodic orbits from time series, originally developed by Lathrop and Kostelich, as initial steps before adding the delayed feedback control input. We show that our control method can be applied to the Hénon map and an intermittent androgen suppression (IAS) therapy model, which is a model for therapy of advanced prostate cancer. The fact that our method can be applied to the IAS therapy model indicates that our control strategy may be useful in the therapy of advanced prostate cancer.
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.
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
NASA Astrophysics Data System (ADS)
Keppens, R.; Porth, O.; Galsgaard, K.; Frederiksen, J. T.; Restante, A. L.; Lapenta, G.; Parnell, C.
2013-09-01
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.
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.
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.
Takada, Ryu; Munetaka, Daigo; Kobayashi, Shoji; Suemitsu, Yoshikazu; Nara, Shigetoshi
2007-09-01
Chaotic dynamics in a recurrent neural network model and in two-dimensional cellular automata, where both have finite but large degrees of freedom, are investigated from the viewpoint of harnessing chaos and are applied to motion control to indicate that both have potential capabilities for complex function control by simple rule(s). An important point is that chaotic dynamics generated in these two systems give us autonomous complex pattern dynamics itinerating through intermediate state points between embedded patterns (attractors) in high-dimensional state space. An application of these chaotic dynamics to complex controlling is proposed based on an idea that with the use of simple adaptive switching between a weakly chaotic regime and a strongly chaotic regime, complex problems can be solved. As an actual example, a two-dimensional maze, where it should be noted that the spatial structure of the maze is one of typical ill-posed problems, is solved with the use of chaos in both systems. Our computer simulations show that the success rate over 300 trials is much better, at least, than that of a random number generator. Our functional simulations indicate that both systems are almost equivalent from the viewpoint of functional aspects based on our idea, harnessing of chaos. PMID:19003512
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.
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
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.
Fractal and Small-World Networks Formed by Self-Organized Critical Dynamics
NASA Astrophysics Data System (ADS)
Watanabe, Akitomo; Mizutaka, Shogo; Yakubo, Kousuke
2015-11-01
We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.
The Geometry and Dynamics of a Propagating Front in a Chaotic Flow Field
NASA Astrophysics Data System (ADS)
Paul, Mark
There are many important problems regarding transport in complex fluid flows with implications in science, nature, and technology. Examples include the combustion of pre-mixed gases in a turbulent flow, the complex patterns of reagents in a chemical system, the spread of a forest fire, and the outbreak of an epidemic. This talk explores the transport and dynamics of a reacting species in a chaotic fluid flow field. Large-scale parallel numerical simulations are used to explore the dynamics of propagating fronts in complex three-dimensional time-dependent fluid flows for the precise conditions of the laboratory. It is shown that a chaotic flow field enhances the front propagation when compared with a purely cellular flow field. This enhancement is quantified by computing measures of the spreading rate of the products and by quantifying the complexity of the three-dimensional front geometry for a range of chaotic flow conditions.
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...
Allegrini, Paolo; Paradisi, Paolo; Menicucci, Danilo; Gemignani, Angelo
2010-01-01
Resting-state EEG signals undergo rapid transition processes (RTPs) that glue otherwise stationary epochs. We study the fractal properties of RTPs in space and time, supporting the hypothesis that the brain works at a critical state. We discuss how the global intermittent dynamics of collective excitations is linked to mentation, namely non-constrained non-task-oriented mental activity. PMID:21423370
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.
[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.
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.
Nonlinear enhancement of the fractal structure in the escape dynamics of Bose-Einstein condensates
Mitchell, Kevin A.; Ilan, Boaz
2009-10-15
We consider the escape dynamics of an ensemble of Bose-Einstein-condensed atoms from an optical-dipole trap consisting of two overlapping Gaussian wells. Earlier theoretical studies (based on a model of quantum evolution using ensembles of classical trajectories) predicted that self-similar fractal features could be visible in this system by measuring the escaping flux as a function of time for varying initial conditions. Here, direct numerical quantum simulations show the clear influence of quantum interference on the escape data. Fractal features are still evident in the data, albeit with interference fringes superposed. Furthermore, the nonlinear influence of atom-atom interactions is also considered, in the context of the (2+1)-dimensional Gross-Pitaevskii equation. Of particular note is that an attractive nonlinear interaction enhances the resolution of fractal structures in the escape data. Thus, the interplay between nonlinear focusing and dispersion results in dynamics that resolve the underlying classical fractal more faithfully than the noninteracting quantum (or classical) dynamics.
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
NASA Astrophysics Data System (ADS)
Ge, Zheng-Ming; Leu, Jia-Haur; Lin, Tsung-Nan
The paper is to present the detailed dynamic analysis of a vertically vibrating and rotating elliptic tube containing a particle. By subjecting to an external periodic excitation, it has shown that the system exhibits both regular and chaotic motions. By using the Lyapunov direct method and Chetaev’s theorem, the stability and instability of the relative equilibrium position of the particle in the tube can be determined. The center manifold theorem is applied to verify the conditions of stability when system is under the critical case. The effects of the changes of parameters in the system can be found in the bifurcation and parametric diagrams. By applying various numerical results such as phase plane, Poincaré map and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. Further, chaotic behavior can be verified by using Lyapunov exponents and Lyapunov dimensions.
Analysis and circuitry realization of a novel three-dimensional chaotic system
NASA Astrophysics Data System (ADS)
Abooee, A.; Yaghini-Bonabi, H. A.; Jahed-Motlagh, M. R.
2013-05-01
In this paper a new three-dimensional chaotic system is introduced. Some basic dynamical properties are analyzed to show chaotic behavior of the presented system. These properties are covered by dissipation of system, instability of equilibria, strange attractor, Lyapunov exponents, fractal dimension and sensitivity to initial conditions. Through altering one of the system parameters, various dynamical behaviors are observed which included chaos, periodic and convergence to an equilibrium point. Eventually, an analog circuit is designed and implemented experimentally to realize the chaotic system.
Observation of chaotic dynamics of coupled nonlinear oscillators
NASA Astrophysics Data System (ADS)
van Buskirk, R.; Jeffries, C.
1985-05-01
Experimental data are employed as bases for theoretically modelling the behavior of a finite number of driven nonlinear coupled oscillators. Attention is focused on Si p-n junction resonators exposed to an external inductance. A junction oscillator displays period doubling, Hopf figuracions to quasi-periodicity, entrainment horns and breakup of the invariant torus. Calculated and measured data are compared, with favorable results, by means of Poincare' sections, bifurcation diagrams and parameter phase space diagrams for the drive voltage and frequency. Fractal dimensions 2.03 and 2.33 are expressed in Poincare' sections to illustrate the behavior of single and dual coupled resonators which experience a breakup of the strange attractor.
Chaotic dynamics of cardioventilatory coupling in humans: effects of ventilatory modes
Mangin, Laurence; Clerici, Christine; Similowski, Thomas; Poon, Chi-Sang
2009-01-01
Cardioventilatory coupling (CVC), a transient temporal alignment between the heartbeat and inspiratory activity, has been studied in animals and humans mainly during anesthesia. The origin of the coupling remains uncertain, whether or not ventilation is a main determinant in the CVC process and whether the coupling exhibits chaotic behavior. In this frame, we studied sedative-free, mechanically ventilated patients experiencing rapid sequential changes in breathing control during ventilator weaning during a switch from a machine-controlled assistance mode [assist-controlled ventilation (ACV)] to a patient-driven mode [inspiratory pressure support (IPS) and unsupported spontaneous breathing (USB)]. Time series were computed as R to start inspiration (RI) and R to the start of expiration (RE). Chaos was characterized with the noise titration method (noise limit), largest Lyapunov exponent (LLE) and correlation dimension (CD). All the RI and RE time series exhibit chaotic behavior. Specific coupling patterns were displayed in each ventilatory mode, and these patterns exhibited different linear and chaotic dynamics. When switching from ACV to IPS, partial inspiratory loading decreases the noise limit value, the LLE, and the correlation dimension of the RI and RE time series in parallel, whereas decreasing intrathoracic pressure from IPS to USB has the opposite effect. Coupling with expiration exhibits higher complexity than coupling with inspiration during mechanical ventilation either during ACV or IPS, probably due to active expiration. Only 33% of the cardiac time series (RR interval) exhibit complexity either during ACV, IPS, or USB making the contribution of the cardiac signal to the chaotic feature of the coupling minimal. We conclude that 1) CVC in unsedated humans exhibits a complex dynamic that can be chaotic, and 2) ventilatory mode has major effects on the linear and chaotic features of the coupling. Taken together these findings reinforce the role of
Effects of correlation among stored patterns on associative dynamics of chaotic neural network
NASA Astrophysics Data System (ADS)
Iwai, Toshiya; Matsuzaki, Fuminari; Kuroiwa, Jousuke; Miyake, Shogo
2005-12-01
We numerically investigate the effects of correlation among stored patterns on the associative dynamics in a chaotic neural network model. In the model, there are two kinds of parameters: one is a measure of the Hopfield like behavior of the retrieval process and another controls the chaotic behavior. The parameter dependence of the associative dynamics is also examined. The following results are found. (i) Two dimensional parameter space is divided into two kinds of associative states by a distinct boundary. One is the retrieval state of the association such as the Hopfield like retrieval state, and another is the wandering state of the associative dynamics where the network retrieves stored patterns and their reverse patterns. (ii) The area of the wandering state becomes larger as the degree of correlation becomes larger. (iii) As the degree of correlation becomes larger, both the recall ratio of correlated patterns and the transition frequency between correlated patterns becomes larger in the wandering state. (iv) The whole region of the wandering state in the parameter space is not necessarily chaotic from the view point of the Lyapunov dimension, but most of the region of the wandering state is chaotic.
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.
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.
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.
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
Chaotic dynamics and basin erosion in nanomagnets subject to time-harmonic magnetic fields
NASA Astrophysics Data System (ADS)
d'Aquino, M.; Quercia, A.; Serpico, C.; Bertotti, G.; Mayergoyz, I. D.; Perna, S.; Ansalone, P.
2016-04-01
Magnetization dynamics in uniformly magnetized particles subject to time-harmonic (AC) external fields is considered. The study is focused on the behavior of the AC-driven dynamics close to saddle equilibria. It happens that such dynamics has chaotic nature at moderately low power level, due to the heteroclinic tangle phenomenon which is produced by the combined effect of AC-excitations and saddle type dynamics. By using analytical theory for the threshold AC excitation amplitudes necessary to create the heteroclinic tangle together with numerical simulations, we quantify and show how the tangle produces the erosion of the safe basin around the stable equilibria.
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.
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.
Mixmaster universe: A chaotic Farey tale
NASA Astrophysics Data System (ADS)
Cornish, Neil J.; Levin, Janna J.
1997-06-01
When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer-dependent answers. Here we resolve the conflict by using observer-independent fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the six-dimensional minisuperspace of the full Einstein equations and in a two-dimensional discretization of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy, and multifractal dimensions. As all of these quantities are coordinate or gauge independent, there is no longer any ambiguity-the mixmaster universe is indeed chaotic.
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
Magma Mixing by Chaotic Dynamics: Results from a New Experimental Device
NASA Astrophysics Data System (ADS)
de Campos, C. P.; Perugini, D.; Ingrisch, W. E.; Dingwell, D. B.; Poli, G.
2009-12-01
In this work we present a new experimental device, based on the Journal Bearing System (JBS), to perform chaotic mixing of high viscosity melts under controlled fluid-dynamics and temperature conditions. The system consists of an outer cylinder, hosting the melts of interest and, an inner cylinder, eccentrically located, whose motions are independent. This way the development of chaotic streamlines in the mixing system is induced. An experiment was performed, using as end-members a peralkaline haplogranite (HPG8) and a mafic melt, corresponding to the 1 atm eutectic composition in the An-Di binary system. The two melts were stirred together in the JBS for ca. two hours at 1,400°C under laminar fluid dynamic condition (Re of the order of 10-7). Viscosity ratio between the two melts, at the beginning of the experiment, was of the order of 1,000. Analyses of experimental samples revealed, at short length scale (of the order of μm), a complex pattern of mixing structures. These consisted of an intimate distribution of filaments of the two melts, a typical feature in rocks produced by magma mixing processes. Stretching and folding dynamics between both melts induced chaotic flow fields and generated wide compositional interfaces. This way, chemical diffusion processes acted efficiently, producing melts with highly heterogeneous compositions. Despite a short running time, very low Re and a high viscosity ratio, a clear modulation of compositional fields has been obtained (fig.1). This indicates that chaotic mixing can be a very efficient process in enhancing the compositional variability in igneous systems, even under extreme rheological conditions and laminar fluid-dynamics. The excellence of our experimental device to replicate natural magma mixing features may open new frontiers in the study of this important petrological and volcanological process. Figure 1
NASA Astrophysics Data System (ADS)
de Campos, Cristina; Perugini, Diego; Ertel-Ingrisch, Werner; Dingwell, Donald B.; Poli, Giampiero
2010-05-01
A new experimental device has been developed to perform chaotic mixing between high viscosity melts under controlled fluid-dynamic conditions. The apparatus is based on the Journal Bearing System (JBS). It consists of an outer cylinder hosting the melts of interest and an inner cylinder, which is eccentrically located. Both cylinders can be independently moved to generate chaotic streamlines in the mixing system. Two experiments were performed using as end-members different proportions of a peralkaline haplogranite and a mafic melt, corresponding to the 1 atm eutectic composition in the An-Di binary system. The two melts were stirred together in the JBS for ca. two hours, at 1,400° C and under laminar fluid dynamic condition (Re of the order of 10-7). The viscosity ratio between the two melts, at the beginning of the experiment, was of the order of 103. Optical analyses of experimental samples revealed, at short length scale (of the order of μm), a complex pattern of mixed structures. These consisted of an intimate distribution of filaments; a complex inter-fingering of the two melts. Such features are typically observed in rocks thought to be produced by magma mixing processes. Stretching and folding dynamics between the melts induced chaotic flow fields and generated wide compositional interfaces. In this way, chemical diffusion processes become more efficient, producing melts with highly heterogeneous compositions. A remarkable modulation of compositional fields has been obtained by performing short time-scale experiments and using melts with a high viscosity ratio. This indicates that chaotic mixing of magmas can be a very efficient process in modulating compositional variability in igneous systems, especially under high viscosity ratios and laminar fluid-dynamic regimes. Our experimental device may replicate magma mixing features, observed in natural rocks, and therefore open new frontiers in the study of this important petrologic and volcanological process.
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.
Cycles, randomness, and transport from chaotic dynamics to stochastic processes
NASA Astrophysics Data System (ADS)
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.
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.
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.
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.
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.}
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
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.
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.
Dynamics of chaotic systems with attractive and repulsive couplings.
Chen, Yuehua; Xiao, Jinghua; Liu, Weiqing; Li, Lixiang; Yang, Yixian
2009-10-01
Together with attractive couplings, repulsive couplings play crucial roles in determining important evolutions in natural systems, such as in learning and oscillatory processes of neural networks. The complex interactions between them have great influence on the systems. A detailed understanding of the dynamical properties under this type of couplings is of practical significance. In this paper, we propose a model to investigate the dynamics of attractive and repulsive couplings, which give rise to rich phenomena, especially for amplitude death (AD). The relationship among various dynamics and possible transitions to AD are illustrated. When the system is in the maximally stable AD, we observe the transient behavior of in-phase (high frequency) and out-of-phase (low frequency) motions. The mechanism behind the phenomenon is given. PMID:19905414
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.
NASA Astrophysics Data System (ADS)
Xia, H. M.; Shu, C.; Wan, S. Y. M.; Chew, Y. T.
2006-01-01
A micromixer is a key component of various microfluidic systems, such as microreactors and μ-total analysis systems. One important strategy for passive mixer design is to generate chaotic advection using channel geometry, which usually has spatially periodic structures. In this paper, the influence of the Reynolds number on chaotic mixing in such mixers is studied with three mixer models. Characterization of the mixer with dynamical system techniques is also studied. The influence of fluid inertial effects on the occurrence of chaotic advection is first discussed. It is found that at low Re(Re < 1), the flow could become reversible in the mixer, which raises the difficulty to generate chaotic advection. In this case, specific fluid manipulations, such as stretching and folding processes, are necessary. This study also proposes a characterization method using Lyapunov exponent (λ) and Poincaré mapping information, which allows us to analyze the mixing performance of the mixer with one single mixer unit. Results show that it objectively reflects the dynamical properties of the mixers, such as being globally chaotic, partially chaotic or stable. So it can be used as an analytical tool to differentiate, evaluate and optimize various chaotic micromixers.
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).
NASA Astrophysics Data System (ADS)
Che, Yanqiu; Yang, Tingting; Li, Ruixue; Li, Huiyan; Han, Chunxiao; Wang, Jiang; Wei, Xile
2015-09-01
In this paper, we propose a dynamic delayed feedback control approach or desynchronization of chaotic-bursting synchronous activities in an ensemble of globally coupled neuronal oscillators. We demonstrate that the difference signal between an ensemble's mean field and its time delayed state, filtered and fed back to the ensemble, can suppress the self-synchronization in the ensemble. These individual units are decoupled and stabilized at the desired desynchronized states while the stimulation signal reduces to the noise level. The effectiveness of the method is illustrated by examples of two different populations of globally coupled chaotic-bursting neurons. The proposed method has potential for mild, effective and demand-controlled therapy of neurological diseases characterized by pathological synchronization.
Wang, Zhiheng; Huo, Zhanqiang; Shi, Wenbo
2015-01-01
With rapid development of computer technology and wide use of mobile devices, the telecare medicine information system has become universal in the field of medical care. To protect patients' privacy and medial data's security, many authentication schemes for the telecare medicine information system have been proposed. Due to its better performance, chaotic maps have been used in the design of authentication schemes for the telecare medicine information system. However, most of them cannot provide user's anonymity. Recently, Lin proposed a dynamic identity based authentication scheme using chaotic maps for the telecare medicine information system and claimed that their scheme was secure against existential active attacks. In this paper, we will demonstrate that their scheme cannot provide user anonymity and is vulnerable to the impersonation attack. Further, we propose an improved scheme to fix security flaws in Lin's scheme and demonstrate the proposed scheme could withstand various attacks. PMID:25486894
Spatial-temporal dynamics of chaotic behavior in cultured hippocampal networks.
Chen, Wenjuan; Li, Xiangning; Pu, Jiangbo; Luo, Qingming
2010-06-01
Using multiple nonlinear techniques, we revealed the existence of chaos in the spontaneous activity of neuronal networks in vitro. The spatial-temporal dynamics of these networks indicated that emergent transition between chaotic behavior and superburst occurred periodically in low-frequency oscillations. An analysis of network-wide activity indicated that chaos was synchronized among different sites. Moreover, we found that the degree of chaos increased as the number of active sites in the network increased during long-term development (over three months in vitro). The chaotic behavior of the dissociated networks had similar spatial-temporal characteristics (rapid transition, periodicity, and synchronization) as the intact brain; however, the degree of chaos depended on the number of active sites at the mesoscopic level. This work could provide insight into neural coding and neurocybernetics. PMID:20866436
Exploring the Spatiotemporal Dynamics of Covariant Lyapunov Vectors for Chaotic Convection
NASA Astrophysics Data System (ADS)
Xu, Mu; Paul, Mark
Covariant Lyapunov vectors provide access to fundamental features of chaos in high-dimensional systems that are driven far-from-equilibrium. We explore the spatiotemporal dynamics of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection to provide new physical insights. We use the covariant Lyapunov vectors to quantify the transition from hyperbolic to non-hyperbolic dynamics, to determine the degree of Oseledec splitting exhibited by the dynamics, and to shed light upon upon the tangled nature of the Lyapunov vectors. In this talk, we will explore the spatiotemporal dynamics of the Lyapunov vectors and their relation with the chaotic pattern dynamics of the flow field. Our results suggest that the Lyapunov vectors contain two distinct spatiotemporal features consisting of highly localized regions near defect structures and a spatially distributed checkerboard pattern. We will explore the connection between these features and the ideas of physical and spurious modes that may compose the dynamics. This research was funded by NSF Grant No. DMS-1125234.
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
Dynamics of fractal cluster colloidal gels with embedded active Janus particles
NASA Astrophysics Data System (ADS)
Solomon, Michael; Szakasits, Megan; Zhang, Wenxuan
We find that fractal cluster gels of colloids in which platinum-coated Janus particles have been embedded exhibit enhanced mobility when the Janus particles are made active by the addition of hydrogen peroxide. Gelation is induced through addition of a divalent salt, magnesium chloride, to an initially stable suspension of Janus and polystyrene colloids, each of size about 1 micron. After the gels have been created, the embedded Janus colloids are activated by hydrogen peroxide, which is delivered to the system through a porous hydrogel membrane. We vary the ratio of active to passive colloids in the gels from about 1:20 to 1:8. Changes in structure and dynamics are visualized by two channel confocal laser scanning microscopy. By image analysis, we determine the particle positions and compute the mean squared displacement (MSD) of all particles in the gel. We measure the mobility enhancement in the fractal gels as a function of hydrogen peroxide concentration and Janus particle concentration and discuss the results in terms of the force provided by each active particle to the fractal gel network.
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.
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.
Stabilization of chaotic and non-permanent food-web dynamics
NASA Astrophysics Data System (ADS)
Williams, R. J.; Martinez, N. D.
2004-03-01
Several decades of dynamical analyses of food-web networks[CITE] have led to important insights into the effects of complexity, omnivory and interaction strength on food-web stability[CITE]. Several recent insights[CITE] are based on nonlinear bioenergetic consumer-resource models[CITE] that display chaotic behavior in three species food chains[CITE] which can be stabilized by omnivory[CITE] and weak interaction of a fourth species[CITE]. We slightly relax feeding on low-density prey in these models by modifying standard food-web interactions known as “typeII” functional responses[CITE]. This change drastically alters the dynamics of realistic systems containing up to ten species. Our modification stabilizes chaotic dynamics in three species systems and reduces or eliminates extinctions and non-persistent chaos[CITE] in ten species systems. This increased stability allows analysis of systems with greater biodiversity than in earlier work and suggests that dynamic stability is not as severe a constraint on the structure of large food webs as previously thought. The sensitivity of dynamical models to small changes in the predator-prey functional response well within the range of what is empirically observed suggests that functional response is a crucial aspect of species interactions that must be more precisely addressed in empirical studies.
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
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
Wind tunnel experiments on chaotic dynamics of a flexible tube row in a cross flow
Muntean, G.; Moon, F.C.
1994-12-31
Flow visualization and dynamics measurements of flexible cylindrical tubes in a cross-flow are described. Five tubes mounted on flexible supports were subjected to cross flow in a low turbulence wind tunnel. Dynamic measurements of the tube motion are presented. The data suggests that a low dimensional attractor exists for tube flutter under impact constraints using fractal dimension calculations. There is also qualitative evidence for single tube flutter in-line with the flow. In another set of experiments, a flow visualization technique is used to examine the flow behind the vibrating cylinders. Four different configurations of the jet flow behind the cylinders are observed. Coupling of the jet dynamics and tube motion seems apparent from the video data. These experiments are being used to try and construct a low order nonlinear model for the tube-flow dynamics.
Fractal dynamics in self-evaluation reveal self-concept clarity.
Wong, Alexander E; Vallacher, Robin R; Nowak, Andrzej
2014-10-01
The structural account of self-esteem and self-evaluation maintains that they are distinct constructs. Trait self-esteem is stable and is expressed over macro timescales, whereas state self-evaluation is unstable and experienced on micro timescales. We compared predictions based on the structural account with those derived from a dynamical systems perspective on the self, which maintains that self-esteem and self-evaluation are hierarchically related and share basic dynamic properties. Participants recorded a 3-minute narrative about themselves, then used the mouse paradigm (Vallacher, Nowak, Froehlich, & Rockloff, 2002) to track the momentary self-evaluation in their narrative. Multiple methods converged to reveal fractal patterns in the resultant temporal patterns, indicative of nested timescales that link micro and macro selfevaluation and thus supportive of the dynamical account. The fractal dynamics were associated with participants' self-concept clarity, suggesting that the hierarchical relation between macro self-evaluation (self-esteem) and momentary self-evaluation is predicted by the coherence of self-concept organization. PMID:25196705
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.
Modulated point-vortex pairs on a rotating sphere: Dynamics and chaotic advection
NASA Astrophysics Data System (ADS)
Drótos, Gábor; Tél, Tamás; Kovács, Gergely
2013-06-01
The dynamics of modulated point-vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). For sufficiently close vortices (dipoles) the trajectories of their center-of-mass are shown to correspond to those of a point particle moving freely on a rotating sphere. For finite size vortex pairs, a qualitative similarity to the geodesic dynamics is found. The advection dynamics generated by vortex pairs on a rotating sphere is found to be chaotic. In the short time dynamics we point out a transition from closed to open chaotic advection, which implies that the transport properties of the flow might drastically be altered by changing the initial conditions of the pair on the sphere. Due to spherical topology, for long times, even the open advection patterns are found to gradually cross over to that corresponding to a homogeneous closed mixing. This pattern extends along a zonal band, whereas short term closed mixing remains always bounded to the moving pair.
Modulated point-vortex pairs on a rotating sphere: dynamics and chaotic advection.
Drótos, Gábor; Tél, Tamás; Kovács, Gergely
2013-06-01
The dynamics of modulated point-vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). For sufficiently close vortices (dipoles) the trajectories of their center-of-mass are shown to correspond to those of a point particle moving freely on a rotating sphere. For finite size vortex pairs, a qualitative similarity to the geodesic dynamics is found. The advection dynamics generated by vortex pairs on a rotating sphere is found to be chaotic. In the short time dynamics we point out a transition from closed to open chaotic advection, which implies that the transport properties of the flow might drastically be altered by changing the initial conditions of the pair on the sphere. Due to spherical topology, for long times, even the open advection patterns are found to gradually cross over to that corresponding to a homogeneous closed mixing. This pattern extends along a zonal band, whereas short term closed mixing remains always bounded to the moving pair. PMID:23848782
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
Chaotic dynamics of dilute thermal atom clouds on stationary optical Bessel beams
NASA Astrophysics Data System (ADS)
Castañeda, J. A.; Pérez-Pascual, R.; Jáuregui, R.
2013-07-01
We characterize the semiclassical dynamics of dilute thermal atom clouds located in three-dimensional optical lattices generated by stationary optical Bessel beams. The dynamics of the cold atoms is explored in the quasi-Hamiltonian regime that arises using laser beams with far-off resonance detuning. Although the transverse structure of Bessel beams exhibits a complex topological structure, it is found that the longitudinal motion along the main propagation axis of the beam is the detonator of a high sensitivity of the atoms' motion to the initial conditions. This effect would not be properly described by bidimensional models. We show that an experimental implementation can be highly simplified by an analysis of the behaviour of the dynamical system under scale transformations. Experimentally feasible signatures of the chaotic dynamics of the atom clouds are also identified.
Lifetime statistics in chaotic dielectric microresonators
Schomerus, Henning; Wiersig, Jan; Main, Joerg
2009-05-15
We discuss the statistical properties of lifetimes of electromagnetic quasibound states in dielectric microresonators with fully chaotic ray dynamics. Using the example of a resonator of stadium geometry, we find that a recently proposed random-matrix model very well describes the lifetime statistics of long-lived resonances, provided that two effective parameters are appropriately renormalized. This renormalization is linked to the formation of short-lived resonances, a mechanism also known from the fractal Weyl law and the resonance-trapping phenomen0008.
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.
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.
Haotic, Fractal, and Nonlinear Signal Processing. Proceedings
Katz, R.A.
1996-10-01
These proceedings include papers presented at the Third Technical Conference on Nonlinear Dynamics and Full{minus}Spectrum Processing held in Mystic, Connecticut. The Conference focus was on the latest advances in chaotic, fractal and nonlinear signal processing methods. Topics of discussion covered in the Conference include: mathematical frontiers; predictability and control of chaos, detection and classification with applications in acoustics; advanced applied signal processing methods(linear and nonlinear); stochastic resonance; machinery diagnostics; turbulence; geophysics; medicine; and recent novel approaches to modeling nonlinear systems. There were 58 papers in the conference and all have been abstracted for the Energy Science and Technology database. (AIP)
Chaotic dynamics of one-dimensional systems with periodic boundary conditions
NASA Astrophysics Data System (ADS)
Kumar, Pankaj; Miller, Bruce N.
2014-12-01
We provide appropriate tools for the analysis of dynamics and chaos for one-dimensional systems with periodic boundary conditions. Our approach allows for the investigation of the dependence of the largest Lyapunov exponent on various initial conditions of the system. The method employs an effective approach for defining the phase-space distance appropriate for systems with periodic boundaries and allows for an unambiguous test-orbit rescaling in the phase space required to calculate the Lyapunov exponents. We elucidate our technique by applying it to investigate the chaotic dynamics of a one-dimensional plasma with periodic boundaries. Exact analytic expressions are derived for the electric field and potential using Ewald sums, thereby making it possible to follow the time evolution of the plasma in simulations without any special treatment of the boundary. By employing a set of event-driven algorithms, we calculate the largest Lyapunov exponent, the radial distribution function, and the pressure by following the evolution of the system in phase space without resorting to numerical manipulation of the equations of motion. Simulation results are presented and analyzed for the one-dimensional plasma with a view to examining the dynamical and chaotic behavior exhibited by small and large versions of the system.
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.
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
Regular and Chaotic Flow Behavior and Orientational Dynamics of Tumbling Nematics
NASA Astrophysics Data System (ADS)
Hess, S.; Heidenreich, S.; Ilg, P.; Kröger, M.
2006-05-01
We consider liquid crystalline polymers under plane Couette flow and investigate the influence of fluctuating shear rates on the orientational dynamics. With help of phase portraits and time evolution diagrams of the alignment tensor components, we discuss the effect of fluctuations on the flow-aligned, isotropic and periodic solutions. To explore the effect of fluctuations on the chaotic behavior we calculated the greatest Lyapunov exponent for different fluctuation strengths. We found that fluctuations of the shear rate in general have little effect on the dynamics of tumbling nematics. Further we present a new amended potential modeling the isotropic-to-nematic transition. In contrast to the Landau-de Gennes potential our potential has the advantage to restrict the order parameter to physically admissible values. In the end we present some results of the orientational dynamics for a spatially inhomogeneous system.
Dynamic interpretation of atomic and molecular spectra in the chaotic regime
Taylor, H.S.; Zakrzewski, J.
1988-10-01
A quantum partitioning theory is given for extracting dynamic information from the high-resolution spectra of highly excited atoms and molecules that is relatively simple to apply. The presented approach is applicable whenever the classical counterpart of the system studied is chaotic. The theory allows a picture of the underlying non-statistically-describable part of the dynamics to be obtained from the spectra. The theory presented effectively uses and unifies many aspects of classical trajectory approaches, Feshbach resonant-scattering partitioning theory, semiclassical periodic-orbit theory, ''scars'' theory, bright- and dark-state concepts, and Fourier transforms of the spectra. The power of the theory is demonstrated quantitatively by interpreting the dynamics underlying the absorption spectra of the hydrogen atom in a strong uniform magnetic field.
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.
Experimental and theoretical evidence for the chaotic dynamics of complex structures
NASA Astrophysics Data System (ADS)
Agop, M.; Dimitriu, D. G.; Niculescu, O.; Poll, E.; Radu, V.
2013-04-01
This paper presents the experimental results on the formation, dynamics and evolution towards chaos of complex space charge structures that emerge in front of a positively biased electrode immersed in a quiescent plasma. In certain experimental conditions, we managed to obtain the so-called multiple double layers (MDLs) with non-concentric configuration. Our experiments show that the interactions between each MDL's constituent entities are held responsible for the complex dynamics and eventually for its transition to chaos through cascades of spatio-temporal sub-harmonic bifurcations. Further, we build a theoretical model based on the fractal approximation (scale relativity theory) in order to reproduce the experimental results (plasma self-structuring and scenario of evolution to chaos). Comparing the experimental results with the theoretical ones, we observe a good correlation between them.
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…
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
High-frequency chaotic dynamics enabled by optical phase-conjugation
NASA Astrophysics Data System (ADS)
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.
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
Zunino, L; Soriano, M C; Rosso, O A
2012-10-01
In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations. PMID:23214666
Cheng, Mengfan; Deng, Lei; Li, Hao; Liu, Deming
2014-03-10
We propose a scheme whereby a time domain fractional Fourier transform (FRFT) is used to post process the optical chaotic carrier generated by an electro-optic oscillator. The time delay signature of the delay dynamics is successfully masked by the FRFT when some conditions are satisfied. Meanwhile the dimension space of the physical parameters is increased. Pseudo random binary sequence (PRBS) with low bit rate (hundreds of Mbps) is introduced to control the parameters of the FRFT. The chaotic optical carrier, FRFT parameters and the PRBS are covered by each other so that the eavesdropper has to search the whole key space to crack the system. The scheme allows enhancing the security of communication systems based on delay dynamics without modifying the chaotic source. In this way, the design of chaos based communication systems can be implemented in a modular manner. PMID:24663864
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.
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
A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics.
Salari, Lucia; Rondoni, Lamberto; Giberti, Claudio; Klages, Rainer
2015-07-01
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here, we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process. PMID:26232964
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.
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.
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)
NASA Astrophysics Data System (ADS)
Small, Michael; Walker, David M.; Tordesillas, Antoinette; Tse, Chi K.
2013-03-01
For a given observed time series, it is still a rather difficult problem to provide a useful and compelling description of the underlying dynamics. The approach we take here, and the general philosophy adopted elsewhere, is to reconstruct the (assumed) attractor from the observed time series. From this attractor, we then use a black-box modelling algorithm to estimate the underlying evolution operator. We assume that what cannot be modeled by this algorithm is best treated as a combination of dynamic and observational noise. As a final step, we apply an ensemble of techniques to quantify the dynamics described in each model and show that certain types of dynamics provide a better match to the original data. Using this approach, we not only build a model but also verify the performance of that model. The methodology is applied to simulations of a granular assembly under compression. In particular, we choose a single time series recording of bulk measurements of the stress ratio in a biaxial compression test of a densely packed granular assembly—observed during the large strain or so-called critical state regime in the presence of a fully developed shear band. We show that the observed behavior may best be modeled by structures capable of exhibiting (hyper-) chaotic dynamics.
Micro and MACRO Fractals Generated by Multi-Valued Dynamical Systems
NASA Astrophysics Data System (ADS)
Banakh, T.; Novosad, N.
2014-08-01
Given a multi-valued function Φ : X \\mumap X on a topological space X we study the properties of its fixed fractal \\malteseΦ, which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals \\maltese Φ and \\maltese {Φ -1} for a contracting compact-valued function Φ : X \\mumap X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.
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
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
NASA Astrophysics Data System (ADS)
Druzgalski, Clara; Mani, Ali
2014-11-01
We have investigated the transport dynamics of an electrokinetic instability that occurs when ions are driven from bulk fluids to ion-selective membranes due to externally applied electric fields. This phenomenon is relevant to a wide range of electrochemical applications including electrodialysis for fresh water production. Using data from our 3D DNS, we show how electroconvective instability, arising from concentration polarization, results in a chaotic flow that significantly alters the net ion transport rate across the membrane surface. The 3D DNS results, which fully resolve the spatiotemporal scales including the electric double layers, enable visualization of instantaneous snapshots of current density directly on the membrane surface, as well as analysis of transport statistics such as concentration variance and fluctuating advective fluxes. Furthermore, we present a full spectral analysis revealing broadband spectra in both concentration and flow fields and deduce the key parameter controlling the range of contributing scales.
NASA Astrophysics Data System (ADS)
Yuan, Fang; Wang, Guang-Yi; Wang, Xiao-Yuan
2015-06-01
To develop real world memristor application circuits, an equivalent circuit model which imitates memductance (memory conductance) of the HP memristor is presented. The equivalent circuit can be used for breadboard experiments for various application circuit designs of memristor. Based on memductance of the realistic HP memristor and Chua’s circuit a new chaotic oscillator is designed. Some basic dynamical behaviors of the oscillator, including equilibrium set, Lyapunov exponent spectrum, and bifurcations with various circuit parameters are investigated theoretically and numerically. To confirm the correction of the proposed oscillator an analog circuit is designed using the proposed equivalent circuit model of an HP memristor, and the circuit simulations and the experimental results are given. Project supported by the National Natural Science Foundation of China (Grant Nos. 61271064 and 60971046), the Natural Science Foundation of Zhejiang Province, China (Grant No. LZ12F01001), and the Program for Zhejiang Leading Team of Science and Technology Innovation, China (Grant No. 2010R50010-07).
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.
NASA Astrophysics Data System (ADS)
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
2014-12-15
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. PMID:25554054
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.
Chaos vs linear instability in the Vlasov equation: A fractal analysis characterization
Atalmi, A.; Baldo, M.; Burgio, G.F.; Rapisarda, A.
1996-05-01
In this paper we discuss the most recent results concerning the Vlasov dynamics inside the spinodal region. The chaotic behavior which follows an initial regular evolution is characterized through the calculation of the fractal dimension of the distribution of the final modes excited. The ambiguous role of the largest Lyapunov exponent for unstable systems is also critically reviewed. This investigation seems to confirm the crucial role played by deterministic chaos in nuclear multifragmentation. {copyright} {ital 1996 The American Physical Society.}
Conduction block and chaotic dynamics in an asymmetrical model of coupled cardiac cells.
Landau, M; Lorente, P
1997-05-01
The initiation and propagation of the cardiac impulse depends on intrinsic properties of cells, geometrical arrangements, and intercellular coupling resistances. To address the issue of the interplay between these factors in a simple way, we have used a system, based on the van Capelle and Dürrer model, including a pacemaker and a non-pacemaker cell linked by an ohmic coupling resistance. The influence of asymmetrical cell sizes and electronic load was investigated by using numerical simulations and continuation-bifurcation techniques. The loading of a small pacemaker cell by a large non-pacemaker one (pacemaker: non-pacemaker size ratio = 0.3) was expressed as a pronounced early repolarization in the pacemaker cell and a quite long latency for the impulse propagation. Using coupling resistance as the continuation parameter, three behavioral zones were detected from low to high coupling resistance values: a zone of total quiescence (0:0), a zone of effective entertainment (1:1), and a zone of total block (1:0 pattern). At the boundary between 1:1 and 1:0 patterns, for relatively high coupling resistance values, a cascade of period doubling bifurcations emerged, corresponding to discrete changes of propagation patterns leading into irregular dynamics. Another route to irregular dynamics was also observed in the parameter space. The high sensitivity of the detected irregular dynamics to initial conditions and the positive value of the maximum Lyapunov exponent allowed us to identify these dynamics as being chaotic. Since neither intermediate block patterns nor irregular dynamics were observed with larger size ratios, we suggest that the interplay between resting membrane conductance of the non-pacemaker cell and intercellular coupling may bring about these rhythmic disturbances. PMID:9176640
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
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)
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.
2012-09-01
A set of Maple routines is presented, fully compatible with the new releases of Maple (14 and higher). The package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. The novelty that the package presents an optional C interface is maintained. This allows for fast numerical integration, even for the totally inexperienced Maple user, without any C expertise being required. Finally, the package provides the routines to calculate the fractal dimension of boundaries (via box counting). New version program summary Program Title: Ndynamics Catalogue identifier: %Leave blank, supplied by Elsevier. Licensing provisions: no. Programming language: Maple, C. Computer: Intel(R) Core(TM) i3 CPU M330 @ 2.13 GHz. Operating system: Windows 7. RAM: 3.0 GB Keywords: Dynamical systems, Box counting, Fractal dimension, Symbolic computation, Differential equations, Maple. Classification: 4.3. Catalogue identifier of previous version: ADKH_v1_0. Journal reference of previous version: Comput. Phys. Commun. 119 (1999) 256. Does the new version supersede the previous version?: Yes. Nature of problem Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries. Solution method The default method of integration is a fifth-order Runge-Kutta scheme, but any method of integration present on the Maple system is available via an argument when calling the routine. A box counting [1] method is used to calculate the fractal dimension [2] of the boundaries. Reasons for the new version The Ndynamics package met a demand of our research community for a flexible and friendly environment for analyzing dynamical systems. All the user has to do is create his/her own Maple session, with the system to
Impacts of Riparian Zone Plant Water Use on Fractal Dynamics of Groundwater Levels
NASA Astrophysics Data System (ADS)
Zhu, J.; Young, M.
2011-12-01
In areas where plants directly tap groundwater for their water supply, hydrographs from the water table typically display diurnal fluctuations superimposed on other larger and smaller trends during the plant growing season. In this work, we investigate groundwater system dynamics in relation to plant water use and near-river stage fluctuations in riparian zones where phreatophytes exist. Using detrended fluctuation analysis (DFA), we examine the influence of regular diurnal fluctuations due to phreatophyte water use on long temporal scaling properties of groundwater level variations. We found that groundwater use by phreatophytes, at the field site on the Colorado River, USA, results in distinctive slope changes in the logarithm plots of root-mean-square fluctuations of the detrended time series vs. time scales of groundwater level dynamics. For groundwater levels monitored at wells close to the river, we identified one slope change at ~1 day in the fractal scaling characteristics of groundwater level variations. When time scale exceeds 1 day, the scaling properties decrease from persistent to 1/f noise, where f is the frequency. For groundwater levels recorded at wells further from the river, the slope of the straight line fit (i.e., scaling exponent) is smallest when the time scale is between 1 to ~3 days. When the time scale is <1 day, groundwater variations become persistent. When the time scale is between 1 - 3 days, the variations are close to white noise, but return to persistent when the time scale is >~3 days.
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
Fractality, chaos, and reactions in imperfectly mixed open hydrodynamical flows
NASA Astrophysics Data System (ADS)
Péntek, Á.; Károlyi, G.; Scheuring, I.; Tél, T.; Toroczkai, Z.; Kadtke, J.; Grebogi, C.
1999-12-01
We investigate the dynamics of tracer particles in time-dependent open flows. If the advection is passive the tracer dynamics is shown to be typically transiently chaotic. This implies the appearance of stable fractal patterns, so-called unstable manifolds, traced out by ensembles of particles. Next, the advection of chemically or biologically active tracers is investigated. Since the tracers spend a long time in the vicinity of a fractal curve, the unstable manifold, this fractal structure serves as a catalyst for the active process. The permanent competition between the enhanced activity along the unstable manifold and the escape due to advection results in a steady state of constant production rate. This observation provides a possible solution for the so-called “paradox of plankton”, that several competing plankton species are able to coexists in spite of the competitive exclusion predicted by classical studies. We point out that the derivation of the reaction (or population dynamics) equations is analog to that of the macroscopic transport equations based on a microscopic kinetic theory whose support is a fractal subset of the full phase space.
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.
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.
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.
Regular and chaotic dynamics of a fountain in a stratified fluid
NASA Astrophysics Data System (ADS)
Druzhinin, O. A.; Troitskaya, Yu. I.
2012-06-01
In the present paper, we study by direct numerical simulation (DNS) and theoretical analysis, the dynamics of a fountain penetrating a pycnocline (a sharp density interface) in a density-stratified fluid. A circular, laminar jet flow of neutral buoyancy is considered, which propagates vertically upwards towards the pycnocline level, penetrates a distance into the layer of lighter fluid, and further stagnates and flows down under gravity around the up-flowing core thus creating a fountain. The DNS results show that if the Froude number (Fr) is small enough, the fountain top remains axisymmetric and steady. However, if Fr is increased, the fountain top becomes unsteady and oscillates in a circular flapping (CF) mode, whereby it retains its shape and moves periodically around the jet central axis. If Fr is increased further, the fountain top rises and collapses chaotically in a bobbing oscillation mode (or B-mode). The development of these two modes is accompanied by the generation of different patterns of internal waves (IW) in the pycnocline. The CF-mode generates spiral internal waves, whereas the B-mode generates IW packets with a complex spatial distribution. The dependence of the amplitude of the fountain-top oscillations on Fr is well described by a Landau-type two-mode-competition model.
Regular and chaotic dynamics of a fountain in a stratified fluid.
Druzhinin, O A; Troitskaya, Yu I
2012-06-01
In the present paper, we study by direct numerical simulation (DNS) and theoretical analysis, the dynamics of a fountain penetrating a pycnocline (a sharp density interface) in a density-stratified fluid. A circular, laminar jet flow of neutral buoyancy is considered, which propagates vertically upwards towards the pycnocline level, penetrates a distance into the layer of lighter fluid, and further stagnates and flows down under gravity around the up-flowing core thus creating a fountain. The DNS results show that if the Froude number (Fr) is small enough, the fountain top remains axisymmetric and steady. However, if Fr is increased, the fountain top becomes unsteady and oscillates in a circular flapping (CF) mode, whereby it retains its shape and moves periodically around the jet central axis. If Fr is increased further, the fountain top rises and collapses chaotically in a bobbing oscillation mode (or B-mode). The development of these two modes is accompanied by the generation of different patterns of internal waves (IW) in the pycnocline. The CF-mode generates spiral internal waves, whereas the B-mode generates IW packets with a complex spatial distribution. The dependence of the amplitude of the fountain-top oscillations on Fr is well described by a Landau-type two-mode-competition model. PMID:22757523
NASA Astrophysics Data System (ADS)
Laroche, C.; Labbé, R.; Pétrélis, F.; Fauve, S.
2012-02-01
We show that electric motors and dynamos can be used to illustrate most elementary instabilities or bifurcations discussed in courses on nonlinear oscillators and dynamical systems. These examples are easier to understand and display a richer behavior than the ones commonly used from mechanics, electronics, hydrodynamics, lasers, chemical reactions, and population dynamics. In particular, an electric motor driven by a dynamo can display stationary, Hopf, and codimension-two bifurcations by tuning the driving speed of the dynamo and the electric current in the stator of the electric motor. When the dynamo is driven at constant torque instead of constant rotation rate, chaotic reversals of the generated current and of the angular rotation of the motor are observed. Simple deterministic models are presented which capture the observed dynamical regimes.
Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale
NASA Astrophysics Data System (ADS)
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)
Nara, Shigetoshi
2003-09-01
Complex dynamics including chaos in systems with large but finite degrees of freedom are considered from the viewpoint that they would play important roles in complex functioning and controlling of biological systems including the brain, also in complex structure formations in nature. As an example of them, the computer experiments of complex dynamics occurring in a recurrent neural network model are shown. Instabilities, itinerancies, or localization in state space are investigated by means of numerical analysis, for instance by calculating correlation functions between neurons, basin visiting measures of chaotic dynamics, etc. As an example of functional experiments with use of such complex dynamics, we show the results of executing a memory search task which is set in a typical ill-posed context. We call such useful dynamics "constrained chaos," which might be called "chaotic itinerancy" as well. These results indicate that constrained chaos could be potentially useful in complex functioning and controlling for systems with large but finite degrees of freedom typically observed in biological systems and may be such that working in a delicate balance between converging dynamics and diverging dynamics in high dimensional state space depending on given situation, environment and context to be controlled or to be processed.
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
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
NASA Astrophysics Data System (ADS)
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.
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
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)
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.
Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-01-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented. PMID:23558425
NASA Astrophysics Data System (ADS)
Kotulski, Zbigniew; Szczepaski, Janusz
In the paper we propose a new method of constructing cryptosystems utilising a nonpredictability property of discrete chaotic systems. We formulate the requirements for such systems to assure their safety. We also give examples of practical realisation of chaotic cryptosystems, using a generalisation of the method presented in [7]. The proposed algorithm of encryption and decryption is based on multiple iteration of a certain dynamical chaotic system. We assume that some part of the initial condition is a plain message. As the secret key we assume the system parameter(s) and additionally another part of the initial condition.
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
Turner, L.
1996-11-01
Adhering to the lore that vorticity is a critical ingredient of fluid turbulence, a triad of coupled helicity (vorticity) states of the incompressible Navier-Stokes fluid are followed. Effects of the remaining states of the fluid on the triad are then modeled as a simple driving term. Numerical solution of the equations yield attractors that seem strange and chaotic. This suggests that the unpredictability of nonlinear fluid dynamics (i.e., turbulence) may be traced back to the most primordial structure of the Navier-Stokes equation; namely, the driven triadic interaction. {copyright} {ital 1996 The American Physical Society.}
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.
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.
NASA Astrophysics Data System (ADS)
Chen, Heng-Hui
2004-06-01
An analysis of stability and chaotic dynamics is presented by a single-axis rate gyro subjected to linear feedback control loops. This rate gyro is supposed to be mounted on a space vehicle which undergoes an uncertain angular velocity ωZ( t) around its spin axis. And simultaneously acceleration ω˙X(t) occurs with respect to the output axis. The necessary and sufficient conditions of stability for the autonomous case, whose vehicle undergoes a steady rotation, were provided by Routh-Hurwitz theory. Also, the degeneracy conditions of the non-hyperbolic point were derived and the dynamics of the resulting system on the center manifold near the double-zero degenerate point by using center manifold and normal form methods were examined. The stability of the non-linear non-autonomous system was investigated by Liapunov stability and instability theorems. As the electrical time constant is much smaller than the mechanical time constant, the singularly perturbed system can be obtained by the singular perturbation theory. The Liapunov stability of this system by studying the reduced and boundary-layer systems was also analyzed. Numerical simulations were performed to verify the analytical results. The stable regions of the autonomous system were obtained in parametric diagrams. For the non-autonomous case in which ωZ( t) oscillates near boundary of stability, periodic, quasiperiodic and chaotic motions were demonstrated by using time history, phase plane and Poincaré maps.
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.
Anomalous Diffusion and Mixing of Chaotic Orbits in Hamiltonian Dynamical Systems
NASA Astrophysics Data System (ADS)
Ishizaki, R.; Horita, T.; Mori, H.
1993-05-01
Anomalous behaviors of the diffusion and mixing of chaotic orbits due to the intermittent sticking to the islands of normal tori and accelerator-mode tori in a widespread chaotic sea are studied numerically and theoretically for Hamiltonian systems with two degrees of freedom. The probability distribution functions for the coarse-grained velocity (characterizing the diffusion) and the coarse-grained expansion rate (characterizing the mixing) turn out to obey an anomalous scaling law which is quite different from the Gaussian. The scaling law is confirmed for both diffusion and mixing by numerical experiments on the heating map introduced by Karney, which exhibits remarkable statistical properties more clearly than the standard map. Its scaling exponents for the two cases, however, are found to be different from each other.
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.
Are earthquakes deterministic or chaotic?
NASA Astrophysics Data System (ADS)
Rundle, John B.; Julian, Bruce R.; Turcotte, Donald L.
During the last decade, physicists and applied mathematicians have made substantial headway in understanding the dynamics of complex nonlinear systems. Progress has been possible due to the development of several new tools, including the renormalization group approach, phase portraits, and scaling methods (fractals). At the same time, mathematical geophysicists interested in earthquakes have begun to utilize these same concepts to generate models of faults and fractures.In order to bring these scientific communities together, it was decided to convene the workshop, Physics of Earthquake Faults: Deterministic or Chaotic?, held February 12-15, at the Asilomar conference center near Monterey, Calif. Thirty-six Earth scientists met with 15 physicists and applied mathematicians to discuss how recent advances in nonlinear systems might be applied to better understand earthquakes. Funding was provided by the Geodynamics Branch of the National Aeronautics and Space Administration, the National Science Foundation, and the Office of Basic Energy Sciences of the U.S. Department of Energy. Organizational and logistical support were provided by the U.S. Geological Survey.
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.
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.
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.…
NASA Astrophysics Data System (ADS)
Egorov, E. N.; Hramov, A. E.
2006-08-01
The effect of the strength of the focusing magnetic field on chaotic dynamic processes occurring in an electron beam with a virtual cathode, as well as on the processes whereby the structures form in the beam and interact with each other, is studied by means of two-dimensional numerical simulations based on solving a self-consistent set of Vlasov-Maxwell equations. It is shown that, as the focusing magnetic field is decreased, the dynamics of an electron beam with a virtual cathode becomes more complicated due to the formation and interaction of spatiotemporal longitudinal and transverse structures in the interaction region of a vircator. The optimum efficiency of the interaction of an electron beam with the electromagnetic field of the vircator is achieved at a comparatively weak external magnetic field and is determined by the fundamentally two-dimensional nature of the motion of the beam electrons near the virtual cathode.
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.
Cheng, Jun; Sun, Jing; Huang, Yun; Feng, Jia; Zhou, Junhu; Cen, Kefa
2013-12-01
To extract lipids from wet microalgae through cell disruption, the effects of microwave treatment on the dynamic cell wall microstructures were investigated. The fractal dimension of raw, untreated microalgal cells was 1.46. The disruption level of microalgal cell walls was enhanced when microwave treatment temperature increased from 80 to 120°C, resulting in an increase in microalgal cell fractal dimension from 1.61 to 1.91. The cell wall thickness and pore diameters in cell walls increased from 0.11 to 0.59 μm and from 0.005 to 0.18 μm, respectively, when microwave treatment time increased from 0 to 20 min. The outer pectin layers of cell walls gradually detached and the porosity of inner cellulose layers increased when microwave treatment time increased to 26 min. The initial point of disruption appeared at the maximum curvature (approximately 1.01×10(7) m(-1)) of cell walls. Numbers of short-chain and saturated lipids increased because of microwave electromagnetic effect. PMID:24152788
NASA Astrophysics Data System (ADS)
Li, Damei; Lu, Jun-An; Wu, Xiaoqun; Chen, Guanrong
2006-11-01
To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos
``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.
Hayashi, Kenta; Gotoda, Hiroshi; Gentili, Pier Luigi
2016-05-01
The convective motions within a solution of a photochromic spiro-oxazine being irradiated by UV only on the bottom part of its volume, give rise to aperiodic spectrophotometric dynamics. In this paper, we study three nonlinear properties of the aperiodic time series: permutation entropy, short-term predictability and long-term unpredictability, and degree distribution of the visibility graph networks. After ascertaining the extracted chaotic features, we show how the aperiodic time series can be exploited to implement all the fundamental two-inputs binary logic functions (AND, OR, NAND, NOR, XOR, and XNOR) and some basic arithmetic operations (half-adder, full-adder, half-subtractor). This is possible due to the wide range of states a nonlinear system accesses in the course of its evolution. Therefore, the solution of the convective photochemical oscillator results in hardware for chaos-computing alternative to conventional complementary metal-oxide semiconductor-based integrated circuits. PMID:27249942
NASA Astrophysics Data System (ADS)
Hayashi, Kenta; Gotoda, Hiroshi; Gentili, Pier Luigi
2016-05-01
The convective motions within a solution of a photochromic spiro-oxazine being irradiated by UV only on the bottom part of its volume, give rise to aperiodic spectrophotometric dynamics. In this paper, we study three nonlinear properties of the aperiodic time series: permutation entropy, short-term predictability and long-term unpredictability, and degree distribution of the visibility graph networks. After ascertaining the extracted chaotic features, we show how the aperiodic time series can be exploited to implement all the fundamental two-inputs binary logic functions (AND, OR, NAND, NOR, XOR, and XNOR) and some basic arithmetic operations (half-adder, full-adder, half-subtractor). This is possible due to the wide range of states a nonlinear system accesses in the course of its evolution. Therefore, the solution of the convective photochemical oscillator results in hardware for chaos-computing alternative to conventional complementary metal-oxide semiconductor-based integrated circuits.
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.
NASA Astrophysics Data System (ADS)
Cushman, J. H.; O'Malley, D.; Park, M.
2009-04-01
We construct a family of stochastic processes with independent, nonstationary increments and arbitrary, but apriori specified mean square displacement. The family of processes is shown to be an extension of Brownian motion. If the time derivative of the variance of the process is homogeneous, then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of Brownian motion. For two particles initially separated by a distance x, the finite-size Lyapunov exponent (FSLE), measures the average rate of exponential separation to a distance ax. An analytical expression is developed for the FSLE of the extended Brownian processes and numerical examples presented. The construction of the extended Brownian processes illustrates that contrary to what has been stated in the literature, a power-law mean-square displacement is not related to a breakdown in the classical CLT.
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, i&tacute;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 (Muuḱil Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoiŕ models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mes&hacute; cells was designed and calibrated in the studied area. The statistically sound power law relations were
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
Capturing correlations in chaotic diffusion by approximation methods.
Knight, Georgie; Klages, Rainer
2011-10-01
We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line that contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher-order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, while the third method approximates Markov partitions and transition matrices by using a slight variation of the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence, and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in the case of dynamics where exact results for the diffusion coefficient are not available. PMID:22181115
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. PMID:25900114
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.
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
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
Riemann zeros, prime numbers, and fractal potentials.
van Zyl, Brandon P; Hutchinson, David A W
2003-06-01
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels. PMID:16241330
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.
Chaotic Nonlinear Prime Number Function
NASA Astrophysics Data System (ADS)
Mateos, Luis A.
2011-06-01
Dynamical systems in nature, such as heartbeat patterns, DNA sequence pattern, prime number distribution, etc., exhibit nonlinear (chaotic) space-time fluctuations and exact quantification of the fluctuation pattern for predictability purposes has not yet been achieved [1]. In this paper a chaotic-nonlinear prime number function P(s) is developed, from which prime numbers are generated and decoded while composite numbers are encoded over time following the Euler product methodology, which works on sequences progressively culled from multiples of the preceding primes. By relating this P(s) to a virtually closed 2D number line manifold, it is possible to represent the evolving in time of nonlinear (chaotic) systems to a final value where the system becomes stable, becomes linear. This nonlinear prime number function is proposed as a chaotic model system able to describe chaotic systems.
Initial-state dependence of the quench dynamics in integrable quantum systems. III. Chaotic states
NASA Astrophysics Data System (ADS)
He, Kai; Rigol, Marcos
2013-04-01
We study sudden quantum quenches in which the initial states are selected to be either eigenstates of an integrable Hamiltonian that is nonmappable to a noninteracting one or a nonintegrable Hamiltonian, while the Hamiltonian after the quench is always integrable and mappable to a noninteracting one. By studying weighted energy densities and entropies, we show that quenches starting from nonintegrable (chaotic) eigenstates lead to an “ergodic” sampling of the eigenstates of the final Hamiltonian, while those starting from the integrable eigenstates do not (or at least it is not apparent for the system sizes accessible to us). This goes in parallel with the fact that the distribution of conserved quantities in the initial states is thermal in the nonintegrable cases and nonthermal in the integrable ones, and means that, in general, thermalization occurs in integrable systems when the quench starts form an eigenstate of a nonintegrable Hamiltonian (away from the edges of the spectrum), while it fails (or requires larger system sizes than those studied here to become apparent) for quenches starting at integrable points. We test those conclusions by studying the momentum distribution function of hard-core bosons after a quench.
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
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.
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.
Christov, Ivan C.; Lueptow, Richard M.; Ottino, Julio M.; Sturman, Rob
2014-01-01
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
2014-01-01
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
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.
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
Chaotic advection in blood flow.
Schelin, A B; Károlyi, Gy; de Moura, A P S; Booth, N A; Grebogi, C
2009-07-01
In this paper we argue that the effects of irregular chaotic motion of particles transported by blood can play a major role in the development of serious circulatory diseases. Vessel wall irregularities modify the flow field, changing in a nontrivial way the transport and activation of biochemically active particles. We argue that blood particle transport is often chaotic in realistic physiological conditions. We also argue that this chaotic behavior of the flow has crucial consequences for the dynamics of important processes in the blood, such as the activation of platelets which are involved in the thrombus formation. PMID:19658798
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
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.
Plate falling in a fluid: Regular and chaotic dynamics of finite-dimensional models
NASA Astrophysics Data System (ADS)
Kuznetsov, Sergey P.
2015-05-01
Results are reviewed concerning the planar problem of a plate falling in a resisting medium studied with models based on ordinary differential equations for a small number of dynamical variables. A unified model is introduced to conduct a comparative analysis of the dynamical behaviors of models of Kozlov, Tanabe-Kaneko, Belmonte-Eisenberg-Moses and Andersen-Pesavento-Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models manifests certain similarities caused by the same inherent symmetry and by the universal nature of the phenomena involved in nonlinear dynamics (fixed points, limit cycles, attractors, and bifurcations).
Cary, J.R.
1992-11-30
Substantial progress was in several areas of accelerator dynamics. For developing understanding of longitudinal adiabatic dynamics, and for creating efficiency enhancements of recirculating free-electron lasers, was substantially completed. A computer code for analyzing the critical KAM tori that bound the dynamic aperture in circular machines was developed. Studies of modes that arise due to the interaction of coating beams with a narrow-spectrum impedance have begun. During this research educational and research ties with the accelerator community at large have been strengthened.
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.
Chaos-based encryption for fractal image coding
NASA Astrophysics Data System (ADS)
Yuen, Ching-Hung; Wong, Kwok-Wo
2012-01-01
A chaos-based cryptosystem for fractal image coding is proposed. The Rényi chaotic map is employed to determine the order of processing the range blocks and to generate the keystream for masking the encoded sequence. Compared with the standard approach of fractal image coding followed by the Advanced Encryption Standard, our scheme offers a higher sensitivity to both plaintext and ciphertext at a comparable operating efficiency. The keystream generated by the Rényi chaotic map passes the randomness tests set by the United States National Institute of Standards and Technology, and so the proposed scheme is sensitive to the key.
Bose-Einstein condensates on tilted lattices: Coherent, chaotic, and subdiffusive dynamics
Kolovsky, Andrey R.; Gomez, Edgar A.; Korsch, Hans Juergen
2010-02-15
The dynamics of a (quasi-) one-dimensional interacting atomic Bose-Einstein condensate in a tilted optical lattice is studied in a discrete mean-field approximation, i.e., in terms of the discrete nonlinear Schroedinger equation. If the static field is varied, the system shows a plethora of dynamical phenomena. In the strong field limit, we demonstrate the existence of (almost) nonspreading states which remain localized on the lattice region populated initially and show coherent Bloch oscillations with fractional revivals in the momentum space (so-called quantum carpets). With decreasing field, the dynamics becomes irregular, however, still confined in configuration space. For even weaker fields, we find subdiffusive dynamics with a wave-packet width growing as t{sup 1/4}.
Regular versus chaotic dynamics in closed systems of interacting Fermi particles
Izrailev, F.M.; Castaneda-Mendoza, A.
2005-07-08
We discuss dynamical properties of strongly interacting Fermi-particles. Main attention is paid to the evolution of wave packets in the many-particle basis of non-interacting particles. Specifically, we analyze the time dependence of the return probability and the Shannon entropy of packets. We start with the model of two-body random interaction which allows us to obtain analytical expression for the time dependence of the above quantities. Analytical results are compared with numerical data obtained in direct simulation of the wave packet dynamics. To understand to what extent these results are generic, we have considered the spin model of a quantum computation with a non-random (dynamical) interaction between spins. We have found that the linear increase of the Shannon entropy observed in the two-body random model, occurs, under some conditions, in the dynamical model. Finally, we have analyzed the role of weak external perturbation taken in the form of static disorder.
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)
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.
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
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.
Regular and Chaotic Dynamics of Periodic and Quasi-Periodic Motions
NASA Astrophysics Data System (ADS)
Celletti, Alessandra
We review some basic topics from Dynamical System theory, which are of interest in Space Manifold Dynamics. We start by recalling some notions related to equilibrium points. Floquet theorem leads to the introduction of Lyapunov exponents. Nearly-integrable systems are very common in Celestial Mechanics; their study motivated the development of perturbation theories as well as of KAM and Nekhoroshev's theorem. The Lindstedt-Poincaré technique allows to look for periodic orbits. Finally, we recall the derivation of the Lagrangian points in the circular and elliptic, planar, restricted three-body problem. Each section is almost self-contained and can be read independently from the others.
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.
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). PMID
NASA Astrophysics Data System (ADS)
Wuorinen, Charles
2015-03-01
Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...
The study of effects of small perturbations on chaotic systems
Grebogi, C.; Yorke, J.A.
1991-12-01
This report discusses the following topics: controlling chaotic dynamical systems; embedding of experimental data; effect of noise on critical exponents of crises; transition to chaotic scattering; and distribution of floaters on a fluid surface. (LSP)
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.
Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators
NASA Astrophysics Data System (ADS)
Pinto, Carla M. A.
We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?
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.
Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions
Gospodinov, Dragomir; Marekova, Elisaveta; Marinov, Alexander
2010-01-21
A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/r{sup D}propor tor{sup -D} where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A 'box-counting' procedure could also be applied giving the 'capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term 'fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis.The fractal coefficients analysis provides some important advantages in examining earthquake spatial
Fractal Dimension in Epileptic EEG Signal Analysis
NASA Astrophysics Data System (ADS)
Uthayakumar, R.
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi's algorithm, advantages of this method include
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).
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 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.
Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics
NASA Astrophysics Data System (ADS)
Kuznetsov, Sergey P.; Kruglov, Vyacheslav P.
2016-03-01
Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale-Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.
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.
On chaotic dynamics in "pseudobilliard" Hamiltonian systems with two degrees of freedom
NASA Astrophysics Data System (ADS)
Eleonsky, V. M.; Korolev, V. G.; Kulagin, N. E.
1997-12-01
A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail.
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.
Fractals in petroleum geology and earth processes
Barton, C.C.; La Pointe, P.R.
1995-12-31
The editors of this book chose a diverse spectrum of papers written by pioneers in the field of fractals and their application to the exploration and production of hydrocarbons. The geology of the Earth`s crust is complex, chaotic, and unpredictable. Fractal geometry can quantify the spatial heterogeneity of the different geologic patterns and ultimately help improve the results of both production and exploration. To this goal the book has accomplished such an objective with diverse, well-chosen contributions from a variety of experts in the field. The book starts with a chapter introducing the basics, with a short historical foot-note by Benoit Mandelbrot, who is considered the {open_quotes}father of fractals.{close_quotes} Mandelbrot emphasized that geologic processes not only exhibit fractal properties but also are strongly connected to the economic system. This paved the way for the next three chapters that deal with the size and spatial distribution of hydrocarbon reserves and their importance in economic evaluations. The following four chapters deal with the fractal processes as related to sedimentologic, stratigraphic, and geomorphologic systems. Chapter five is an interesting one that deals with stratigraphic models and how their fractal processes can be tied with the inter-well correlation and reconstruct depositional environments. The next three chapters are concerned with porous and fractured rocks and how they affect the flow of fluids. The last two chapters (chapters 13 and 14) are of particular interest. Chapter 13 deals with the vertical vs. horizontal well-log variability and application to fractal reservoir modeling. Chapter 14 illustrates how fractal geometry brings mathematical order to geological and geophysical disorder. This is evident when dealing with geophysical modeling and inversion.
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 dynamics of Yang-Mills field as source of particle couplings and masses
Goldfain, E.
1995-04-01
Dynamics of classical uniform Yang-Mills fields is explored from the viewpoint of universal route to chaos in nonlinear systems. The author shows how the path to nonintegrable behavior of the field is equivalent to the period doubling bifurcation of the logistic map. Universal scalings of the growth parameter yield the full set of Standard Model couplings. Hamiltonian formulation in action-angle variables leads to the physics of phase transitions in classical lattice models. The ground state phase diagram of the system with {open_quotes}antiferromagnetic{close_quotes} interaction is known to exhibit a devil`s staircase form. Linking the staircase attributes to the asymptotic freedom of the gauge coupling yields an universal mass equation. Critical exponent is found to depend on the number of field flavors. Further solving the model for various stability plateaus renders the spectrum of particle masses in the low energy framework. Agreement between theory and experimental results is confirmed for the photon/graviton pair, weak bosons, leptons and quarks. The approach offers an intriguing explanation of the dymanical origin of the physical mass and on the internal hierarchy of particle families.
Elastic properties of inhomogeneous media with chaotic structure.
Novikov, V V; Wojciechowski, K W; Belov, D V; Privalko, V P
2001-03-01
The elastic properties of an inhomogeneous medium with chaotic structure were derived within the framework of a fractal model using the iterative averaging approach. The predicted values of a critical index for the bulk elastic modulus and of the Poisson ratio in the vicinity of a percolation threshold were in fair agreement with the available experimental data for inhomogeneous composites. PMID:11308722
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
NASA Astrophysics Data System (ADS)
Wang, C.; Sun, B. A.; Wang, W. H.; Bai, H. Y.
2016-02-01
We study serrated flow dynamics during brittle-to-ductile transition induced by tuning the sample aspect ratio in a Zr-based metallic glass. The statistical analysis reveals that the serrated flow dynamics transforms from a chaotic state characterized by Gaussian-distribution serrations corresponding to stick-slip motion of randomly generated and uncorrelated single shear band and brittle behavior, into a self-organized critical state featured by intermittent scale-free distribution of shear avalanches corresponding to a collective motion of multiple shear bands and ductile behavior. The correlation found between serrated flow dynamics and plastic deformation might shed light on the plastic deformation dynamic and mechanism in metallic glasses.
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.
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
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
Equilibration of quantum chaotic systems.
Zhuang, Quntao; Wu, Biao
2013-12-01
The quantum ergordic theorem for a large class of quantum systems was proved by von Neumann [Z. Phys. 57, 30 (1929)] and again by Reimann [Phys. Rev. Lett. 101, 190403 (2008)] in a more practical and well-defined form. However, it is not clear whether the theorem applies to quantum chaotic systems. With a rigorous proof still elusive, we illustrate and verify this theorem for quantum chaotic systems with examples. Our numerical results show that a quantum chaotic system with an initial low-entropy state will dynamically relax to a high-entropy state and reach equilibrium. The quantum equilibrium state reached after dynamical relaxation bears a remarkable resemblance to the classical microcanonical ensemble. However, the fluctuations around equilibrium are distinct: The quantum fluctuations are exponential while the classical fluctuations are Gaussian. PMID:24483425
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.
Chaotic Systems with Absorption
NASA Astrophysics Data System (ADS)
Altmann, Eduardo G.; Portela, Jefferson S. E.; Tél, Tamás
2013-10-01
Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate κ in terms of the natural conditionally invariant measure of the system, (ii) an increased multifractality when compared to the spectrum of dimensions Dq obtained without taking absorption and return times into account, and (iii) a generalization of the Kantz-Grassberger formula that expresses D1 in terms of κ, the positive Lyapunov exponent, the average return time, and a new quantity, the reflection rate. Simulations in the cardioid billiard confirm these results.
NASA Astrophysics Data System (ADS)
Hausdorff, Jeffrey M.
2009-06-01
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.
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
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. PMID:26966561
NASA Astrophysics Data System (ADS)
Salazar, F. J. T.; Macau, E. E. N.; Winter, O. C.
In the frame of the equilateral equilibrium points exploration, numerous future space missions will require maximization of payload mass, simultaneously achieving reasonable transfer times. To fulfill this request, low-energy non-Keplerian orbits could be used to reach L4 and L5 in the Earth-Moon system instead of high energetic transfers. Previous studies have shown that chaos in physical systems like the restricted three-body Earth-Moon-particle problem can be used to direct a chaotic trajectory to a target that has been previously considered. In this work, we propose to transfer a spacecraft from a circular Earth Orbit in the chaotic region to the equilateral equilibrium points L4 and L5 in the Earth-Moon system, exploiting the chaotic region that connects the Earth with the Moon and changing the trajectory of the spacecraft (relative to the Earth) by using a gravity assist maneuver with the Moon. Choosing a sequence of small perturbations, the time of flight is reduced and the spacecraft is guided to a proper trajectory so that it uses the Moon's gravitational force to finally arrive at a desired target. In this study, the desired target will be an orbit about the Lagrangian equilibrium points L4 or L5. This strategy is not only more efficient with respect to thrust requirement, but also its time transfer is comparable to other known transfer techniques based on time optimization.
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.
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
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
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
Reinventing the wheel: The chaotic sandwheel
NASA Astrophysics Data System (ADS)
Tongen, Anthony; Thelwell, Roger J.; Becerra-Alonso, David
2013-02-01
The Malkus chaotic waterwheel, a tool to mechanically demonstrate Lorenzian dynamics, motivates the study of a chaotic sandwheel. We model the sandwheel in parallel with the waterwheel when possible, noting where methods may be extended and where no further analysis seems feasible. Numerical simulations are used to compare and contrast the behavior of the sandwheel with the waterwheel. Simulations confirm that the sandwheel retains many of the elements of chaotic Lorenzian dynamics. However, bifurcation diagrams show dramatic differences in where the order-chaos-order transitions occur.
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)
Chaotic vibrations of nonlinearly supported tubes in crossflow
Cai, Y.; Chen, S.S.
1992-02-01
By means of the unsteady-flow theory and a bilinear mathematical model, a theoretical study is presented for chaotic vibrations associated with the fluidelastic instability of nonlinearly supported tubes in a crossflow. Effective tools, including phase portraits, power spectral density, Poincare maps, Lyapunov exponent, fractal dimension, and bifurcation diagrams, are utilized to distinguish periodic and chaotic motions when the tubes vibrate in the instability region. The results show periodic and chaotic motions in the region corresponding to fluid-damping-controlled instability. Nonlinear supports, with symmetric or asymmetric gaps, significantly affect the distribution of periodic, quasiperiodic, and chaotic motions of a tube exposed to various flow velocities in the instability region of the tube-support-plate-inactive mode.
Chaotic vibrations of tubes with nonlinear supports in crossflow
Cai, Y.; Chen, S.S.
1992-12-01
By means of the unsteady flow theory and a bilinear mathematical model, a theoretical study is presented for chaotic vibrations associated with the fluidelastic instability of nonlinearly supported tubes in a crossflow. A series of effective tools, including phase portraits, power spectral density, Poincar`e maps, Lyapunov exponent, fractal dimension, and bifurcation diagrams, are utilized to distinguish periodic and chaotic motions when the tubes vibrate in the instability region. Results show periodic and chaotic motions in the region corresponding to the fluid damping controlled instability. Nonlinear supports, with symmetric or asymmetric gaps, significantly affect the distributions of periodic, quasiperiodic and chaotic motions of the tube with various flow velocity in the instability region of the TSP(tube-support-plate)-inactive mode.
Chaotic vibrations of tubes with nonlinear supports in crossflow
Cai, Y.; Chen, S.S.
1992-01-01
By means of the unsteady flow theory and a bilinear mathematical model, a theoretical study is presented for chaotic vibrations associated with the fluidelastic instability of nonlinearly supported tubes in a crossflow. A series of effective tools, including phase portraits, power spectral density, Poincar'e maps, Lyapunov exponent, fractal dimension, and bifurcation diagrams, are utilized to distinguish periodic and chaotic motions when the tubes vibrate in the instability region. Results show periodic and chaotic motions in the region corresponding to the fluid damping controlled instability. Nonlinear supports, with symmetric or asymmetric gaps, significantly affect the distributions of periodic, quasiperiodic and chaotic motions of the tube with various flow velocity in the instability region of the TSP(tube-support-plate)-inactive mode.
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.…
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
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.
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
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
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
Meyer, M; Rahmel, A; Marconi, C; Grassi, B; Skinner, J E; Cerretelli, P
1998-01-01
The dynamics of heartbeat interval time series over large time scales were studied by a modified random walk analysis introduced recently as Detrended Fluctuation Analysis. In this analysis, the intrinsic fractal long-range power-law correlation properties of beat-to-beat fluctuations generated by the dynamical system (i.e., cardiac rhythm generator), after decomposition from extrinsic uncorrelated sources, can be quantified by the scaling exponent (alpha) which, in healthy subjects, for time scales of approximately 10(4) beats is approximately 1.0. The effects of chronic hypoxia were determined from serial heartbeat interval time series of digitized twenty-four-hour ambulatory ECGs recorded in nine healthy subjects (mean age thirty-four years old) at sea level and during a sojourn at 5,050 m for thirty-four days (EvK2-CNR Pyramid Laboratory, Sagarmatha National Park, Nepal). The group averaged alpha exponent (+/- SD) was 0.99 +/- 0.04 (range 0.93-1.04). Longitudinal assessment of alpha in individual subjects did not reveal any effect of exposure to chronic high altitude hypoxia. The finding of alpha approximately 1 indicating scale-invariant long-range power-law correlations (1/f noise) of heartbeat fluctuations would reflect a genuinely self-similar fractal process that typically generates fluctuations on a wide range of time scales. Lack of a characteristic time scale along with the absence of any effect from exposure to chronic hypoxia on scaling properties suggests that the neuroautonomic cardiac control system is preadapted to hypoxia which helps prevent excessive mode-locking (error tolerance) that would restrict its functional responsiveness (plasticity) to hypoxic or other physiological stimuli. PMID:9594353
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. PMID:22938916
Fractal scaling of microbial colonies affects growth
NASA Astrophysics Data System (ADS)
Károlyi, György
2005-03-01
The growth dynamics of filamentary microbial colonies is investigated. Fractality of the fungal or actinomycetes colonies is shown both theoretically and in numerical experiments to play an important role. The growth observed in real colonies is described by the assumption of time-dependent fractality related to the different ages of various parts of the colony. The theoretical results are compared to a simulation based on branching random walks.
Evolution to a small-world network with chaotic units
NASA Astrophysics Data System (ADS)
Gong, P.; van Leeuwen, C.
2004-07-01
We investigated the mutually supporting role of chaotic activity and evolving structure in a complex network. An initially randomly coupled network with chaotic activation is adaptively rewired according to dynamic coherence between its units. The evolving network reaches a small-world structure. Meanwhile, collective network activity tends to an intermittent dynamic clustering regime. Spontaneous chaotic activity and adaptively evolving structure jointly enhance signal propagation capacity.
Intrinsic irreversibility and the validity of the kinetic description of chaotic systems
Hasegawa, H.H.; Driebe, D.J. International Solvay Institutes for Physics and Chemistry, 1050 Brussels )
1994-09-01
Irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities. The generalized representation is valid for one-dimensional systems when the initial probability density satisfies certain physical conditions'' of smoothness. The formalism is first applied to the one-dimensional multi-Bernoulli map, which is a simple map displaying deterministic diffusion. The two-dimensional, invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered. The generalized representation is constructed using a resolvent formalism. The eigenstates of the diffusive systems are seen to be of a fractal nature.
Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems
NASA Astrophysics Data System (ADS)
Hildebrand, R.; Lokutsievskiy, L. V.; Zelikin, M. I.
2013-03-01
Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).
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.
Image encryption with chaotically coupled chaotic maps
NASA Astrophysics Data System (ADS)
Pisarchik, A. N.; Zanin, M.
2008-10-01
We present a novel secure cryptosystem for direct encryption of color images, based on chaotically coupled chaotic maps. The proposed cipher provides good confusion and diffusion properties that ensures extremely high security because of the chaotic mixing of pixels’ colors. Information is mixed and distributed over a complete image using a complex strategy that makes known plaintext attack unfeasible. The encryption algorithm guarantees the three main goals of cryptography: strong cryptographic security, short encryption/decryption time, and robustness against noise and other external disturbances. Due to the high speed, the proposed cryptosystem is suitable for application in real-time communication systems.
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. PMID:26565163
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.
Chaotic systems with absorption.
Altmann, Eduardo G; Portela, Jefferson S E; Tél, Tamás
2013-10-01
Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate κ in terms of the natural conditionally invariant measure of the system, (ii) an increased multifractality when compared to the spectrum of dimensions D(q) obtained without taking absorption and return times into account, and (iii) a generalization of the Kantz-Grassberger formula that expresses D(1) in terms of κ, the positive Lyapunov exponent, the average return time, and a new quantity, the reflection rate. Simulations in the cardioid billiard confirm these results. PMID:24138240
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.
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)
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.
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)
Fractal Geometry of Architecture
NASA Astrophysics Data System (ADS)
Lorenz, Wolfgang E.
In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately, scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects of the twentieth century called for an overall idea that is mirrored in every single detail, but also Gothic cathedrals and Indian temples offer self-similarity. This study mainly focuses upon the question whether this concept of self-similarity makes architecture with fractal properties more diverse and interesting than Euclidean Modern architecture. The first part gives an introduction and explains Fractal properties in various natural and architectural objects, presenting the underlying structure by computer programmed renderings. In this connection, differences between the fractal, architectural concept and true, mathematical Fractals are worked out to become aware of limits. This is the basis for dealing with the problem whether fractal-like architecture, particularly facades, can be measured so that different designs can be compared with each other under the aspect of fractal properties. Finally the usability of the Box-Counting Method, an easy-to-use measurement method of Fractal Dimension is analyzed with regard to architecture.
Grebogi, C.; Yorke, J.A.
1991-12-01
This report discusses the following topics: controlling chaotic dynamical systems; embedding of experimental data; effect of noise on critical exponents of crises; transition to chaotic scattering; and distribution of floaters on a fluid surface. (LSP)
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
Fractal Plate Reconstructions, Incorporating Asymmetric Spreading and Kinematics
NASA Astrophysics Data System (ADS)
Pilger, R.
2011-12-01
Accretionary plate boundaries - spreading centers and associated transform faults - possess fractal structure like coastlines. Their apparent length demonstrates self-similarity over a range of scales, maximizing multiplicity (entropy) in a recursive chaotic process. Further, optimal, combined oceanic plate reconstructions, incorporating asymmetric accretion over a range of ages produce fractal structure. The minimum fractal configuration as a function of the reconstruction parameters approximates the Lagrangian constraint (the information) in the maximum entropy formalism. The optimal fractal spectrum itself represents maximum entropy of the reconstructed data describing the spreading center for the preferred rotation parameters. Because fractals are intrinsically discontinuous (and analytic derivatives are unavailable), conventional non-linear least squares approaches are inapplicable. Instead, derivative-free, iterative conjugate gradient and simplex algorithms are utilized. In order to allow for kinematic calculations and integrated reconstructions of diverse data ages, parameters are spline-interpolated, rate-normalized, pseudo-vectors. The new formalism provides a unique fitting criterion and algorithm for simultaneous plate and spreading-center reconstruction and kinematics. It also provides a fractal template for reconstructions of other tectonic types.
Unified model of fractal conductance fluctuations for diffusive and ballistic semiconductor devices
NASA Astrophysics Data System (ADS)
Marlow, C. A.; Taylor, R. P.; Martin, T. P.; Scannell, B. C.; Linke, H.; Fairbanks, M. S.; Hall, G. D. R.; Shorubalko, I.; Samuelson, L.; Fromhold, T. M.; Brown, C. V.; Hackens, B.; Faniel, S.; Gustin, C.; Bayot, V.; Wallart, X.; Bollaert, S.; Cappy, A.
2006-05-01
We present an experimental comparison of magnetoconductance fluctuations measured in the ballistic, quasiballistic, and diffusive scattering regimes of semiconductor devices. In contradiction to expectations, we show that the spectral content of the magnetoconductance fluctuations exhibits an identical fractal behavior for these scattering regimes and that this behavior is remarkably insensitive to device boundary properties. We propose a unified model of fractal conductance fluctuations in the ballistic, quasiballistic, and diffusive transport regimes, in which the generic fractal behavior is generated by a subtle interplay between boundary and material-induced chaotic scattering events.
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.
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
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.
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.
Fractal structures in stenoses and aneurysms in blood vessels
Schelin, Adriane B.; Károlyi, György; de Moura, Alessandro P. S.; Booth, Nuala A.; Grebogi, Celso
2010-01-01
Recent advances in the field of chaotic advection provide the impetus to revisit the dynamics of particles transported by blood flow in the presence of vessel wall irregularities. The irregularity, being either a narrowing or expansion of the vessel, mimicking stenoses or aneurysms, generates abnormal flow patterns that lead to a peculiar filamentary distribution of advected particles, which, in the blood, would include platelets. Using a simple model, we show how the filamentary distribution depends on the size of the vessel wall irregularity, and how it varies under resting or exercise conditions. The particles transported by blood flow that spend a long time around a disturbance either stick to the vessel wall or reside on fractal filaments. We show that the faster flow associated with exercise creates widespread filaments where particles can get trapped for a longer time, thus allowing for the possible activation of such particles. We argue, based on previous results in the field of active processes in flows, that the non-trivial long-time distribution of transported particles has the potential to have major effects on biochemical processes occurring in blood flow, including the activation and deposition of platelets. One aspect of the generality of our approach is that it also applies to other relevant biological processes, an example being the coexistence of plankton species investigated previously. PMID:21078637
NASA Astrophysics Data System (ADS)
Samson, A. M.; Kotomtseva, L. A.; Grigor'eva, E. V.
1989-02-01
A theoretical study of the dynamics of a laser with a bleachable filter revealed chaotic lasing regimes and ranges of bistable states of parameters close to those found in reality. It is shown how a transition to chaos occurs as a result of period-doubling bifurcation. A study is reported of the degree of chaos and of the structure of the resultant strange attractor by calculation of its fractal dimensionality and of the Lyapunov indices.
Introduction to Controversial Topics in Nonlinear Science: Is the Normal Heart Rate Chaotic?
NASA Astrophysics Data System (ADS)
Glass, Leon
2009-06-01
In June 2008, the editors of Chaos decided to institute a new section to appear from time to time that addresses timely and controversial topics related to nonlinear science. The first of these deals with the dynamical characterization of human heart rate variability. We asked authors to respond to the following questions: Is the normal heart rate chaotic? If the normal heart rate is not chaotic, is there some more appropriate term to characterize the fluctuations (e.g., scaling, fractal, multifractal)? How does the analysis of heart rate variability elucidate the underlying mechanisms controlling the heart rate? Do any analyses of heart rate variability provide clinical information that can be useful in medical assessment (e.g., in helping to assess the risk of sudden cardiac death)? If so, please indicate what additional clinical studies would be useful for measures of heart rate variability to be more broadly accepted by the medical community. In addition, as a challenge for analysis methods, PhysioNet [A. L. Goldberger et al., "PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals," Circulation 101, e215-e220 (2000)] provided data sets from 15 patients of whom five were normal, five had heart failure, and five had atrial fibrillation (http://www.physionet.org/challenge/chaos/). This introductory essay summarizes the main issues and introduces the essays that respond to these questions.
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
NASA Astrophysics Data System (ADS)
Altmann, Eduardo G.; Portela, Jefferson S. E.; Tél, Tamás
2013-04-01
There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. A unified treatment of leaking systems is provided and applications to different physical problems, in both the classical and quantum pictures, are reviewed. The treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption and/or transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the limaçon family of billiards illustrate the main text. Regarding billiard dynamics, it is emphasized that a correct discrete-time representation can be given only in terms of the so-called true-time maps, while traditional Poincaré maps lead to erroneous results. Perron-Frobenius-type operators are generalized so that they describe true-time maps with partial leaks.
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.
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.
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.
Electromagnetic fields in fractal continua
NASA Astrophysics Data System (ADS)
Balankin, Alexander S.; Mena, Baltasar; Patiño, Julián; Morales, Daniel
2013-04-01
Fractal continuum electrodynamics is developed on the basis of a model of three-dimensional continuum ΦD3⊂E3 with a fractal metric. The generalized forms of Maxwell equations are derived employing the local fractional vector calculus related to the Hausdorff derivative. The difference between the fractal continuum electrodynamics based on the fractal metric of continua with Euclidean topology and the electrodynamics in fractional space Fα accounting the fractal topology of continuum with the Euclidean metric is outlined. Some electromagnetic phenomena in fractal media associated with their fractal time and space metrics are discussed.
On the stability of fractal globules.
Schram, Raoul D; Barkema, Gerard T; Schiessel, Helmut
2013-06-14
The fractal globule, a self-similar compact polymer conformation where the chain is spatially segregated on all length scales, has been proposed to result from a sudden polymer collapse. This state has gained renewed interest as one of the prime candidates for the non-entangled states of DNA molecules inside cell nuclei. Here, we present Monte Carlo simulations of collapsing polymers. We find through studying polymers of lengths between 500 and 8000 that a chain collapses into a globule, which is neither fractal, nor as entangled as an equilibrium globule. To demonstrate that the non-fractalness of the conformation is not just the result of the collapse dynamics, we study in addition the dynamics of polymers that start from fractal globule configurations. Also in this case the chain moves quickly to the weakly entangled globule where the polymer is well mixed. After a much longer time the chain entangles reach its equilibrium conformation, the molten globule. We find that the fractal globule is a highly unstable conformation that only exists in the presence of extra constraints such as cross-links. PMID:23781815
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
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.
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
Casimir force between integrable and chaotic pistons
Alvarez, Ezequiel; Mazzitelli, Francisco D.; Wisniacki, Diego A.; Monastra, Alejandro G.
2010-11-15
We have computed numerically the Casimir force between two identical pistons inside a very long cylinder, considering different shapes for the pistons. The pistons can be considered quantum billiards, whose spectrum determines the vacuum force. The smooth part of the spectrum fixes the force at short distances and depends only on geometric quantities like the area or perimeter of the piston. However, correcting terms to the force, coming from the oscillating part of the spectrum which is related to the classical dynamics of the billiard, could be qualitatively different for classically integrable or chaotic systems. We have performed a detailed numerical analysis of the corresponding Casimir force for pistons with regular and chaotic classical dynamics. For a family of stadium billiards, we have found that the correcting part of the Casimir force presents a sudden change in the transition from regular to chaotic geometries. This suggests that there could be signatures of quantum chaos in the Casimir effect.
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
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
Size-Dependent Transition to High-Dimensional Chaotic Dynamics in a Two-Dimensional Excitable Medium
Strain, M.C.; Greenside, H.S.
1998-03-01
The spatiotemporal dynamics of an excitable medium with multiple spiral defects is shown to vary smoothly with system size from short-lived transients for small systems to extensive chaos for large systems. A comparison of the Lyapunov dimension density with the average spiral defect density suggests an average dimension per spiral defect varying between 3 and 7. We discuss some implications of these results for experimental studies of excitable media. {copyright} {ital 1998} {ital The American Physical Society}
Size-Dependent Transition to High-Dimensional Chaotic Dynamics in a Two-Dimensional Excitable Medium
NASA Astrophysics Data System (ADS)
Strain, Matthew C.; Greenside, Henry S.
1998-03-01
The spatiotemporal dynamics of an excitable medium with multiple spiral defects is shown to vary smoothly with system size from short-lived transients for small systems to extensive chaos for large systems. A comparison of the Lyapunov dimension density with the average spiral defect density suggests an average dimension per spiral defect varying between 3 and 7. We discuss some implications of these results for experimental studies of excitable media.
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)
Persistence intervals of fractals
NASA Astrophysics Data System (ADS)
Máté, Gabriell; Heermann, Dieter W.
2014-07-01
Objects and structures presenting fractal like behavior are abundant in the world surrounding us. Fractal theory provides a great deal of tools for the analysis of the scaling properties of these objects. We would like to contribute to the field by analyzing and applying a particular case of the theory behind the P.H. dimension, a concept introduced by MacPherson and Schweinhart, to seek an intuitive explanation for the relation of this dimension and the fractality of certain objects. The approach is based on recently elaborated computational topology methods and it proves to be very useful for investigating scaling hidden in dimensions lower than the “native” dimension in which the investigated object is embedded. We demonstrate the applicability of the method with two examples: the Sierpinski gasket-a traditional fractal-and a two dimensional object composed of short segments arranged according to a circular structure.
Fractal funcitons and multiwavelets
Massopust, P.R.
1997-04-01
This paper reviews how elements from the theory of fractal functions are employed to construct scaling vectors and multiwavelets. Emphasis is placed on the one-dimensional case, however extensions to IR{sup m} are indicated.
ERIC Educational Resources Information Center
Bannon, Thomas J.
1991-01-01
Discussed are several different transformations based on the generation of fractals including self-similar designs, the chaos game, the koch curve, and the Sierpinski Triangle. Three computer programs which illustrate these concepts are provided. (CW)
NASA Astrophysics Data System (ADS)
Ryzhov, Eugene
2015-11-01
Vortex motion in shear flows is of great interest from the point of view of nonlinear science, and also as an applied problem to predict the evolution of vortices in nature. Considering applications to the ocean and atmosphere, it is well-known that these media are significantly stratified. The simplest way to take stratification into account is to deal with a two-layer flow. In this case, vortices perturb the interface, and consequently, the perturbed interface transits the vortex influences from one layer to another. Our aim is to investigate the dynamics of two point vortices in an unbounded domain where a shear and rotation are imposed as the leading order influence from some generalized perturbation. The two vortices are arranged within the bottom layer, but an emphasis is on the upper-layer fluid particle motion. Point vortices induce singular velocity fields in the layer they belong to, however, in the other layers of a multi-layer flow, they induce regular velocity fields. The main feature is that singular velocity fields prohibit irregular dynamics in the vicinity of the singular points, but regular velocity fields, provided optimal conditions, permit irregular dynamics to extend almost in every point of the corresponding phase space.
Fractal properties of lysozyme: a neutron scattering study.
Lushnikov, S G; Svanidze, A V; Gvasaliya, S N; Torok, G; Rosta, L; Sashin, I L
2009-03-01
The spatial structure and dynamics of hen egg white lysozyme have been investigated by small-angle and inelastic neutron scattering. Analysis of the results was carried using the fractal approach, which allowed determination of the fractal and fracton dimensions of lysozyme, i.e., consideration of the protein structure and dynamics by using a unified approach. Small-angle neutron scattering studies of thermal denaturation of lysozyme have revealed changes in the fractal dimension in the vicinity of the thermal denaturation temperature that reflect changes in the spatial organization of protein. PMID:19391977
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.
Chaotic neurochips for fuzzy computing
NASA Astrophysics Data System (ADS)
Szu, Harold H.; Zadeh, Lotfi A.; Hsu, Charles C.; DeWitte, Joseph T., Jr.; Moon, Gyu; Gobovic, Desa; Zaghloul, Mona E.
1994-03-01
A massive chaotic neural network (CNN) is demonstrated with a fixed-point Hebbian synaptic weight dynamic: an instantaneous input, and a piecewise negative logic output. The variable slope of the output versus the input becomes a software control of the collective chaos hardware. Two applications are given. The mean synaptic weight field plays an important role for fast pattern recognition capability in examples of both the habituation and the novelty detections. Another novel usage of CNN is to be a bridge between neural learning and learnable fuzzy logic.
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.
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.
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
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
NASA Astrophysics Data System (ADS)
Shibata, Kazuaki; Horio, Yoshihiko; Aihara, Kazuyuki
The quadratic assignment problem (QAP) is one of the NP-hard combinatorial optimization problems. An exponential chaotic tabu search using a 2-opt algorithm driven by chaotic neuro-dynamics has been proposed as one heuristic method for solving QAPs. In this paper we first propose a new local search, the double-assignment method, suitable for the exponential chaotic tabu search, which adopts features of the Lin-Kernighan algorithm. We then introduce chaotic neuro-dynamics into the double-assignment method to propose a novel exponential chaotic tabu search. We further improve the proposed exponential chaotic tabu search with the double-assignment method by enhancing the effect of chaotic neuro-dynamics.
A Chaotic Circuit Based on Bouali's Equations
NASA Astrophysics Data System (ADS)
Bouali, S.; Buscarino, A.; Fortuna, L.; Frasca, M.; Gambuzza, L. V.
2011-09-01
In this work, the electronic implementation of the Bouali's equations exhibiting a rich repertoire of nonlinear dynamical phenomena is introduced. The robust plug and play chaotic circuit is designed to be easily realized using standard components in a rigorous, fast and inexpensive way. We find that experimental results display periodicity, bifurcations and chaos that match with high accuracy the corresponding theoretical values.
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. PMID:25373135
Mutual stabilization of chaotic systems through entangled cupolets
NASA Astrophysics Data System (ADS)
Morena, Matthew Allan
Recent experimental and theoretical work has detected signatures of chaotic behavior in nearly every physical science, including quantum entanglement. In some instances, chaos either plays a significant role or, as an underlying presence, explains perplexing observations. There are certain properties of chaotic systems which are consistently encountered and become focal points of the investigations. For instance, chaotic systems typically admit a dense set of unstable periodic orbits around an attractor. These orbits collectively provide a rich source of qualitative information about the associated system and their abundance has been utilized in a variety of applications. We begin this thesis by describing a control scheme that stabilizes the unstable periodic orbits of chaotic systems and we go on to discuss several properties of these orbits. This technique allows for the creation of thousands of periodic orbits, known as cupolets ( Chaotic Unstable Periodic Orbit-lets ). We then present several applications of cupolets for investigating chaotic systems. First, we demonstrate an effective technique that combines cupolets with algebraic graph theory in order to transition between their orbits. This also induces certainty into the control of nonlinear systems and effectively provides an efficient algorithm for the steering and targeting of chaotic systems. Next, we establish that many higher-order cupolets are amalgamations of simpler cupolets, possibly through bifurcations. From a sufficiently large set of cupolets, we obtain a hierarchal subset of fundamental cupolets from which other cupolets may be assembled and dynamical invariants approximated. We then construct an independent coordinate system aligned to the local dynamical geometry and that reveals the local stretching and folding dynamics which characterize chaotic behavior. This partitions the dynamical landscape into regions of high or low chaoticity, thereby supporting prediction capabilities. Finally
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.
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. PMID:24697373
Controlled transitions between cupolets of chaotic systems
Morena, Matthew A.; Short, Kevin M.; Cooke, Erica E.
2014-01-01
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. PMID:24697373
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)
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.
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.
Entanglement entropy on fractals
NASA Astrophysics Data System (ADS)
Faraji Astaneh, Amin
2016-03-01
We use the heat kernel method to calculate the entanglement entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cutoff parameter is (generally) a fractional number, which, indeed, is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log-periodic oscillatory behavior in the expression of entropy which has root in the complex dimension of the fractal. We finally indicate that the holographic calculation in a certain hyperscaling-violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyperscaling-violating theory with the spectral dimension of the fractal. We provide additional support by comparing the behavior of the thermal entropy in terms of the temperature, computed for two geometries, the fractal geometry and the hyperscaling-violating background.
NASA Astrophysics Data System (ADS)
Cervantes, F.; Gonzalez, J.; Real, C.; Hoyos, L.
2012-12-01
ABSTRACT: Chaotic invariants like fractal dimensions are used to characterize non-linear time series. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. In this work four fractal dimension estimation algorithms are applied to non-linear time series. The algorithms employed are the Higuchi's algorithm, the Petrosian's algorithm, the Katz's Algorithm and the Box counting method. The analyzed time series are associated with natural phenomena, the Dst a geomagnetic index which monitors the world wide magnetic storm; the Dst index is a global indicator of the state of the Earth's geomagnetic activity. The time series used in this work show a behavior self-similar, which depend on the time scale of measurements. It is also observed that fractal dimensions may not be constant over all time scales.
Exact folded-band chaotic oscillator.
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. PMID:22757520
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.
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
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 atomic-level percolation in metallic glasses.
Chen, David Z; Shi, Crystal Y; An, Qi; Zeng, Qiaoshi; Mao, Wendy L; Goddard, William A; Greer, Julia R
2015-09-18
Metallic glasses are metallic alloys that exhibit exotic material properties. They may have fractal structures at the atomic level, but a physical mechanism for their organization without ordering has not been identified. We demonstrated a crossover between fractal short-range (<2 atomic diameters) and homogeneous long-range structures using in situ x-ray diffraction, tomography, and molecular dynamics simulations. A specific class of fractal, the percolation cluster, explains the structural details for several metallic-glass compositions. We postulate that atoms percolate in the liquid phase and that the percolating cluster becomes rigid at the glass transition temperature. PMID:26383945
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 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
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.
Chaotic Hierarchy in High Dimensions
NASA Astrophysics Data System (ADS)
Postnov, D. E.; Balanov, A. G.; Sosnovtseva, O. V.; Mosekilde, E.
The paper suggests a new mechanism for the development of higher-order chaos in accordance with the concept of a chaotic hierarchy. A discrete-time model is proposed which demonstrates how the creation of coexisting chaotic attractors combined with boundary crises can produce a continued growth of the Lyapunov dimension of the resulting chaotic behavior.
De Vera, Luis; Santana, Alejandro; Gonzalez, Julian J
2008-10-01
Both nonlinear and fractal properties of beat-to-beat R-R interval variability signal (RRV) of freely moving lizards (Gallotia galloti) were studied in baseline and under autonomic nervous system blockade. Nonlinear techniques allowed us to study the complexity, chaotic behavior, nonlinearity, stationarity, and regularity over time of RRV. Scaling behavior of RRV was studied by means of fractal techniques. The autonomic nervous system blockers used were atropine, propranolol, prazosin, and yohimbine. The nature of RRV was linear in baseline and under beta-, alpha(1)- and alpha(2)-adrenoceptor blockades. Atropine changed the linear nature of RRV to nonlinear and increased its stationarity, regularity and fractality. Propranolol increased the complexity and chaotic behavior, and decreased the stationarity, regularity, and fractality of RRV. Both prazosin and yohimbine did not change any of the nonlinear and fractal properties of RRV. It is suggested that 1) the use of both nonlinear and fractal analysis is an appropriate approach for studying cardiac period variability in reptiles; 2) the cholinergic activity, which seems to make the alpha(1)-, alpha(2)- and beta-adrenergic activity interaction unnecessary, determines the linear behavior in basal RRV; 3) fractality, as well as both RRV regularity and stationarity over time, may result from the balance between cholinergic and beta-adrenergic activities opposing actions; 4) beta-adrenergic activity may buffer both the complexity and chaotic behavior of RRV, and 5) neither the alpha(1)- nor the alpha(2)-adrenergic activity seem to be involved in the mediation of either nonlinear or fractal components of RRV. PMID:18685061
A comparison of spectral and chaotic analysis of electrochemical noise
Legat, A.; Govekar, E.
1996-12-31
Potential and current fluctuations spontaneously generated by corrosion reactions are known as electrochemical noise. In certain cases, good correlation can be obtained between the results of the spectral analysis of electrochemical noise and corrosion rate and type. However, because of the chaotic nature of corrosion processes, a special mathematical treatment may be needed. In the present study, the electrochemical noise measured on various metals was treated by methods known from the theory of chaos, and the results were compared with the results of spectral analysis. It has been shown that the chaotic characteristics of electrochemical noise are related to corrosion type, whereas the rate of corrosion has no influence on the fractal dimensions of the noise.
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
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.
FRACTAL DIMENSION OF GALAXY ISOPHOTES
Thanki, Sandip; Rhee, George; Lepp, Stephen E-mail: grhee@physics.unlv.edu
2009-09-15
In this paper we investigate the use of the fractal dimension of galaxy isophotes in galaxy classification. We have applied two different methods for determining fractal dimensions to the isophotes of elliptical and spiral galaxies derived from CCD images. We conclude that fractal dimension alone is not a reliable tool but that combined with other parameters in a neural net algorithm the fractal dimension could be of use. In particular, we have used three parameters to segregate the ellipticals and lenticulars from the spiral galaxies in our sample. These three parameters are the correlation fractal dimension D {sub corr}, the difference between the correlation fractal dimension and the capacity fractal dimension D {sub corr} - D {sub cap}, and, thirdly, the B - V color of the galaxy.
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 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
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)
Hsü, K J; Hsü, A J
1990-01-01
Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-log linear plot. PMID:11607061
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.
Design of Grid Multiscroll Chaotic Attractors via Transformations
NASA Astrophysics Data System (ADS)
Ai, Xingxing; Sun, Kehui; He, Shaobo; Wang, Huihai
Three transformation approaches for generating grid multiscroll chaotic attractors are presented through theoretical analysis and numerical simulation. Three kinds of grid multiscroll chaotic attractors are generated based on one-dimensional multiscroll Chua system. The dynamics of the multiscroll chaotic attractors are analyzed by means of equilibrium points, eigenvalues, the largest Lyapunov exponent and complexity. As the experimental verification, we implemented the circular grid multiscroll attractor on DSP platform. The simulation and experimental results are consistent well with that of theoretical analysis, and it shows that the design approaches are effective.
Chaotic and hyperchaotic attractors of a complex nonlinear system
NASA Astrophysics Data System (ADS)
Mahmoud, Gamal M.; Al-Kashif, M. A.; Farghaly, A. A.
2008-02-01
In this paper, we introduce a complex nonlinear hyperchaotic system which is a five-dimensional system of nonlinear autonomous differential equations. This system exhibits both chaotic and hyperchaotic behavior and its dynamics is very rich. Based on the Lyapunov exponents, the parameter values at which this system has chaotic, hyperchaotic attractors, periodic and quasi-periodic solutions and solutions that approach fixed points are calculated. The stability analysis of these fixed points is carried out. The fractional Lyapunov dimension of both chaotic and hyperchaotic attractors is calculated. Some figures are presented to show our results. Hyperchaos synchronization is studied analytically as well as numerically, and excellent agreement is found.
Design of Chaotic Pseudo-Random Bit Generator and its Applications in Stream-Cipher Cryptography
NASA Astrophysics Data System (ADS)
Wang, Xing-Yuan; Wang, Xiao-Juan
Because of the sensitivity of chaotic systems on initial conditions/control parameters, when chaotic systems are realized in a discrete space with finite states, the dynamical properties will be far different from the ones described by the continuous chaos theory, and some degradation will arise. This problem will cause the chaotic trajectory eventually periodic. In order to solve the problem, a new binary stream-cipher algorithm based on one-dimensional piecewise linear chaotic map is proposed in this paper. In the process of encryption and decryption, we employ a secret variant to perturb the chaotic trajectory and the control parameter to lengthen the cycle-length of chaotic trajectory. In addition, we design a nonlinear principle to generate a pseudo-random chaotic bit sequence as key stream. Cryptanalysis shows that the cryptosystem is of high security.
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.
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.
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.
Building Chaotic Model From Incomplete Time Series
NASA Astrophysics Data System (ADS)
Siek, Michael; Solomatine, Dimitri
2010-05-01
This paper presents a number of novel techniques for building a predictive chaotic model from incomplete time series. A predictive chaotic model is built by reconstructing the time-delayed phase space from observed time series and the prediction is made by a global model or adaptive local models based on the dynamical neighbors found in the reconstructed phase space. In general, the building of any data-driven models depends on the completeness and quality of the data itself. However, the completeness of the data availability can not always be guaranteed since the measurement or data transmission is intermittently not working properly due to some reasons. We propose two main solutions dealing with incomplete time series: using imputing and non-imputing methods. For imputing methods, we utilized the interpolation methods (weighted sum of linear interpolations, Bayesian principle component analysis and cubic spline interpolation) and predictive models (neural network, kernel machine, chaotic model) for estimating the missing values. After imputing the missing values, the phase space reconstruction and chaotic model prediction are executed as a standard procedure. For non-imputing methods, we reconstructed the time-delayed phase space from observed time series with missing values. This reconstruction results in non-continuous trajectories. However, the local model prediction can still be made from the other dynamical neighbors reconstructed from non-missing values. We implemented and tested these methods to construct a chaotic model for predicting storm surges at Hoek van Holland as the entrance of Rotterdam Port. The hourly surge time series is available for duration of 1990-1996. For measuring the performance of the proposed methods, a synthetic time series with missing values generated by a particular random variable to the original (complete) time series is utilized. There exist two main performance measures used in this work: (1) error measures between the actual
Polynomial chaotic inflation in supergravity
Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T. E-mail: fumi@tuhep.phys.tohoku.ac.jp
2013-08-01
We present a general polynomial chaotic inflation model in supergravity, for which the predicted spectral index and tensor-to-scalar ratio can lie within the 1σ region allowed by the Planck results. Most importantly, the predicted tensor-to-scalar ratio is large enough to be probed in the on-going and future B-mode experiments. We study the inflaton dynamics and the subsequent reheating process in a couple of specific examples. The non-thermal gravitino production from the inflaton decay can be suppressed in a case with a discrete Z{sub 2} symmetry. We find that the reheating temperature can be naturally as high as O(10{sup 9−10}) GeV, sufficient for baryon asymmetry generation through (non-)thermal leptogenesis.
Basin topology in dissipative chaotic scattering.
Seoane, Jesús M; Aguirre, Jacobo; Sanjuán, Miguel A F; Lai, Ying-Cheng
2006-06-01
Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Henon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical evidence and a geometric theory for the structural stability of the complex basin topology. PMID:16822004
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.
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.
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.
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.
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.
Fractal polyzirconosiloxane cluster coatings
Sugama, T.
1992-08-01
Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-[3-(triethoxysilyl)propyl]-4,5-dihydroimidazole, Zr(OC{sub 3}H{sub 7}){sub 4}, methanol, and water. When amorphous PZSs were applied to aluminum as protective coatings against NaCl-induced corrosion, the effective film was that derived from the sol having a pH near the isoelectric point in the positive zeta potential region. The following four factors played an important role in assembling the protective PZS coating films: (1) a proper rate of condensation, (2) a moderate ratio of Si-O-Si to Si-O-Zr linkages formed in the PZS network, (3) hydrophobic characteristics, and (4) a specific microstructural geometry, in which large fractal clusters were linked together.
Fractal polyzirconosiloxane cluster coatings
Sugama, T.
1992-01-01
Fractal polyzirconosiloxane (PZS) cluster films were prepared through the hydrolysis-polycondensation-pyrolysis synthesis of two-step HCl acid-NaOH base catalyzed sol precursors consisting of N-(3-(triethoxysilyl)propyl)-4,5-dihydroimidazole, Zr(OC{sub 3}H{sub 7}){sub 4}, methanol, and water. When amorphous PZSs were applied to aluminum as protective coatings against NaCl-induced corrosion, the effective film was that derived from the sol having a pH near the isoelectric point in the positive zeta potential region. The following four factors played an important role in assembling the protective PZS coating films: (1) a proper rate of condensation, (2) a moderate ratio of Si-O-Si to Si-O-Zr linkages formed in the PZS network, (3) hydrophobic characteristics, and (4) a specific microstructural geometry, in which large fractal clusters were linked together.
Hypogenetic chaotic jerk flows
NASA Astrophysics Data System (ADS)
Li, Chunbiao; Sprott, Julien Clinton; Xing, Hongyan
2016-03-01
Removing the amplitude or polarity information in the feedback loop of a jerk structure shows that special nonlinearities with partial information in the variable can also lead to chaos. Some striking properties are found for this kind of hypogenetic chaotic jerk flow, including multistability of symmetric coexisting attractors from an asymmetric structure, hidden attractors with respect to equilibria but with global attraction, easy amplitude control, and phase reversal which is convenient for chaos applications.
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 Astrophysics Data System (ADS)
Martin, Demetri
2015-03-01
Demetri Maritn prepared this palindromic poem as his project for Michael Frame's fractal geometry class at Yale. Notice the first, fourth, and seventh words in the second and next-to-second lines are palindromes, the first two and last two lines are palindromes, the middle line, "Be still if I fill its ebb" minus its last letter is a palindrome, and the entire poem is a palindrome...
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.
NASA Astrophysics Data System (ADS)
Touma, J.; Wisdom, J.
1993-02-01
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.
Using Chaotic System in Encryption
NASA Astrophysics Data System (ADS)
Findik, Oğuz; Kahramanli, Şirzat
In this paper chaotic systems and RSA encryption algorithm are combined in order to develop an encryption algorithm which accomplishes the modern standards. E.Lorenz's weather forecast' equations which are used to simulate non-linear systems are utilized to create chaotic map. This equation can be used to generate random numbers. In order to achieve up-to-date standards and use online and offline status, a new encryption technique that combines chaotic systems and RSA encryption algorithm has been developed. The combination of RSA algorithm and chaotic systems makes encryption system.