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Sample records for nonlinear hamiltonian systems

  1. Forced Oscillations of Nonlinear Hamiltonian Systems, II.

    DTIC Science & Technology

    1979-12-01

    Rabinowitz (J8], 9]). The author obtained similar results ([6]), by using a variational method devised b.: F. Clarke and himself for convex subquadratic...and satisfying, for some constants bl > a’ > 0 and 5 > 2: (39) a’ ixi-21Y 2 _< (V"(x)yy) < b’ lxiy-21yj 2, all x ev , y c ip Then for any T > 0, there is...34Linear operators", Wiley. [6] I. Ekeland, "Periodic Hamiltonian trajectories and a theorem of P. Rabinowitz ", 1978, to appear in Journal of Differential

  2. Nonlinear pulsations of a Hamiltonian system of the fourth order by a nonlinear trigonometric series

    NASA Astrophysics Data System (ADS)

    Miroshnikov, George

    2011-11-01

    Dynamics of Hamiltonian systems is the key issue of solitary waves since the initial-value problems on free surfaces and interfaces are reduced to Hamiltonian problems in the reference frame moving with the wave. The Hamiltonian approach covers applications at high Reynolds numbers, which range from the famous irrotational Boussinesq-Rayleigh solitary wave to the rotational waves with a uniform vorticity. The Hamiltonian system with a polynomial potential of the fourth order is studied in the asymmetric case of subcritical periodic pulsations by using a nonlinear trigonometric series in even powers of cosine. The series solutions are computed symbolically and compared with the numerical solution using the Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant. It is shown that the series solutions with uniform convergence are superior to the numeric solutions with local convergence. The qualitative comparison of the theoretical solutions with the experimental profiles of the Geminga pulsar is also provided.

  3. Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

    SciTech Connect

    Morrison, P.J.; Vanneste, J.

    2016-05-15

    A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria of systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.

  4. Stabilizing feedback controls for nonlinear Hamiltonian systems and nonconservative bilinear systems in elasticity

    NASA Astrophysics Data System (ADS)

    Singh, S. N.

    1982-03-01

    Using the invariance principle of LaSalle (1962) sufficient conditions for the existence of linear and nonlinear control laws for local and global asymptotic stability of nonlinear Hamiltonian systems are derived. An instability theorem is also presented which identifies the control laws from the given class which cannot achieve asymptotic stability. Some of the stability results are based on certain results for the univalence of nonlinear maps. A similar approach for the stabilization of bilinear systems which include nonconservative systems in elasticity is used and a necessary and sufficient condition for stabilization is obtained. An application to attitude control of a gyrostat Satellite is presented.

  5. Hamiltonian-Driven Adaptive Dynamic Programming for Continuous Nonlinear Dynamical Systems.

    PubMed

    Yang, Yongliang; Wunsch, Donald; Yin, Yixin

    2017-02-01

    This paper presents a Hamiltonian-driven framework of adaptive dynamic programming (ADP) for continuous time nonlinear systems, which consists of evaluation of an admissible control, comparison between two different admissible policies with respect to the corresponding the performance function, and the performance improvement of an admissible control. It is showed that the Hamiltonian can serve as the temporal difference for continuous-time systems. In the Hamiltonian-driven ADP, the critic network is trained to output the value gradient. Then, the inner product between the critic and the system dynamics produces the value derivative. Under some conditions, the minimization of the Hamiltonian functional is equivalent to the value function approximation. An iterative algorithm starting from an arbitrary admissible control is presented for the optimal control approximation with its convergence proof. The implementation is accomplished by a neural network approximation. Two simulation studies demonstrate the effectiveness of Hamiltonian-driven ADP.

  6. Linear and nonlinear response of the Vlasov system with nonintegrable Hamiltonian

    NASA Astrophysics Data System (ADS)

    Ogawa, Shun

    2017-07-01

    Linear and nonlinear response formulas taking into account all Casimir invariants are derived without use of angle-action variables of a single-particle (mean-field) Hamiltonian. This article deals mainly with the Vlasov system in a spatially inhomogeneous quasistationary state whose associating single-particle Hamiltonian is not integrable and has only one integral of the motion, the Hamiltonian itself. The basic strategy is to restrict the form of perturbation so that it keeps Casimir invariants within a linear order, and the single particle's probabilistic density function is smooth with respect to the single particle's Hamiltonian. The theory is applied for a spatially two-dimensional system and is confirmed by numerical simulations. A nonlinear response formula is also derived in a similar manner.

  7. Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems

    NASA Astrophysics Data System (ADS)

    Eckmann, Jean-Pierre; Pillet, Claude-Alain; Rey-Bellet, Luc

    1999-04-01

    We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system

  8. On stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems.

    PubMed

    Zhu, Wei-qiu; Ying, Zu-guang

    2004-11-01

    A stochastic optimal control strategy for partially observable nonlinear quasi Hamiltonian systems is proposed. The optimal control forces consist of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimated error of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy.

  9. From Nonlinear to Hamiltonian via Feedback1

    DTIC Science & Technology

    2002-01-01

    distribution unlimited. 13. Abstract Mechanical control systems are a very important class of nonlinear control systems . They posses a rich mathematical...methodologies developed for mechanical control systel logically rendering nonlinear control systems , mechanical by a proper choice of feedback. In particular, w...OF PA Nonlinear mechanical control systems , Hamiltonian Control Systems x 16. PRICE CODE 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19

  10. Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian

    SciTech Connect

    Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar

    2015-09-15

    The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.

  11. Symplectic exponential Runge-Kutta methods for solving nonlinear Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Mei, Lijie; Wu, Xinyuan

    2017-06-01

    Symplecticity is also an important property for exponential Runge-Kutta (ERK) methods in the sense of structure preservation once the underlying problem is a Hamiltonian system, though ERK methods provide a good performance of higher accuracy and better efficiency than classical Runge-Kutta (RK) methods in dealing with stiff problems: y‧ (t) = My + f (y). On account of this observation, the main theme of this paper is to derive and analyze the symplectic conditions for ERK methods. Using the fundamental analysis of geometric integrators, we first establish one class of sufficient conditions for symplectic ERK methods. It is shown that these conditions will reduce to the conventional ones when M → 0, and this means that these conditions of symplecticity are extensions of the conventional ones in the literature. Furthermore, we also present a new class of structure-preserving ERK methods possessing the remarkable property of symplecticity. Meanwhile, the revised stiff order conditions are proposed and investigated in detail. Since the symplectic ERK methods are implicit and iterative solutions are required in practice, we also investigate the convergence of the corresponding fixed-point iterative procedure. Finally, the numerical experiments, including a nonlinear Schrödinger equation, a sine-Gordon equation, a nonlinear Klein-Gordon equation, and the well-known Fermi-Pasta-Ulam problem, are implemented in comparison with the corresponding symplectic RK methods and the prominent numerical results definitely coincide with the theories and conclusions made in this paper.

  12. Hamiltonian theory of nonlinear waves in planetary rings

    NASA Technical Reports Server (NTRS)

    Stewart, G. R.

    1987-01-01

    The derivation of a Hamiltonian field theory for nonlinear density waves in Saturn's rings is discussed. Starting with a Hamiltonian for a discrete system of gravitating streamlines, an averaged Hamiltonian is obtained by successive applications of Lie transforms. The transformation may be carried out to any desired order in q, where q is the nonlinearity parameter defined in the work of Shu, et al (1985) and Borderies et al (1985). Subsequent application of the Wentzel-Kramer-Brillouin Method approximation yields an asymptotic field Hamiltonian. Both the nonlinear dispersion relation and the wave action transport equation are easily derived from the corresponding Lagrangian by the standard variational principle.

  13. Hamiltonian Approach to Nonlinear Travelling Whistler Waves

    SciTech Connect

    Webb, G.M.; McKenzie, J.F.; Dubinin, E.; Sauer, K.

    2005-08-01

    A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is discussed. The model is based on the equations of two-fluid electron-proton plasmas. In the cold gas limit, the complete system of equations reduces to two coupled differential equations for the transverse electron speed u and a phase variable {phi} = {phi}p - {phi}e representing the difference in the phases of the transverse complex velocities of the protons and the electrons. Two integrals of the equations are obtained. The Hamiltonian integral H, is used to classify the trajectories in the ({phi}, w) phase plane, where {phi} and w = u2 are the canonical coordinates. Periodic, oscillation solitary wave and compacton solutions are obtained, depending on the value of the Hamiltonian integral H and the Alfven Mach number M of the travelling wave. The individual electron and proton phase variables {phi}e and {phi}p are determined in terms of {phi} and w. An alternative Hamiltonian formulation in which {phi}-tilde = {phi}p + {phi}e is the new independent variable replacing x is used to write the travelling wave solutions parametrically in terms of {phi}-tilde.

  14. Hamiltonian chaos in nonlinear optical polarization dynamics

    NASA Astrophysics Data System (ADS)

    David, D.; Holm, D. D.; Tratnik, M. V.

    1990-03-01

    This paper applies Hamiltonian methods to the Stokes representation of the one-beam and two-beam problems of polarized optical pulses propagating as travelling waves in nonlinear media. We treat these two dynamical systems as follows. First, we use the reduction method of Marsden and Weinstein to map each of the systems to the two-dimensional sphere, S 2. The resulting reduced systems are then analyzed from the viewpoints of their stability properties and of bifurcations with symmetry; in particular, several degenerate bifurcations are found and described. We also establish the presence of chaotic dynamics in these systems by demonstrating the existence of Smale horseshoe maps in the three- and four-dimensional cases, as well as Arnold diffusion in the higher-dimensional cases. The method we use to establish such complex dynamics is the Mel'nikov technique, as extended by Holmes and Marsden, and Wiggins for the higher-dimensional cases. These results apply to perturbations of homoclinic and heteroclinic orbits of the reduced integrable problems for static, as well as travelling-wave, solutions describing either a single opt ical beam, or two such beams counterpropagating. Thus, we show that these optics problems exhibit complex dynamics and predict the experimental consequences of this dynamics.

  15. Fourier series expansion for nonlinear Hamiltonian oscillators.

    PubMed

    Méndez, Vicenç; Sans, Cristina; Campos, Daniel; Llopis, Isaac

    2010-06-01

    The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relationship. We compare our results with other recent approaches such as variational methods or heuristic approximations, in particular the Ren-He's method. Based on its application to the Duffing oscillator, the nonlinear pendulum and the eardrum equation, it is shown that the Fourier series expansion method is the most accurate.

  16. A partial Hamiltonian approach for current value Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Naz, R.; Mahomed, F. M.; Chaudhry, Azam

    2014-10-01

    We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.

  17. Invariants for time-dependent Hamiltonian systems.

    PubMed

    Struckmeier, J; Riedel, C

    2001-08-01

    An exact invariant is derived for n-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special Ansatz for the invariant and determine its time-dependent coefficients. In the second approach, we perform a two-step canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. The invariant is found to contain a function of time f(2)(t), defined as a solution of a linear third-order differential equation whose coefficients depend in general on the explicitly known configuration space trajectory that follows from the system's time evolution. It is shown that the invariant can be interpreted as the time integral of an energy balance equation. Our result is applied to a one-dimensional, time-dependent, damped non-linear oscillator, and to a three-dimensional system of Coulomb-interacting particles that are confined in a time-dependent quadratic external potential. We finally show that our results can be used to assess the accuracy of numerical simulations of time-dependent Hamiltonian systems.

  18. Implicit variational principle for contact Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Wang, Kaizhi; Wang, Lin; Yan, Jun

    2017-02-01

    We establish an implicit variational principle for the contact Hamiltonian systems generated by the Hamiltonian H(x, u, p) with respect to the contact 1-form α =\\text{d}u-p\\text{d}x under Tonelli and Lipschitz continuity conditions.

  19. Generalized index for Hamiltonian systems with applications

    NASA Astrophysics Data System (ADS)

    Zevin, Alexandr A.

    2005-09-01

    Some classes of linear Hamiltonian equations with periodic coefficients (nondegenerate, strongly stable, completely unstable) are determined by the disposition of the Floquet multipliers. Periodic solutions of nonlinear equations (e.g. elliptic or hyperbolic solutions) are also defined by the multipliers of the corresponding variational equation. In this paper, we consider a general set M of linear Hamiltonian equations with multipliers satisfying some arbitrary conditions and a specific condition on the multiplier lying at some point of the unit circle (all known sets admit such a definition). We show that the set M consists of a finite number of subsets Mi which comprise a countable number of domains M_q^i within which any two Hamiltonians can be continuously deformed into each other. The corresponding integer index q is expressed through the eigenvalues of some self-adjoint problem. It is shown that this index (and, therefore, the known indices relating to specific sets) increases on increasing the Hamiltonian. Using the obtained results, some known and new sets are studied from the unified point of view. It is shown that for the sets of nondegenerate and completely unstable equations, the domains M_q^i are directionally convex; for strongly stable equations, necessary and sufficient conditions for directional convexity are found. The results are applied to problems of existence and stability of periodic solutions of nonlinear Hamiltonian equations.

  20. Incomplete Dirac reduction of constrained Hamiltonian systems

    SciTech Connect

    Chandre, C.

    2015-10-15

    First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.

  1. Symmetries and regular behavior of Hamiltonian systems.

    PubMed

    Kozlov, Valeriy V.

    1996-03-01

    The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting "Lagrangian" vector fields, i.e., the symplectic 2-form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals. (c) 1996 American Institute of Physics.

  2. Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes Piano

    NASA Astrophysics Data System (ADS)

    Falaize, Antoine; Hélie, Thomas

    2017-03-01

    This paper deals with the time-domain simulation of an electro-mechanical piano: the Fender Rhodes. A simplified description of this multi-physical system is considered. It is composed of a hammer (nonlinear mechanical component), a cantilever beam (linear damped vibrating component) and a pickup (nonlinear magneto-electronic transducer). The approach is to propose a power-balanced formulation of the complete system, from which a guaranteed-passive simulation is derived to generate physically-based realistic sound synthesis. Theses issues are addressed in four steps. First, a class of Port-Hamiltonian Systems is introduced: these input-to-output systems fulfill a power balance that can be decomposed into conservative, dissipative and source parts. Second, physical models are proposed for each component and are recast in the port-Hamiltonian formulation. In particular, a finite-dimensional model of the cantilever beam is derived, based on a standard modal decomposition applied to the Euler-Bernoulli model. Third, these systems are interconnected, providing a nonlinear finite-dimensional Port-Hamiltonian System of the piano. Fourth, a passive-guaranteed numerical method is proposed. This method is built to preserve the power balance in the discrete-time domain, and more precisely, its decomposition structured into conservative, dissipative and source parts. Finally, simulations are performed for a set of physical parameters, based on empirical but realistic values. They provide a variety of audio signals which are perceptively relevant and qualitatively similar to some signals measured on a real instrument.

  3. Periodic Solutions of Hamiltonian Systems: A Survey.

    DTIC Science & Technology

    1980-12-01

    auto - nomous Hamiltonian system has the form: (0.) aH 8Hp -S-(p,q) q ( where d denotes This system can be represented more concisely as (HS) z = ZHz(Z...oscillazioni periodiche d’une sistema dinamico, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 19, (1934), 234-237. [15] Arnold, V. I

  4. Mapping between dissipative and Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Xing, Jianhua

    2010-09-01

    Theoretical studies of nonequilibrium systems are complicated by the lack of a general framework. In this work we first show that a transformation recently introduced by Ao (2004 J. Phys. A: Math. Gen. 37 L25) is related to previous works of Graham (1977 Z. Phys. B 26 397) and Eyink et al (1996 J. Stat. Phys. 83 385), which can also be viewed as the generalized application of the Helmholtz theorem in vector calculus. We then show that systems described by ordinary stochastic differential equations with white noise can be mapped to thermostated Hamiltonian systems. A steady-state of a dissipative system corresponds to the equilibrium state of the corresponding Hamiltonian system. These results provide a solid theoretical ground for corresponding studies on nonequilibrium dynamics, especially on nonequilibrium steady state. Mapping permits the application of established techniques and results for Hamiltonian systems to dissipative non-Hamiltonian systems, those for thermodynamic equilibrium states to nonequilibrium steady states. We discuss several implications of this work.

  5. Canonical transformations and Hamiltonian evolutionary systems

    SciTech Connect

    Al-Ashhab, Samer

    2012-06-15

    In many Lagrangian field theories, one has a Poisson bracket defined on the space of local functionals. We find necessary and sufficient conditions for a transformation on the space of local functionals to be canonical in three different cases. These three cases depend on the specific dimensions of the vector bundle of the theory and the associated Hamiltonian differential operator. We also show how a canonical transformation transforms a Hamiltonian evolutionary system and its conservation laws. Finally, we illustrate these ideas with three examples.

  6. Tsallis thermostatistics for finite systems: a Hamiltonian approach

    NASA Astrophysics Data System (ADS)

    Adib, Artur B.; Moreira, Andrã© A.; Andrade, José S., Jr.; Almeida, Murilo P.

    2003-05-01

    The derivation of the Tsallis generalized canonical distribution from the traditional approach of the Gibbs microcanonical ensemble is revisited (Phys. Lett. A 193 (1994) 140). We show that finite systems whose Hamiltonians obey a generalized homogeneity relation rigorously follow the nonextensive thermostatistics of Tsallis. In the thermodynamical limit, however, our results indicate that the Boltzmann-Gibbs statistics is always recovered, regardless of the type of potential among interacting particles. This approach provides, moreover, a one-to-one correspondence between the generalized entropy and the Hamiltonian structure of a wide class of systems, revealing a possible origin for the intrinsic nonlinear features present in the Tsallis formalism that lead naturally to power-law behavior. Finally, we confirm these exact results through extensive numerical simulations of the Fermi-Pasta-Ulam chain of anharmonic oscillators.

  7. Dynamics of Hamiltonian Systems and Memristor Circuits

    NASA Astrophysics Data System (ADS)

    Itoh, Makoto; Chua, Leon

    In this paper, we show that any n-dimensional autonomous systems can be regarded as subsystems of 2n-dimensional Hamiltonian systems. One of the two subsystems is identical to the n-dimensional autonomous system, which is called the driving system. Another subsystem, called the response system, can exhibit interesting behaviors in the neighborhood of infinity. That is, the trajectories approach infinity with complicated nonperiodic (chaotic-like) behaviors, or periodic-like behavior. In order to show the above results, we project the trajectories of the Hamiltonian systems onto n-dimensional spheres, or n-dimensional balls by using the well-known central projection transformation. Another interesting behavior is that the transient regime of the subsystems can exhibit Chua corsage knots. We next show that generic memristors can be used to realize the above Hamiltonian systems. Finally, we show that the internal state of two-element memristor circuits can have the same dynamics as n-dimensional autonomous systems.

  8. Analytic solutions for time-dependent Schrödinger equations with linear of nonlinear Hamiltonians

    NASA Astrophysics Data System (ADS)

    Adomian, G.; Efinger, H. J.

    1994-10-01

    The decomposition method is applied to the time-dependent Schrödinger equation for linear or nonlinear Hamiltonian operators, without linearization, perturbation, or numerical methods, to obtain a rapidly converging analytic solution

  9. Hamiltonian Lattice Studies of Pionic Collective Excitations in the Non-linear Sigma Model

    NASA Astrophysics Data System (ADS)

    Chin, Siu A.

    2001-04-01

    The latticization of the non-linear sigma model reduces a chiral meson field theory to an O(4) spin system with quantum fluctuations. By solving the resulting lattice Hamiltonian by Monte Carlo methods, the dynamics and thermodynamics of pions can be determined non-perturbatively. In particular, the mas gap of pionic collective excitations with quantum number of vector mesons can be determined as the chiral phase transition is approached. Results based on a newly discovered 4th order method of solving for the ground state of a quantum many-body Hamitonian will be presented.

  10. Schrödinger and related equations as hamiltonian systems, manifolds of second-order tensors and new ideas of nonlinearity in quantum mechanics

    NASA Astrophysics Data System (ADS)

    Sławianowski, J. J.; Kovalchuk, V.

    2010-01-01

    Considered is the Schrödinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting example of "mechanics" with singular Lagrangians, effectively treatable within the framework of Dirac formalism. We discuss also some modified "Schrödinger" equations involving second-order time derivatives and introduce a kind of nondirect, nonperturbative, geometrically-motivated nonlinearity based on making the scalar product a dynamical quantity. There are some reasons to expect that this might be a new way of describing open dynamical systems and explaining some quantum "paradoxes".

  11. Diffusion in very chaotic hamiltonian systems

    SciTech Connect

    Abarbanel, Henry D. I.; Crawford, John David

    1981-04-20

    In this paper, we study nonintegrable hamiltonian dynamics: H(I,θ) = H0(I) + kH1(I,θ), for large k, that is, far from integrability. An integral representation is given for the conditional probability P(I,θ, t¦I0, θ0, t0) that the system is at I, θ at t, given it was at I0, θ0 at t0. By discretizing time into steps of size ϵ, we show how to evaluate physical observables for large k, fixed ϵ. An explicit calculation of a diffusion coefficient in a two degrees of freedom problem is reported. Finally, passage to ϵ = 0, the original hamiltonian flow, is discussed.

  12. Polynomial approximation of Poincare maps for Hamiltonian system

    NASA Technical Reports Server (NTRS)

    Froeschle, Claude; Petit, Jean-Marc

    1992-01-01

    Different methods are proposed and tested for transforming a non-linear differential system, and more particularly a Hamiltonian one, into a map without integrating the whole orbit as in the well-known Poincare return map technique. We construct piecewise polynomial maps by coarse-graining the phase-space surface of section into parallelograms and using either only values of the Poincare maps at the vertices or also the gradient information at the nearest neighbors to define a polynomial approximation within each cell. The numerical experiments are in good agreement with both the real symplectic and Poincare maps.

  13. Efficient energy-preserving integrators for oscillatory Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Wu, Xinyuan; Wang, Bin; Shi, Wei

    2013-02-01

    In this paper, we focus our attention on deriving and analyzing an efficient energy-preserving formula for the system of nonlinear oscillatory or highly oscillatory second-order differential equations q″(t)+Mq(t)=fq(t), where M is a symmetric positive semi-definite matrix with M≫1 and f(q)=-∇qU(q) is the negative gradient of a real-valued function U(q). This system is a Hamiltonian system with the Hamiltonian H(p,q)=1/2 pTp+1/2 >qTMq+U(q). The energy-preserving formula exactly preserves the Hamiltonian. We analyze in detail the properties of the energy-preserving formula and propose new efficient energy-preserving integrators in the sense of numerical implementation. The convergence analysis of the fixed-point iteration is presented for the implicit integrators proposed in this paper. It is shown that the convergence of implicit Average Vector Field methods is dependent on M, whereas the convergence of the new energy-preserving integrators is independent of M. The Fermi-Pasta-Ulam problem and the sine-Gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well-known Average Vector Field methods in the scientific literature.

  14. Hierarchical structure of noncanonical Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Yoshida, Z.; Morrison, P. J.

    2016-02-01

    Topological constraints play a key role in the self-organizing processes that create structures in macro systems. In fact, if all possible degrees of freedom are actualized on equal footing without constraint, the state of ‘equipartition’ may bear no specific structure. Fluid turbulence is a typical example—while turbulent mixing seems to increase entropy, a variety of sustained vortical structures can emerge. In Hamiltonian formalism, some topological constraints are represented by Casimir invariants (for example, helicities of a fluid or a plasma), and then, the effective phase space is reduced to the Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir invariant; the circulation is an example of such an invariant. In this work, we formulate a systematic method to embed a Hamiltonian system in an extended phase space; we introduce phantom fields and extend the Poisson algebra. A phantom field defines a new Casimir invariant, a cross helicity, which represents a topological constraint that is not integrable in the original phase space. Changing the perspective, a singularity of the extended system may be viewed as a subsystem on which the phantom fields (though they are actual fields, when viewed from the extended system) vanish, i.e., the original system. This hierarchical relation of degenerate Poisson manifolds enables us to see the ‘interior’ of a singularity as a sub Poisson manifold. The theory can be applied to describe bifurcations and instabilities in a wide class of general Hamiltonian systems (Yoshida and Morrison 2014 Fluid Dyn. Res. 46 031412).

  15. Normalization of Hamiltonian and nonlinear stability of the triangular equilibrium points in non-resonance case with perturbations

    NASA Astrophysics Data System (ADS)

    Kishor, Ram; Kushvah, Badam Singh

    2017-09-01

    For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects. In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information as regards a realistic solution of the problem. In the present study, normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc is performed. To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed by computing the fourth order normalized Hamiltonian with the help of Lie transforms. In non-resonance case, nonlinear stability of the system is discussed using the Arnold-Moser theorem. Again, the effects of radiation pressure, oblateness and the presence of the disc are analyzed separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in the nonlinear sense within the stability range 0<μ<μ1=\\bar{μc} due to failure of the Arnold-Moser theorem. However, perturbation parameters affect the values of μ at which D4=0, significantly. This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag. The results are limited to the regular symmetric disc but it can be extended in the future.

  16. Strong coupling expansions for nonintegrable hamiltonian systems

    SciTech Connect

    Abarbanel, Henry D. I.; Crawford, John David

    1982-09-01

    In this paper, we present a method for studying nonintegrable Hamiltonian systems H(I,θ) = H0(I) + kH1(I,θ) (I, θ are action-angle variables) in the regime of large k. Our central tool is the conditional probability P(I,θ,t | I00,t0) that the system is at I. θ at time t given that it resided at I0, θ0 at t0. An integral representation is given for this conditional probability. By discretizing the Hamiltonian equations of motion in small time steps, ϵ, we arrive at a phase volume-preserving mapping which replaces the actual flow. When the motion on the energy surface E = H(I,θ) is bounded we are able to evaluate physical quantities of interest for large k and fixed ϵ. We also discuss the representation of P (I,θ,t | I00t0) when an external random forcing is added in order to smooth the singular functions associated with the deterministic flow. Explicit calculations of a “diffusion” coefficient are given for a non-integrable system with two degrees of freedom. Finally, the limit ϵ → 0, which returns us to the actual flow, is subtle and is discussed.

  17. Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree

    SciTech Connect

    Lepskii, Timur A

    2010-12-07

    Complex Hamiltonian systems with one degree of freedom on C{sup 2} with the standard symplectic structure {omega}C=dz and dw and a polynomial Hamiltonian function f=z{sup 2}+P{sub n}(w), n=1,2,3,4, are studied. Two Hamiltonian systems (M{sub i}, Re{omega}{sub C,i}, H{sub i}=Ref{sub i}), i=1,2, are said to be Hamiltonian equivalent if there exists a complex symplectomorphism M{sub 1}{yields}M{sub 2} taking the vector field sgradH{sub 1} to sgradH{sub 2}. Hamiltonian equivalence classes of systems are described in the case n=1,2,3,4, a completed system is defined for n=3,4, and it is proved that it is Liouville integrable as a real Hamiltonian system. By restricting the real action-angle coordinates defined for the completed system in a neighbourhood of any nonsingular leaf, real canonical coordinates are obtained for the original system. Bibliography: 9 titles.

  18. Generalized rotational Hamiltonians from nonlinear angular momentum algebras

    SciTech Connect

    Ballesteros, A.; Herranz, F. J.; Civitarese, O.; Reboiro, M.

    2007-04-15

    Higgs algebras are used to construct rotational Hamiltonians. The correspondence between the spectrum of a triaxial rotor and the spectrum of a cubic Higgs algebra is demonstrated. It is shown that a suitable choice of the parameters of the polynomial algebra allows for a precise identification of rotational properties. The harmonic limit is obtained by a contraction of the algebra, leading to a linear symmetry.

  19. Finite Hamiltonian Systems: Linear Transformations and Aberrations

    NASA Astrophysics Data System (ADS)

    Wolf, Kurt Bernardo

    2008-11-01

    In finite Hamiltonian systems, the operators of position, momentum, and energy have a finite number N of eigenvalues. These operators can be naturally realized as generators of the Lie algebra su(2), in a representation of spin j, of dimension N = 2j+1. Time evolution is rotation of a phase space sphere, whose projections perform the harmonic motion of an oscillator. The (centrally extended) group of rigid—linear—motions of this phase space is then U(2). On the other hand, N-point wavefunctions—signals—can be subject to a U(N) group of unitary matrices, containing the linear U(2); aberrations are transformations outside that subgroup. As in geometric optics, we classify the aberration multiplets by order and weight. Their action on phase space is displayed by means of a Wigner function on the sphere, to be compared with the corresponding geometric canonical transformations.

  20. A Hamiltonian-Free Description of Single Particle Dynamics for Hopelessly Complex Periodic Systems

    SciTech Connect

    Forest, E.

    1990-01-01

    We develop a picture of periodic systems which does not rely on the Hamiltonian of the system but on maps between a finite number of time locations. Moser or Deprit-like normalizations are done directly on the maps thereby avoiding the complex time-dependent theory. We redefine linear and nonlinear Floquet variables entirely in terms of maps. This approach relies heavily on the Lie representation of maps introduced by Dragt and Finn. One might say that although we do not use the Hamiltonian in the normalization transformation, we are using Lie operators which are themselves, in some sense, pseudo-Hamiltonians for the maps they represent. Our techniques find application in accelerator dynamics or in any field where the Hamiltonian is periodic but hopelessly complex, such as magnetic field design in stellarators.

  1. Sqeezing generated by a nonlinear master equation and by amplifying-dissipative Hamiltonians

    NASA Technical Reports Server (NTRS)

    Dodonov, V. V.; Marchiolli, M. A.; Mizrahi, Solomon S.; Moussa, M. H. Y.

    1994-01-01

    In the first part of this contribution we show that the master equation derived from the generalized version of the nonlinear Doebner-Goldin equation leads to the squeezing of one of the quadratures. In the second part we consider two familiar Hamiltonians, the Bateman- Caldirola-Kanai and the optical parametric oscillator; going back to their classical Lagrangian form we introduce a stochastic force and a dissipative factor. From this new Lagrangian we obtain a modified Hamiltonian that treats adequately the simultaneous amplification and dissipation phenomena, presenting squeezing, too.

  2. Generic perturbations of linear integrable Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Bounemoura, Abed

    2016-11-01

    In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ɛ -1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).

  3. Uncertainty relation for non-Hamiltonian quantum systems

    SciTech Connect

    Tarasov, Vasily E.

    2013-01-15

    General forms of uncertainty relations for quantum observables of non-Hamiltonian quantum systems are considered. Special cases of uncertainty relations are discussed. The uncertainty relations for non-Hamiltonian quantum systems are considered in the Schroedinger-Robertson form since it allows us to take into account Lie-Jordan algebra of quantum observables. In uncertainty relations, the time dependence of quantum observables and the properties of this dependence are discussed. We take into account that a time evolution of observables of a non-Hamiltonian quantum system is not an endomorphism with respect to Lie, Jordan, and associative multiplications.

  4. Synchronization in chaotic Hamiltonian systems and a geophysical application.

    PubMed

    Hannachi, A

    1999-07-01

    This paper addresses the question of the rate of synchronization of two identical systems as a function of the inserting time interval Delta t between inserted variables of the driving system in the role of the same variables of the driven system in a simplified Hamiltonian system and its application to a simplified geophysical model. We start by analyzing the synchronization in a simplified two-degree Hamiltonian system. The synchronization rate turns out to be a decreasing function of the inserting time interval Delta t up to a certain limit Delta t(o) where the process reverses and the synchronization rate becomes slower as the inserting frequency decreases. The key point of the analysis uses a second-order Taylor expansion of the system resolvent which indicates that synchronization rate is basically of order O(Delta t(2)) for small Delta t. The study is then extended to include a simplified geophysical system. A nonlinear one-dimensional shallow-water model on a periodic domain meant to represent a latitudinal circle around 45 degrees N is used. It is found that when the zonal wind is inserted, the maximum synchronization rate is obtained when the inserting time interval is approximately 4 h. When the meridional wind is inserted, it is obtained at slightly less than 4 h. It is shown, in particular, that the synchronization rate depends on the latitude (or the Coriolis parameter). A low-order simplified dynamical system derived from the one-dimensional shallow-water model is used to show that this optimum time interval Delta t(o) when the zonal wind and the geopotential, for example, are inserted varies approximately as square root of [2]/2 Omega sin phi to accuracy O(Delta t(3)). Analyses performed with a linear version of the shallow-water model reveal that this latter can be used to explain the observed convergence behavior in the nonlinear model. The only point is the choice of the stationary state for linearization purposes. It is then suggested that in

  5. Hyperbolic tori in Hamiltonian systems with slowly varying parameter

    SciTech Connect

    Medvedev, Anton G

    2013-05-31

    This paper looks at a Hamiltonian system which depends periodically on a parameter. For each value of the parameter the system is assumed to have a hyperbolic periodic solution. Using the methods in KAM-theory it is proved that if the Hamiltonian is perturbed so that the value of the parameter varies with constant small frequency, then the nonautonomous system will have hyperbolic 2-tori in the extended phase space. Bibliography: 12 titles.

  6. Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function.

    PubMed

    Gao, Hanhong; Tian, Lei; Zhang, Baile; Barbastathis, George

    2010-12-15

    We present an iterative method for simulating beam propagation in nonlinear media using Hamiltonian ray tracing. The Wigner distribution function of the input beam is computed at the entrance plane and is used as the initial condition for solving the Hamiltonian equations. Examples are given for the study of periodic self-focusing, spatial solitons, and Gaussian-Schell model in Kerr-effect media. Simulation results show good agreement with the split-step beam propagation method. The main advantage of ray tracing, even in the nonlinear case, is that ray diagrams are intuitive and easy to interpret in terms of traditional optical engineering terms, such as aberrations, ray-intercept plots, etc.

  7. Two time physics and Hamiltonian Noether theorem for gauge systems

    SciTech Connect

    Nieto, J. A.; Ruiz, L.; Silvas, J.; Villanueva, V. M.

    2006-09-25

    Motivated by two time physics theory we revisited the Noether theorem for Hamiltonian constrained systems. Our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints.

  8. Nonlinear Systems.

    ERIC Educational Resources Information Center

    Seider, Warren D.; Ungar, Lyle H.

    1987-01-01

    Describes a course in nonlinear mathematics courses offered at the University of Pennsylvania which provides an opportunity for students to examine the complex solution spaces that chemical engineers encounter. Topics include modeling many chemical processes, especially those involving reaction and diffusion, auto catalytic reactions, phase…

  9. Nonlinear Systems.

    ERIC Educational Resources Information Center

    Seider, Warren D.; Ungar, Lyle H.

    1987-01-01

    Describes a course in nonlinear mathematics courses offered at the University of Pennsylvania which provides an opportunity for students to examine the complex solution spaces that chemical engineers encounter. Topics include modeling many chemical processes, especially those involving reaction and diffusion, auto catalytic reactions, phase…

  10. Applications of Noether conservation theorem to Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Mouchet, Amaury

    2016-09-01

    The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the boundary conditions under a canonical transformation and this paper proposes to address this issue. Then, the unified treatment of Hamiltonian systems offered by Noether's approach is illustrated on several examples, including classical field theory and quantum dynamics.

  11. Applications of Noether conservation theorem to Hamiltonian systems

    SciTech Connect

    Mouchet, Amaury

    2016-09-15

    The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the boundary conditions under a canonical transformation and this paper proposes to address this issue. Then, the unified treatment of Hamiltonian systems offered by Noether’s approach is illustrated on several examples, including classical field theory and quantum dynamics.

  12. On the stability of periodic Hamiltonian systems with one degree of freedom in the case of degeneracy

    NASA Astrophysics Data System (ADS)

    Bardin, Boris S.; Lanchares, Victor

    2015-11-01

    We deal with the stability problem of an equilibrium position of a periodic Hamiltonian system with one degree of freedom. We suppose the Hamiltonian is analytic in a small neighborhood of the equilibrium position, and the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study the stability in the sense of Lyapunov. In general, the stability character of the equilibrium depends on nonzero terms of the lowest order N ( N >2) in the Hamiltonian normal form, and the stability problem can be solved by using known criteria.

  13. Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory

    NASA Astrophysics Data System (ADS)

    Cang, Shijian; Wu, Aiguo; Wang, Zenghui; Chen, Zengqiang

    Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

  14. Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    de León, M.; Sardón, C.

    2017-06-01

    In this paper, we apply the geometric Hamilton-Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms. The interest of a geometric Hamilton-Jacobi equation is the primordial observation that if a hamiltonian vector field X H can be projected into a configuration manifold by means of a 1-form dW , then the integral curves of the projected vector field X_HdW can be transformed into integral curves of X H provided that W is a solution of the Hamilton-Jacobi equation. In this way, we use the geometric Hamilton-Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton-Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz-Smorodinsky oscillators and for explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.

  15. Limit of small exits in open Hamiltonian systems.

    PubMed

    Aguirre, Jacobo; Sanjuán, Miguel A F

    2003-05-01

    The nature of open Hamiltonian systems is analyzed, when the size of the exits decreases and tends to zero. Fractal basins appear typically in open Hamiltonian systems, but we claim that in the limit of small exits, the invariant sets tend to fill up the whole phase space with the strong consequence that a new kind of basin appears, where the unpredictability grows indefinitely. This means that for finite, arbitrarily small accuracy, we can find uncertain basins, where any information about the future of the system is lost. This total indeterminism had only been reported in dissipative systems, in particular in the so-called intermingled riddled basins, as well as in the riddledlike basins. We show that this peculiar, behavior is a general feature of open Hamiltonian systems.

  16. Bounded stabilisation of stochastic port-Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Satoh, Satoshi; Saeki, Masami

    2014-08-01

    This paper proposes a stochastic bounded stabilisation method for a class of stochastic port-Hamiltonian systems. Both full-actuated and underactuated mechanical systems in the presence of noise are considered in this class. The proposed method gives conditions for the controller gain and design parameters under which the state remains bounded in probability. The bounded region and achieving probability are both assignable, and a stochastic Lyapunov function is explicitly provided based on a Hamiltonian structure. Although many conventional stabilisation methods assume that the noise vanishes at the origin, the proposed method is applicable to systems under persistent disturbances.

  17. Filtering by nonlinear systems.

    PubMed

    Campos Cantón, E; González Salas, J S; Urías, J

    2008-12-01

    Synchronization of nonlinear systems forced by external signals is formalized as the response of a nonlinear filter. Sufficient conditions for a nonlinear system to behave as a filter are given. Some examples of generalized chaos synchronization are shown to actually be special cases of nonlinear filtering.

  18. Continuation of periodic orbits in symmetric Hamiltonian and conservative systems

    NASA Astrophysics Data System (ADS)

    Galan-Vioque, J.; Almaraz, F. J. M.; Macías, E. F.

    2014-12-01

    We present and review results on the continuation and bifurcation of periodic solutions in conservative, reversible and Hamiltonian systems in the presence of symmetries. In particular we show how two-point boundary value problem continuation software can be used to compute families of periodic solutions of symmetric Hamiltonian systems. The technique is introduced with a very simple model example (the mathematical pendulum), justified with a theoretical continuation result and then applied to two non trivial examples: the non integrable spring pendulum and the continuation of the figure eight solution of the three body problem.

  19. Integrable Hamiltonian systems on low-dimensional Lie algebras

    SciTech Connect

    Korotkevich, Aleksandr A

    2009-12-31

    For any real Lie algebra of dimension 3, 4 or 5 and any nilpotent algebra of dimension 6 an integrable Hamiltonian system with polynomial coefficients is found on its coalgebra. These systems are constructed using Sadetov's method for constructing complete commutative families of polynomials on a Lie coalgebra. Bibliography: 17 titles.

  20. Non-Hamiltonian systems separable by Hamilton Jacobi method

    NASA Astrophysics Data System (ADS)

    Marciniak, Krzysztof; Błaszak, Maciej

    2008-05-01

    We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.

  1. Response of MDOF strongly nonlinear systems to fractional Gaussian noises.

    PubMed

    Deng, Mao-Lin; Zhu, Wei-Qiu

    2016-08-01

    In the present paper, multi-degree-of-freedom strongly nonlinear systems are modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems (including quasi-non-integrable, completely integrable and non-resonant, completely integrable and resonant, partially integrable and non-resonant, and partially integrable and resonant Hamiltonian systems) driven by fractional Gaussian noise is introduced. The averaged fractional stochastic differential equations (SDEs) are derived. The simulation results for some examples show that the averaged SDEs can be used to predict the response of the original systems and the simulation time for the averaged SDEs is less than that for the original systems.

  2. Response of MDOF strongly nonlinear systems to fractional Gaussian noises

    SciTech Connect

    Deng, Mao-Lin; Zhu, Wei-Qiu

    2016-08-15

    In the present paper, multi-degree-of-freedom strongly nonlinear systems are modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems (including quasi-non-integrable, completely integrable and non-resonant, completely integrable and resonant, partially integrable and non-resonant, and partially integrable and resonant Hamiltonian systems) driven by fractional Gaussian noise is introduced. The averaged fractional stochastic differential equations (SDEs) are derived. The simulation results for some examples show that the averaged SDEs can be used to predict the response of the original systems and the simulation time for the averaged SDEs is less than that for the original systems.

  3. Hamiltonian Noether theorem for gauge systems and two time physics

    NASA Astrophysics Data System (ADS)

    Villanueva, V. M.; Nieto, J. A.; Ruiz, L.; Silvas, J.

    2005-08-01

    The Noether theorem for Hamiltonian constrained systems is revisited. In particular, our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints. We apply our results to the relativistic point particle, to the Friedberg et al model and, with special emphasis, to two time physics.

  4. New bi-Hamiltonian systems on the plane

    NASA Astrophysics Data System (ADS)

    Tsiganov, A. V.

    2017-06-01

    We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and recursion operators are also presented in the framework of the Jacobi method.

  5. Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups

    DTIC Science & Technology

    1981-07-28

    478. V. I. Arnold [1964]. Inst"ility of dynamical systenms with several degrees of freedom, Dokl . Akad . Riuk. SSSR 156,9-12. V. I. Ar’nold [1966...a rigid body, Trans, oscow Math. Soc. 41, 287. S.L. Ziglin [1981]. Branching of solutions and nonexistence of integrals in Hamiltonian systems. Doklady Akad . Nauk . SSSR 257, 26-29. - J. I

  6. Evolution of a Spin System Under a Periodic Hamiltonian

    NASA Astrophysics Data System (ADS)

    Goldman, M.

    The expression of the density matrix for a spin system subjected to a periodic Hamiltonian is derived in the form of an expansion in powers of the inverse modulation frequency, an extension of a method devised by Ruishvili and Menabde and by Mehring. Its application to MAS experiments, as regards the contribution of the dipolar interactions to the sideband intensities, is discussed.

  7. Hamiltonian-Based Model to Describe the Nonlinear Physics of Cascading Failures in Power-Grid Networks

    NASA Astrophysics Data System (ADS)

    Yang, Yang; Motter, Adilson

    A local disturbance to the state of a power-grid system can trigger a protective response that disables some grid components, which leads to further responses, and may finally result in large-scale failures. In this talk, I will introduce a Hamiltonian-like model of cascading failures in power grids. This model includes the state variables of generators, which are determined by the nonlinear swing equations and power-flow equations, as well as the on/off status of the network components. This framework allows us to view a cascading failure in the power grid as a phase-space transition from a fixed point with high energy to a fixed point with lower energy. Using real power-grid networks, I will demonstrate that possible cascade outcomes can be predicted by analyzing the stability of the system's equilibria. This work adds an important new dimension to the current understanding of cascading failures.

  8. First integrals of generalized Ermakov systems via the Hamiltonian formulation

    NASA Astrophysics Data System (ADS)

    Mahomed, K. S.; Moitsheki, R. J.

    2016-07-01

    We obtain first integrals of the generalized two-dimensional Ermakov systems, in plane polar form, via the Hamiltonian approaches. There are two methods used for the construction of the first integrals, viz. the standard Hamiltonian and the partial Hamiltonian approaches. In the first approach, F(𝜃) and G(𝜃) in the Ermakov system are related as G(𝜃) + F‧(𝜃)/2 = 0. In this case, we deduce four first integrals (three of which are functionally independent) which correspond to the Lie algebra sl(2,R) ⊕ A1 in a direct constructive manner. We recover the results of earlier work that uses the relationship between symmetries and integrals. This results in the complete integrability of the Ermakov system. By use of the partial Hamiltonian method, we discover four new cases: F(𝜃) = G(𝜃)(c1sin 𝜃 + c3cos 𝜃)/(c1cos 𝜃 - c3sin 𝜃) with c2c3 = c1c4, c1≠0, c3≠0; F(𝜃) = G(𝜃)(c2sin 𝜃 + c4cos 𝜃)/(c2cos 𝜃 - c4sin 𝜃) with c1 = c3 = 0, c2≠0, c4≠0; F(𝜃) = -G(𝜃)cot 𝜃 with c1 = c2 = 0, c3, c4 arbitrary and F(𝜃) = G(𝜃)tan 𝜃 with c3 = c4 = 0, c1, c2 arbitrary, where the cis are constants in all cases. In the last two cases, we find that there are three operators each which give rise to three first integrals each. In both these cases, we have complete integrability of the Ermakov system. The first two cases each result in two first integrals each. For every case, both for the standard and partial Hamiltonian, the angular momentum type first integral arises and this is a consequence of the operator which depends on a momentum coordinate which is a generalized symmetry in the Lagrangian context.

  9. Model spin-orbit coupling Hamiltonians for graphene systems

    NASA Astrophysics Data System (ADS)

    Kochan, Denis; Irmer, Susanne; Fabian, Jaroslav

    2017-04-01

    We present a detailed theoretical study of effective spin-orbit coupling (SOC) Hamiltonians for graphene-based systems, covering global effects such as proximity to substrates and local SOC effects resulting, for example, from dilute adsorbate functionalization. Our approach combines group theory and tight-binding descriptions. We consider structures with global point group symmetries D6 h, D3 d, D3 h, C6 v, and C3 v that represent, for example, pristine graphene, graphene miniripple, planar boron nitride, graphene on a substrate, and free standing graphone, respectively. The presence of certain spin-orbit coupling parameters is correlated with the absence of the specific point group symmetries. Especially in the case of C6 v—graphene on a substrate, or transverse electric field—we point out the presence of a third SOC parameter, besides the conventional intrinsic and Rashba contributions, thus far neglected in literature. For all global structures we provide effective SOC Hamiltonians both in the local atomic and Bloch forms. Dilute adsorbate coverage results in the local point group symmetries C6 v, C3 v, and C2 v, which represent the stable adsorption at hollow, top and bridge positions, respectively. For each configuration we provide effective SOC Hamiltonians in the atomic orbital basis that respect local symmetries. In addition to giving specific analytic expressions for model SOC Hamiltonians, we also present general (no-go) arguments about the absence of certain SOC terms.

  10. Rapid geometrical chaotization in slow-fast Hamiltonian systems.

    PubMed

    Artemyev, A V; Neishtadt, A I; Zelenyi, L M

    2014-06-01

    In this Rapid Communication we demonstrate effects of a new mechanism of adiabaticity destruction in Hamiltonian systems with a separatrix in the phase space. In contrast to the slow diffusive-like destruction typical for many systems, this new mechanism is responsible for very fast chaotization in a large phase volume. To investigate this mechanism we consider a Hamiltonian system with two degrees of freedom and with a separatrix in the phase plane of fast variables. The fast chaotization is due to an asymmetry of the separatrix and corresponding geometrical jumps of an adiabatic invariant. This system describes the motion of charged particles in a inhomogeneous electromagnetic field with a specific configuration. We show that geometrical jumps of the adiabatic invariant result in a very fast chaotization of particle motion.

  11. Comparative index and Sturmian theory for linear Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Šepitka, Peter; Šimon Hilscher, Roman

    2017-01-01

    The comparative index was introduced by J. Elyseeva (2007) as an efficient tool in matrix analysis, which has fundamental applications in the discrete oscillation theory. In this paper we implement the comparative index into the theory of continuous time linear Hamiltonian systems, study its properties, and apply it to obtain new Sturmian separation theorems as well as new and optimal estimates for left and right proper focal points of conjoined bases of these systems on bounded intervals. We derive our results for general possibly abnormal (or uncontrollable) linear Hamiltonian systems. The results turn out to be new even in the case of completely controllable systems. We also provide several examples, which illustrate our new theory.

  12. Hamiltonian structure for degenerate AKNS systems

    NASA Astrophysics Data System (ADS)

    Corona-Corona, Gulmaro

    1997-01-01

    There is a family of degenerate AKNS systems for which the full theory of generic AKNS systems does not directly extend. The linear space of potentials still has a natural Poisson structure, but the scattering method used by Beals and Sattinger to show complete integrability for the generic AKNS systems fails for the degenerate case. A Poisson structure is not induced on the scattering side as in the generic case. As a consequence, the problem of complete integrability for degenerate AKNS systems still is an open question. In addition, contrary to the generic case, the Lax pair gives flows for degenerate integrable systems that are nonlocal. In general, they do not exist, and they are no longer linear on the scattering side. Necessary conditions for their existence and for linear evolution in the scattering side are found.

  13. Hamiltonian Structure for Degenerate Akns Systems

    NASA Astrophysics Data System (ADS)

    Corona-Corona, Gulmaro

    1995-01-01

    There is a family of degenerate AKNS systems for which the full theory of generic AKNS systems does not directly extend. The linear space of potentials still has a natural Poisson structure. This is studied by the scattering method used by Richard Beals and D.H. Sattinger (Commun. Math. Phys. 138, 409-436, 1991) to show complete integrability for the generic AKNS systems. This method fails for the degenerate case since a Poisson structure is not induced on the scattering side as in the generic case. As a consequence, the problem of complete integrability for degenerate AKNS systems still is an open question. In addition, contrary to the generic case, the Lax pair gives flows for degenerate integrable systems that are nonlocal. In general they do not exist, and they are no longer linear on the scattering side. Necessary conditions for their existence and for linear evolution of the scattering side are found.

  14. Finite-time thermodynamics of port-Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Delvenne, Jean-Charles; Sandberg, Henrik

    2014-01-01

    In this paper, we identify a class of time-varying port-Hamiltonian systems that is suitable for studying problems at the intersection of statistical mechanics and control of physical systems. Those port-Hamiltonian systems are able to modify their internal structure as well as their interconnection with the environment over time. The framework allows us to prove the First and Second Laws of thermodynamics, but also lets us apply results from optimal and stochastic control theory to physical systems. In particular, we show how to use linear control theory to optimally extract work from a single heat source over a finite time interval in the manner of Maxwell’s demon. Furthermore, the optimal controller is a time-varying port-Hamiltonian system, which can be physically implemented as a variable linear capacitor and transformer. We also use the theory to design a heat engine operating between two heat sources in finite-time Carnot-like cycles of maximum power, and we compare those two heat engines.

  15. Integrable and superintegrable Hamiltonian systems with four dimensional real Lie algebras as symmetry of the systems

    SciTech Connect

    Abedi-Fardad, J.; Rezaei-Aghdam, A.; Haghighatdoost, Gh.

    2014-05-15

    We construct integrable and superintegrable Hamiltonian systems using the realizations of four dimensional real Lie algebras as a symmetry of the system with the phase space R{sup 4} and R{sup 6}. Furthermore, we construct some integrable and superintegrable Hamiltonian systems for which the symmetry Lie group is also the phase space of the system.

  16. Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Gong, Yuezheng; Wang, Qi; Wang, Zhu

    2017-03-01

    The proper orthogonal decomposition reduced-order models (POD-ROMs) have been widely used as a computationally efficient surrogate models in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy its Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduce-order modeling approach for the Hamiltonian system, which uses the traditional framework of Galerkin projection-based model reduction but modifies the ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use the POD basis from shifted snapshots to improve the Hamiltonian function approximation. We further derive a rigorous a priori error estimate of the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems etc.

  17. Moreau-Yosida approximation and convergence of Hamiltonian systems on Wasserstein space

    NASA Astrophysics Data System (ADS)

    Kim, Hwa Kil

    In this paper, we study the stability property of Hamiltonian systems on the Wasserstein space. Let H be a given Hamiltonian satisfying certain properties. We regularize H using the Moreau-Yosida approximation and denote it by Hτ. We show that solutions of the Hamiltonian system for Hτ converge to a solution of the Hamiltonian system for H as τ converges to zero. We provide sufficient conditions on H to carry out this process.

  18. Stability, bifurcation, and control of Hamiltonian systems

    SciTech Connect

    Marsden, J.E.; Ratiu, T.S.

    1993-04-01

    Work is being done on dissipation-induced instabilities, gyroscopic stabilization and its destruction by a small damping for both finite dimensional and certain infinite dimensional systems (such as rotating rods, strings), nonabelian and abelian cases, Euler-Lagrange-Poincare equations, the Routhian having a form of a Lagrangian with a gyroscopic term, Euler-Lagrange equations, etc.

  19. Stability, bifurcation, and control of Hamiltonian systems

    SciTech Connect

    Marsden, J.E. . Dept. of Mathematics); Ratiu, T.S. . Dept. of Mathematics)

    1993-01-01

    Work is being done on dissipation-induced instabilities, gyroscopic stabilization and its destruction by a small damping for both finite dimensional and certain infinite dimensional systems (such as rotating rods, strings), nonabelian and abelian cases, Euler-Lagrange-Poincare equations, the Routhian having a form of a Lagrangian with a gyroscopic term, Euler-Lagrange equations, etc.

  20. Subharmonic Solutions Near an Equilibrium Point for Hamiltonian Systems

    DTIC Science & Technology

    1989-04-01

    Thus A = 0 will be continued as A,(6) = A+ + ... j = 1,2. (2.5) Since the matrix V is invertible, W is nilpotent and, V and W commute, the matrix (eV...Hamiltonian system z JA(t)z + JHt..(z, t) where A(t) is a matrix , fl,(z,t) = o(I z 1) and both A and H are periodic in t. On the linear part of the system we...write the Hamiltonian as H(z, t) = 1(A(t)z, z) + II(z, t) (0.2) where A(t) denotes the Hessian matrix of H at z = 0 and ft(z, t) = o(Iz12) represents the

  1. Production and transfer of energy and information in Hamiltonian systems.

    PubMed

    Antonopoulos, Chris G; Bianco-Martinez, Ezequiel; Baptista, Murilo S

    2014-01-01

    We present novel results that relate energy and information transfer with sensitivity to initial conditions in chaotic multi-dimensional Hamiltonian systems. We show the relation among Kolmogorov-Sinai entropy, Lyapunov exponents, and upper bounds for the Mutual Information Rate calculated in the Hamiltonian phase space and on bi-dimensional subspaces. Our main result is that the net amount of transfer from kinetic to potential energy per unit of time is a power-law of the upper bound for the Mutual Information Rate between kinetic and potential energies, and also a power-law of the Kolmogorov-Sinai entropy. Therefore, transfer of energy is related with both transfer and production of information. However, the power-law nature of this relation means that a small increment of energy transferred leads to a relatively much larger increase of the information exchanged. Then, we propose an "experimental" implementation of a 1-dimensional communication channel based on a Hamiltonian system, and calculate the actual rate with which information is exchanged between the first and last particle of the channel. Finally, a relation between our results and important quantities of thermodynamics is presented.

  2. On the quantum mechanics of bicomplex Hamiltonian system

    NASA Astrophysics Data System (ADS)

    Banerjee, Abhijit

    2017-02-01

    We investigate the Schrödinger equation in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. We propose an analytical method to solve bicomplex-version of Schrödinger equation corresponding to the systems of Hamiltonians of both hermitian (self-adjoint) and non-hermitian PT symmetric type. In our approach we extend the existing mathematical formulation of quantum system searching for the exact or quasi-exact solution for ground state energy eigenvalues and associated wave functions acting in bicomplex Hilbert space. The model concerning hermitian Hamiltonians is then applied to the problems of two bicomplex valued polynomial oscillators one involving x2 and another of isotonic type. The ground states and associated energy values for both the oscillators are found to be hyperbolic in nature. The model in connection to the unbroken PT symmetric Hamiltonians is then applied to illustrate the problems of complex and bicomplex valued shifted oscillators.

  3. Production and Transfer of Energy and Information in Hamiltonian Systems

    PubMed Central

    Antonopoulos, Chris G.; Bianco-Martinez, Ezequiel; Baptista, Murilo S.

    2014-01-01

    We present novel results that relate energy and information transfer with sensitivity to initial conditions in chaotic multi-dimensional Hamiltonian systems. We show the relation among Kolmogorov-Sinai entropy, Lyapunov exponents, and upper bounds for the Mutual Information Rate calculated in the Hamiltonian phase space and on bi-dimensional subspaces. Our main result is that the net amount of transfer from kinetic to potential energy per unit of time is a power-law of the upper bound for the Mutual Information Rate between kinetic and potential energies, and also a power-law of the Kolmogorov-Sinai entropy. Therefore, transfer of energy is related with both transfer and production of information. However, the power-law nature of this relation means that a small increment of energy transferred leads to a relatively much larger increase of the information exchanged. Then, we propose an “experimental” implementation of a 1-dimensional communication channel based on a Hamiltonian system, and calculate the actual rate with which information is exchanged between the first and last particle of the channel. Finally, a relation between our results and important quantities of thermodynamics is presented. PMID:24586891

  4. Fluctuation theorem for Hamiltonian systems: Le Chatelier's principle.

    PubMed

    Evans, D J; Searles, D J; Mittag, E

    2001-05-01

    For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields.

  5. Fluctuation theorem for Hamiltonian Systems: Le Chatelier's principle

    NASA Astrophysics Data System (ADS)

    Evans, Denis J.; Searles, Debra J.; Mittag, Emil

    2001-05-01

    For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields.

  6. Path-integral description of combined Hamiltonian and non-Hamiltonian dynamics in quantum dissipative systems

    NASA Astrophysics Data System (ADS)

    Barth, A. M.; Vagov, A.; Axt, V. M.

    2016-09-01

    We present a numerical path-integral iteration scheme for the low-dimensional reduced density matrix of a time-dependent quantum dissipative system. Our approach simultaneously accounts for the combined action of a microscopically modeled pure-dephasing-type coupling to a continuum of harmonic oscillators representing, e.g., phonons, and further environmental interactions inducing non-Hamiltonian dynamics in the inner system represented, e.g., by Lindblad-type dissipation or relaxation. Our formulation of the path-integral method allows for a numerically exact treatment of the coupling to the oscillator modes and moreover is general enough to provide a natural way to include Markovian processes that are sufficiently described by rate equations. We apply this new formalism to a model of a single semiconductor quantum dot which includes the coupling to longitudinal acoustic phonons for two cases: (a) external laser excitation taking into account a phenomenological radiative decay of the excited dot state and (b) a coupling of the quantum dot to a single mode of an optical cavity taking into account cavity photon losses.

  7. Hamiltonian of mean force for damped quantum systems.

    PubMed

    Hilt, Stefanie; Thomas, Benedikt; Lutz, Eric

    2011-09-01

    We consider a quantum system linearly coupled to a reservoir of harmonic oscillators. For finite coupling strengths, the stationary distribution of the damped system deviates from the predictions of standard thermodynamics. With the help of the quantum Hamiltonian of mean force, we quantify this deviation exactly for a harmonic oscillator and provide approximations in the limit of high and low temperatures and weak and strong couplings. Moreover, in the semiclassical regime, we use the quantum Smoluchowski equation to obtain results valid for any potential. We finally give a physical interpretation of the deviation in terms of the initial system-reservoir coupling.

  8. Quantization of non-Hamiltonian and dissipative systems

    NASA Astrophysics Data System (ADS)

    Tarasov, Vasily E.

    2001-09-01

    A generalization of canonical quantization which maps a dynamical operator to a dynamical superoperator is suggested. Weyl quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables is a specific case of suggested quantization. This approach allows to define consistent quantization procedure for non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction (generalized Lorenz-Rossler-Leipnik-Newton equation), the Fokker-Planck-type system and Lorenz-type system are considered.

  9. Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians

    NASA Astrophysics Data System (ADS)

    Fring, Andreas; Frith, Thomas

    2017-01-01

    We propose a procedure to obtain exact analytical solutions to the time-dependent Schrödinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.

  10. Rigorous KAM results around arbitrary periodic orbits for Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Kapela, Tomasz; Simó, Carles

    2017-03-01

    We set up a methodology for computer assisted proofs of the existence and the KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give two examples of application for systems with two and three degrees of freedom. The first example verifies the existence of tiny elliptic islands inside large chaotic domains for a quartic potential. In the 3-body problem we prove the KAM stability of the well-known figure eight orbit and two selected orbits of the so called family of rotating eights. Some additional theoretical and numerical information is also given for the dynamics of both examples.

  11. Renormalization Group Reduction of Non Integrable Hamiltonian Systems

    SciTech Connect

    Stephan I. Tzenov

    2002-05-09

    Based on Renormalization Group method, a reduction of non integratable multi-dimensional Hamiltonian systems has been performed. The evolution equations for the slowly varying part of the angle-averaged phase space density and for the amplitudes of the angular modes have been derived. It has been shown that these equations are precisely the Renormalization Group equations. As an application of the approach developed, the modulational diffusion in one-and-a-half degrees of freedom dynamical system has been studied in detail.

  12. Nonconventional fluctuation dissipation process in non-Hamiltonian dynamical systems

    NASA Astrophysics Data System (ADS)

    Bianucci, Marco

    2016-08-01

    Here, we introduce a statistical approach derived from dynamics, for the study of the geophysical fluid dynamics phenomena characterized by a weak interaction among the variables of interest and the rest of the system. The approach is reminiscent of the one developed some years ago [M. Bianucci, R. Mannella, P. Grigolini and B. J. West, Phys. Rev. E 51, 3002 (1995)] to derive statistical mechanics of macroscopic variables on interest starting from Hamiltonian microscopic dynamics. However, in the present work, we are interested to generalize this approach beyond the context of the foundation of thermodynamics, in fact, we take into account the cases where the system of interest could be non-Hamiltonian (dissipative) and also the interaction with the irrelevant part can be of a more general type than Hamiltonian. As such example, we will refer to a typical case from geophysical fluid dynamics: the complex ocean-atmosphere interaction that gives rise to the El Niño Southern Oscillation (ENSO). Here, changing all the scales, the role of the “microscopic” system is played by the atmosphere, while the ocean (or some ocean variables) plays the role of the intrinsically dissipative macroscopic system of interest. Thus, the chaotic and divergent features of the fast atmosphere dynamics remains in the decaying properties of the correlation functions and of the response function of the atmosphere variables, while the exponential separation of the perturbed (or close) single trajectories does not play a direct role. In the present paper, we face this problem in the frame of a not formal Langevin approach, limiting our discussion to physically based rather than mathematics arguments. Elsewhere, we obtain these results via a much more formal procedure, using the Zwanzing projection method and some elements from the Lie Algebra field.

  13. Dynamics symmetries of Hamiltonian system on time scales

    NASA Astrophysics Data System (ADS)

    Peng, Keke; Luo, Yiping

    2014-04-01

    In this paper, the dynamics symmetries of Hamiltonian system on time scales are studied. We study the symmetries and quantities based on the calculation of variation and Lie transformation group. Particular focus lies in: the Noether symmetry leads to the Noether conserved quantity and the Lie symmetry leads to the Noether conserved quantity if the infinitesimal transformations satisfy the structure equation. As the new application of result, at end of the article, we give a simple example of Noether symmetry and Lie symmetry on time scales.

  14. Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Chen, Zhaoxia; You, Xiong; Shi, Wei; Liu, Zhongli

    2012-01-01

    The ERKN methods proposed by H. Yang et al. [Comput. Phys. Comm. 180 (2009) 1777] are an important improvement of J.M. Franco's ARKN methods for perturbed oscillators [J.M. Franco, Comput. Phys. Comm. 147 (2002) 770]. This paper focuses on the symmetry and symplecticity conditions for ERKN methods solving oscillatory Hamiltonian systems. Two examples of symmetric and symplectic ERKN (SSERKN) methods of orders two and four respectively are constructed. The phase and stability properties of the new methods are analyzed. The results of numerical experiments show the robustness and competence of the SSERKN methods compared with some well-known methods in the literature.

  15. Dynamics symmetries of Hamiltonian system on time scales

    SciTech Connect

    Peng, Keke Luo, Yiping

    2014-04-15

    In this paper, the dynamics symmetries of Hamiltonian system on time scales are studied. We study the symmetries and quantities based on the calculation of variation and Lie transformation group. Particular focus lies in: the Noether symmetry leads to the Noether conserved quantity and the Lie symmetry leads to the Noether conserved quantity if the infinitesimal transformations satisfy the structure equation. As the new application of result, at end of the article, we give a simple example of Noether symmetry and Lie symmetry on time scales.

  16. The symmetry groups of bifurcations of integrable Hamiltonian systems

    SciTech Connect

    Orlova, E I

    2014-11-30

    Two-dimensional atoms are investigated; these are used to code bifurcations of the Liouville foliations of nondegenerate integrable Hamiltonian systems. To be precise, the symmetry groups of atoms with complexity at most 3 are under study. Atoms with symmetry group Z{sub p}⊕Z{sub q} are considered. It is proved that Z{sub p}⊕Z{sub q} is the symmetry group of a toric atom. The symmetry groups of all nonorientable atoms with complexity at most 3 are calculated. The concept of a geodesic atom is introduced. Bibliography: 9 titles.

  17. A hierarchy of super AKNS hierarchy related to Lie superalgebra sl(2|1) and a finite dimensional super Hamiltonian system

    NASA Astrophysics Data System (ADS)

    Zhou, Ruguang

    2015-07-01

    A hierarchy of super AKNS equations associated with a sl(2|1) matrix-valued spectral problem is derived. It is shown that each equation in the hierarchy is bi-super Hamiltonian. Moreover, a new finite dimensional super Hamiltonian system (FDSHS), together with its Lax representation, r-matrix and conversed integrals of motion, is obtained from the spectral problem by binary nonlinearization.

  18. Hamiltonian Dynamics of Spider-Type Multirotor Rigid Bodies Systems

    SciTech Connect

    Doroshin, Anton V.

    2010-03-01

    This paper sets out to develop a spider-type multiple-rotor system which can be used for attitude control of spacecraft. The multirotor system contains a large number of rotor-equipped rays, so it was called a 'Spider-type System', also it can be called 'Rotary Hedgehog'. These systems allow using spinups and captures of conjugate rotors to perform compound attitude motion of spacecraft. The paper describes a new method of spacecraft attitude reorientation and new mathematical model of motion in Hamilton form. Hamiltonian dynamics of the system is investigated with the help of Andoyer-Deprit canonical variables. These variables allow obtaining exact solution for hetero- and homoclinic orbits in phase space of the system motion, which are very important for qualitative analysis.

  19. Hamiltonian Dynamics of Spider-Type Multirotor Rigid Bodies Systems

    NASA Astrophysics Data System (ADS)

    Doroshin, Anton V.

    2010-03-01

    This paper sets out to develop a spider-type multiple-rotor system which can be used for attitude control of spacecraft. The multirotor system contains a large number of rotor-equipped rays, so it was called a ``Spider-type System,'' also it can be called ``Rotary Hedgehog.'' These systems allow using spinups and captures of conjugate rotors to perform compound attitude motion of spacecraft. The paper describes a new method of spacecraft attitude reorientation and new mathematical model of motion in Hamilton form. Hamiltonian dynamics of the system is investigated with the help of Andoyer-Deprit canonical variables. These variables allow obtaining exact solution for hetero- and homoclinic orbits in phase space of the system motion, which are very important for qualitative analysis.

  20. Nonlinear systems in medicine.

    PubMed Central

    Higgins, John P.

    2002-01-01

    Many achievements in medicine have come from applying linear theory to problems. Most current methods of data analysis use linear models, which are based on proportionality between two variables and/or relationships described by linear differential equations. However, nonlinear behavior commonly occurs within human systems due to their complex dynamic nature; this cannot be described adequately by linear models. Nonlinear thinking has grown among physiologists and physicians over the past century, and non-linear system theories are beginning to be applied to assist in interpreting, explaining, and predicting biological phenomena. Chaos theory describes elements manifesting behavior that is extremely sensitive to initial conditions, does not repeat itself and yet is deterministic. Complexity theory goes one step beyond chaos and is attempting to explain complex behavior that emerges within dynamic nonlinear systems. Nonlinear modeling still has not been able to explain all of the complexity present in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thus will augment our understanding of the nature of complex dynamic systems within the human body in health and in disease states. PMID:14580107

  1. Detecting order and chaos in three-dimensional Hamiltonian systems by geometrical methods.

    PubMed

    Ben Zion, Yossi; Horwitz, Lawrence

    2007-10-01

    We use a geometrical method to distinguish between ordered and chaotic motion in three-dimensional Hamiltonian systems. We show that this method gives results in agreement with the computation of Lyapunov characteristic exponents. We discuss some examples of unstable Hamiltonian systems in three dimensions, giving, as a particular illustration, detailed results for a potential obtained from a Hamiltonian obtained from a Yang-Mills system.

  2. Long-range correlations in quantum systems with aperiodic Hamiltonians

    NASA Astrophysics Data System (ADS)

    Lin, Zhifang; Goda, Masaki

    1997-03-01

    An efficient algorithm for the computation of correlation function (CF) at very long distances is presented for quantum systems whose Hamiltonian is formed by the substitution aperiodic sequence alternating over unit intervals in time or space. The algorithm reorganizes the expression of the CF in such a way that the evaluation of the CF at distances equal to some special numbers is related to a family of graphs generated recursively. As examples of applications, we evaluate the CF, over unprecedentedly long time intervals up to order of 1012, for aperiodic two-level systems subject to kicking perturbations that are in the Thue-Morse, the period-doubling, and the Rudin-Shapiro sequences, respectively. Our results show the presence of long-range correlations in all these aperiodic quantum systems.

  3. On the Hamiltonian structure of large deviations in stochastic hybrid systems

    NASA Astrophysics Data System (ADS)

    Bressloff, Paul C.; Faugeras, Olivier

    2017-03-01

    We present a new derivation of the classical action underlying a large deviation principle (LDP) for a stochastic hybrid system, which couples a piecewise deterministic dynamical system in {{{R}}d} with a time-homogeneous Markov chain on some discrete space Γ . We assume that the Markov chain on Γ is ergodic, and that the discrete dynamics is much faster than the piecewise deterministic dynamics (separation of time-scales). Using the Perron–Frobenius theorem and the calculus-of-variations, we show that the resulting action Hamiltonian is given by the Perron eigenvalue of a | Γ | -dimensional linear equation. The corresponding linear operator depends on the transition rates of the Markov chain and the nonlinear functions of the piecewise deterministic system. We compare the Hamiltonian to one derived using WKB methods, and show that the latter is a reduction of the former. We also indicate how the analysis can be extended to a multi-scale stochastic process, in which the continuous dynamics is described by a piecewise stochastic differential equations (SDE). Finally, we illustrate the theory by considering applications to conductance-based models of membrane voltage fluctuations in the presence of stochastic ion channels.

  4. Nonlinear resonance

    NASA Astrophysics Data System (ADS)

    Kevorkian, J.

    This report discusses research in the area of slowly varying nonlinear oscillatory systems. Some of the topics discussed are as follows: adiabatic invariants and transient resonance in very slowly varying Hamiltonian systems; sustained resonance in very slowly varying Hamiltonian systems; free-electron lasers with very slow wiggler taper; and bursting oscillators.

  5. Numerical study on a canonized Hamiltonian system representing reduced magnetohydrodynamics and its comparison with two-dimensional Euler system

    SciTech Connect

    Kaneko, Yuta; Yoshida, Zensho

    2014-03-15

    Introducing a Clebsch-like parameterization, we have formulated a canonical Hamiltonian system on a symplectic leaf of reduced magnetohydrodynamics. An interesting structure of the equations is in that the Lorentz-force, which is a quadratic nonlinear term in the conventional formulation, appears as a linear term −ΔQ, just representing the current density (Q is a Clebsch variable, and Δ is the two-dimensional Laplacian); omitting this term reduces the system into the two-dimensional Euler vorticity equation of a neutral fluid. A heuristic estimate shows that current sheets grow exponentially (even in a fully nonlinear regime) together with the action variable P that is conjugate to Q. By numerical simulation, the predicted behavior of the canonical variables, yielding exponential growth of current sheets, has been demonstrated.

  6. Control of Nonlinear Systems.

    DTIC Science & Technology

    1980-02-26

    6-7 C. Minimum Energy Regulators for Commutative Bilinear Systems .................... ........ 8-9 D. Control Law.s for Certain Aerospace...class of nonlinear systems (3,10]. (c) Minimum energy regulators for commutative bilinear systems [3,10]. (D) Control laws for certain aerospace...With Delay in Control," IEEE Trans. on Auto Contr., Vol. AC-20, pp. 702-704, 1975, and [3].) - !. 8 C. Minimum Energy Regulators for Commutative Bilinear

  7. Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R+4

    NASA Astrophysics Data System (ADS)

    Llibre, Jaume; Xiao, Dongmei

    2017-02-01

    In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R+4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka-Volterra Hamiltonian systems in R+4. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R+4 defined by the hypersurfaces axy + bzw + cx2 y + dxy2 + ez2 w + fzw2 = h, where a , b , c , d , e , f , w and h are real constants.

  8. Lagrangian-Hamiltonian unified formalism for autonomous higher order dynamical systems

    NASA Astrophysics Data System (ADS)

    Prieto-Martínez, Pedro Daniel; Román-Roy, Narciso

    2011-09-01

    The Lagrangian-Hamiltonian unified formalism of Skinner and Rusk was originally stated for autonomous dynamical systems in classical mechanics. It has been generalized for non-autonomous first-order mechanical systems, as well as for first-order and higher order field theories. However, a complete generalization to higher order mechanical systems is yet to be described. In this work, after reviewing the natural geometrical setting and the Lagrangian and Hamiltonian formalisms for higher order autonomous mechanical systems, we develop a complete generalization of the Lagrangian-Hamiltonian unified formalism for these kinds of systems, and we use it to analyze some physical models from this new point of view.

  9. Accelerator-feasible N -body nonlinear integrable system

    NASA Astrophysics Data System (ADS)

    Danilov, V.; Nagaitsev, S.

    2014-12-01

    Nonlinear N -body integrable Hamiltonian systems, where N is an arbitrary number, have attracted the attention of mathematical physicists for the last several decades, following the discovery of some number of these systems. This paper presents a new integrable system, which can be realized in facilities such as particle accelerators. This feature makes it more attractive than many of the previous such systems with singular or unphysical forces.

  10. Accelerator-Feasible N-Body Nonlinear Integrable System

    SciTech Connect

    Danilov, V.; Nagaitsev, S.

    2014-12-23

    Nonlinear N-body integrable Hamiltonian systems, where N is an arbitrary number, attract the attention of mathematical physicists for the last several decades, following the discovery of some number of these systems. This paper presents a new integrable system, which can be realized in facilities such as particle accelerators. This feature makes it more attractive than many of the previous such systems with singular or unphysical forces.

  11. Accelerator-feasible N-body nonlinear integrable system

    DOE PAGES

    Danilov, V.; Nagaitsev, S.

    2014-12-23

    Nonlinear N-body integrable Hamiltonian systems, where N is an arbitrary number, attract the attention of mathematical physicists for the last several decades, following the discovery of some number of these systems. This research presents a new integrable system, which can be realized in facilities such as particle accelerators. This feature makes it more attractive than many of the previous such systems with singular or unphysical forces.

  12. Hamilton-Jacobi theorems for regular reducible Hamiltonian systems on a cotangent bundle

    NASA Astrophysics Data System (ADS)

    Wang, Hong

    2017-09-01

    In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system and regular reduced Hamiltonian systems are given. At first, an important lemma is proved, and it is a modification for the corresponding result of Abraham and Marsden (1978), such that we can prove two types of geometric Hamilton-Jacobi theorem for a Hamiltonian system on the cotangent bundle of a configuration manifold, by using the symplectic form and dynamical vector field. Then these results are generalized to the regular reducible Hamiltonian system with symmetry and momentum map, by using the reduced symplectic form and the reduced dynamical vector field. The Hamilton-Jacobi theorems are proved and two types of Hamilton-Jacobi equations, for the regular point reduced Hamiltonian system and the regular orbit reduced Hamiltonian system, are obtained. As an application of the theoretical results, the regular point reducible Hamiltonian system on a Lie group is considered, and two types of Lie-Poisson Hamilton-Jacobi equation for the regular point reduced system are given. In particular, the Type I and Type II of Lie-Poisson Hamilton-Jacobi equations for the regular point reduced rigid body and heavy top systems are shown, respectively.

  13. Weakly Hamiltonian actions

    NASA Astrophysics Data System (ADS)

    Martínez Torres, David; Miranda, Eva

    2017-05-01

    In this paper we generalize constructions of non-commutative integrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamiltonian actions on symplectic manifolds split into Hamiltonian and non-Hamiltonian factors, and explore generalizations in the Poisson setting.

  14. Coupled nonlinear dynamical systems

    NASA Astrophysics Data System (ADS)

    Sun, Hongyan

    In this dissertation, we study coupled nonlinear dynamical systems that exhibit new types of complex behavior. We numerically and analytically examine a variety of dynamical models, ranging from systems of ordinary differential equations (ODE) with novel elements of feedback to systems of partial differential equations (PDE) that model chemical pattern formation. Chaos, dynamical uncertainty, synchronization, and spatiotemporal pattern formation constitute the primary topics of the dissertation. Following the introduction in Chapter 1, we study chaos and dynamical uncertainty in Chapter 2 with coupled Lorenz systems and demonstrate the existence of extreme complexity in high-dimensional ODE systems. In Chapter 3, we demonstrate that chaos synchronization can be achieved by mutual and multiplicative coupling of dynamical systems. Chapter 4 and 5 focus on pattern formation in reaction-diffusion systems, and we investigate segregation and integration behavior of populations in competitive and cooperative environments, respectively.

  15. Nonlinear Quantum Metrology of Many-Body Open Systems

    NASA Astrophysics Data System (ADS)

    Beau, M.; del Campo, A.

    2017-07-01

    We introduce general bounds for the parameter estimation error in nonlinear quantum metrology of many-body open systems in the Markovian limit. Given a k -body Hamiltonian and p -body Lindblad operators, the estimation error of a Hamiltonian parameter using a Greenberger-Horne-Zeilinger state as a probe is shown to scale as N-[k -(p /2 )], surpassing the shot-noise limit for 2 k >p +1 . Metrology equivalence between initial product states and maximally entangled states is established for p ≥1 . We further show that one can estimate the system-environment coupling parameter with precision N-(p /2 ), while many-body decoherence enhances the precision to N-k in the noise-amplitude estimation of a fluctuating k -body Hamiltonian. For the long-range Ising model, we show that the precision of this parameter beats the shot-noise limit when the range of interactions is below a threshold value.

  16. Fractional Hamiltonian analysis of higher order derivatives systems

    SciTech Connect

    Baleanu, Dumitru; Muslih, Sami I.; Tas, Kenan

    2006-10-15

    The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski's formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part and a damped oscillator are analyzed. The classical results are obtained when fractional derivatives are replaced with the integer order derivatives.

  17. State transformations and Hamiltonian structures for optimal control in discrete systems

    NASA Astrophysics Data System (ADS)

    Sieniutycz, S.

    2006-04-01

    Preserving usual definition of Hamiltonian H as the scalar product of rates and generalized momenta we investigate two basic classes of discrete optimal control processes governed by the difference rather than differential equations for the state transformation. The first class, linear in the time interval θ, secures the constancy of optimal H and satisfies a discrete Hamilton-Jacobi equation. The second class, nonlinear in θ, does not assure the constancy of optimal H and satisfies only a relationship that may be regarded as an equation of Hamilton-Jacobi type. The basic question asked is if and when Hamilton's canonical structures emerge in optimal discrete systems. For a constrained discrete control, general optimization algorithms are derived that constitute powerful theoretical and computational tools when evaluating extremum properties of constrained physical systems. The mathematical basis is Bellman's method of dynamic programming (DP) and its extension in the form of the so-called Carathéodory-Boltyanski (CB) stage optimality criterion which allows a variation of the terminal state that is otherwise fixed in Bellman's method. For systems with unconstrained intervals of the holdup time θ two powerful optimization algorithms are obtained: an unconventional discrete algorithm with a constant H and its counterpart for models nonlinear in θ. We also present the time-interval-constrained extension of the second algorithm. The results are general; namely, one arrives at: discrete canonical equations of Hamilton, maximum principles, and (at the continuous limit of processes with free intervals of time) the classical Hamilton-Jacobi theory, along with basic results of variational calculus. A vast spectrum of applications and an example are briefly discussed with particular attention paid to models nonlinear in the time interval θ.

  18. Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations

    NASA Astrophysics Data System (ADS)

    Ballesteros, Angel; Marrero, Juan C.; Ravanpak, Zohreh

    2017-04-01

    Given a Lie–Poisson completely integrable bi-Hamiltonian system on {{{R}}n} , we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie–Poisson system on a non-abelian Poisson–Lie group {{G}η} of dimension n, where η \\in {R} is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson–Lie) Hamiltonian structures on {{G}η} that underly the dynamics of the deformed system and by making use of the group law on {{G}η} , one may obtain two completely integrable Hamiltonian systems on {{G}η}× {{G}η} . By construction, both systems admit reduction, via the multiplication in {{G}η} , to the deformed bi-Hamiltonian system in {{G}η} . The previous approach is applied to two relevant Lie–Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.

  19. Linearization of Nonlinear Systems.

    DTIC Science & Technology

    1986-11-24

    series. IEEE Trans. Circuits Syst., CAS-32(11):1150-1171, November 1985. [BC85b] S. Boyd and L. 0. Chua. Uniqueness of circuits and systems containing...Control and Information Sciences vol. 58, p10 1- 1 19 , June 1983. [BC85c] S. Boyd and L. 0. Chua. Volterra series for nonlinear circuits . In Proc. IEEE...ISCAS, Tokyo, June 1985. [BCD84] S. Boyd, L. 0. Chua, and C. A. Desoer . Analytical foundations of Volterra series. IMA Journal of Mathematical

  20. Entanglement Hamiltonians in Fermion Systems and the Riemann-Hilbert problem

    NASA Astrophysics Data System (ADS)

    Klich, Israel

    2015-03-01

    In this talk, I will discuss work on entanglement in fermion systems. I will describe recent results on effective entanglement hamiltonians in conformal quantum field theories, and how the free fermion entanglement Hamiltonian in 1d can be obtained by solving a Riemann-Hilbert problem. I will also show how finite size corrections to the Hamiltonian may be obtained by perturbing around the Riemann-Hilbert solutions, as well as explore subtle difference between the Neveu-Schwartz and Ramond sectors of free fermion fields.

  1. Nonlinear Control Systems

    DTIC Science & Technology

    2009-11-18

    in a trim condition is a typical problem of output regulation near an equilibrium setting, tailless or nearly tailless aircraft , such as UCAV’s...control to produce significant nonlinear excursions. Taking advantage of these nonequilibrium nonlinearities in tailless aircraft also promises to...will also have multiple nonlinear axes and a smaller domain of stability than conventional aircraft , involving nonlinear trajectories which cannot be

  2. Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

    NASA Astrophysics Data System (ADS)

    Chanu, Claudia Maria; Degiovanni, Luca; Rastelli, Giovanni

    2017-03-01

    It is natural to investigate if the quantization of integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. In particular, Stäckel systems are studied in detail. Examples in conformally and non-conformally flat manifolds are given.

  3. Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

    NASA Astrophysics Data System (ADS)

    Restuccia, Alvaro; Sotomayor, Adrián

    2016-01-01

    We obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four different real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose field equations are the ɛ-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limit ɛ → 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limit ɛ → ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ɛ → 0 and ɛ → ∞.

  4. From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity

    NASA Astrophysics Data System (ADS)

    Okuyama, Manaka; Takahashi, Kazutaka

    2016-08-01

    Using shortcuts to adiabaticity, we solve the time-dependent Schrödinger equation that is reduced to a classical nonlinear integrable equation. For a given time-dependent Hamiltonian, the counterdiabatic term is introduced to prevent nonadiabatic transitions. Using the fact that the equation for the dynamical invariant is equivalent to the Lax equation in nonlinear integrable systems, we obtain the counterdiabatic term exactly. The counterdiabatic term is available when the corresponding Lax pair exists and the solvable systems are classified in a unified and systematic way. Multisoliton potentials obtained from the Korteweg-de Vries equation and isotropic X Y spin chains from the Toda equations are studied in detail.

  5. Hamiltonian purification

    SciTech Connect

    Orsucci, Davide; Burgarth, Daniel; Facchi, Paolo; Pascazio, Saverio; Nakazato, Hiromichi; Yuasa, Kazuya; Giovannetti, Vittorio

    2015-12-15

    The problem of Hamiltonian purification introduced by Burgarth et al. [Nat. Commun. 5, 5173 (2014)] is formalized and discussed. Specifically, given a set of non-commuting Hamiltonians (h{sub 1}, …, h{sub m}) operating on a d-dimensional quantum system ℋ{sub d}, the problem consists in identifying a set of commuting Hamiltonians (H{sub 1}, …, H{sub m}) operating on a larger d{sub E}-dimensional system ℋ{sub d{sub E}} which embeds ℋ{sub d} as a proper subspace, such that h{sub j} = PH{sub j}P with P being the projection which allows one to recover ℋ{sub d} from ℋ{sub d{sub E}}. The notions of spanning-set purification and generator purification of an algebra are also introduced and optimal solutions for u(d) are provided.

  6. Solution of Nonlinear Systems

    NASA Technical Reports Server (NTRS)

    Turner, L. R.

    1960-01-01

    The problem of solving systems of nonlinear equations has been relatively neglected in the mathematical literature, especially in the textbooks, in comparison to the corresponding linear problem. Moreover, treatments that have an appearance of generality fail to discuss the nature of the solutions and the possible pitfalls of the methods suggested. Probably it is unrealistic to expect that a unified and comprehensive treatment of the subject will evolve, owing to the great variety of situations possible, especially in the applied field where some requirement of human or mechanical efficiency is always present. Therefore we attempt here simply to pose the problem and to describe and partially appraise the methods of solution currently in favor.

  7. Estimating nonlinear interdependences in dynamical systems using cellular nonlinear networks

    NASA Astrophysics Data System (ADS)

    Krug, Dieter; Osterhage, Hannes; Elger, Christian E.; Lehnertz, Klaus

    2007-10-01

    We propose a method for estimating nonlinear interdependences between time series using cellular nonlinear networks. Our approach is based on the nonlinear dynamics of interacting nonlinear elements. We apply it to time series of coupled nonlinear model systems and to electroencephalographic time series from an epilepsy patient, and we show that an accurate approximation of symmetric and asymmetric realizations of a nonlinear interdependence measure can be achieved, thus allowing one to detect the strength and direction of couplings.

  8. Boosting Bayesian parameter inference of nonlinear stochastic differential equation models by Hamiltonian scale separation

    NASA Astrophysics Data System (ADS)

    Albert, Carlo; Ulzega, Simone; Stoop, Ruedi

    2016-04-01

    Parameter inference is a fundamental problem in data-driven modeling. Given observed data that is believed to be a realization of some parameterized model, the aim is to find parameter values that are able to explain the observed data. In many situations, the dominant sources of uncertainty must be included into the model for making reliable predictions. This naturally leads to stochastic models. Stochastic models render parameter inference much harder, as the aim then is to find a distribution of likely parameter values. In Bayesian statistics, which is a consistent framework for data-driven learning, this so-called posterior distribution can be used to make probabilistic predictions. We propose a novel, exact, and very efficient approach for generating posterior parameter distributions for stochastic differential equation models calibrated to measured time series. The algorithm is inspired by reinterpreting the posterior distribution as a statistical mechanics partition function of an object akin to a polymer, where the measurements are mapped on heavier beads compared to those of the simulated data. To arrive at distribution samples, we employ a Hamiltonian Monte Carlo approach combined with a multiple time-scale integration. A separation of time scales naturally arises if either the number of measurement points or the number of simulation points becomes large. Furthermore, at least for one-dimensional problems, we can decouple the harmonic modes between measurement points and solve the fastest part of their dynamics analytically. Our approach is applicable to a wide range of inference problems and is highly parallelizable.

  9. Boosting Bayesian parameter inference of nonlinear stochastic differential equation models by Hamiltonian scale separation.

    PubMed

    Albert, Carlo; Ulzega, Simone; Stoop, Ruedi

    2016-04-01

    Parameter inference is a fundamental problem in data-driven modeling. Given observed data that is believed to be a realization of some parameterized model, the aim is to find parameter values that are able to explain the observed data. In many situations, the dominant sources of uncertainty must be included into the model for making reliable predictions. This naturally leads to stochastic models. Stochastic models render parameter inference much harder, as the aim then is to find a distribution of likely parameter values. In Bayesian statistics, which is a consistent framework for data-driven learning, this so-called posterior distribution can be used to make probabilistic predictions. We propose a novel, exact, and very efficient approach for generating posterior parameter distributions for stochastic differential equation models calibrated to measured time series. The algorithm is inspired by reinterpreting the posterior distribution as a statistical mechanics partition function of an object akin to a polymer, where the measurements are mapped on heavier beads compared to those of the simulated data. To arrive at distribution samples, we employ a Hamiltonian Monte Carlo approach combined with a multiple time-scale integration. A separation of time scales naturally arises if either the number of measurement points or the number of simulation points becomes large. Furthermore, at least for one-dimensional problems, we can decouple the harmonic modes between measurement points and solve the fastest part of their dynamics analytically. Our approach is applicable to a wide range of inference problems and is highly parallelizable.

  10. The isotropic Hamiltonian formalism

    SciTech Connect

    Vaisman, Izu

    2011-02-10

    A Hamiltonian formalism is a procedure that allows to associate a dynamical system to a function and that includes classical Hamiltonian mechanics as a particular case. The present, expository paper gives a survey of the Hamiltonian formalism defined by an isotropic subbundle of TM+T*M, in particular, by a Dirac structure. We discuss reduction and geometric quantization of the Hamiltonian dynamical systems provided by this formalism.

  11. The wave function and minimum uncertainty function of the bound quadratic Hamiltonian system

    NASA Technical Reports Server (NTRS)

    Yeon, Kyu Hwang; Um, Chung IN; George, T. F.

    1994-01-01

    The bound quadratic Hamiltonian system is analyzed explicitly on the basis of quantum mechanics. We have derived the invariant quantity with an auxiliary equation as the classical equation of motion. With the use of this invariant it can be determined whether or not the system is bound. In bound system we have evaluated the exact eigenfunction and minimum uncertainty function through unitary transformation.

  12. Local energy conservation law for a spatially-discretized Hamiltonian Vlasov-Maxwell system

    NASA Astrophysics Data System (ADS)

    Xiao, Jianyuan; Qin, Hong; Liu, Jian; Zhang, Ruili

    2017-06-01

    Because of the unparalleled long-term conservative property, the structure-preserving geometric algorithm for the Vlasov-Maxwell (VM) equations is currently an active research topic. We show that spatially discretized Hamiltonian systems for the VM equations admit a local energy conservation law in space-time. This is accomplished by proving that a sum-free and only locally non-zero scalar field can always be written as the divergence of a vector field that is only locally non-zero. The result demonstrates that the Hamiltonian discretization of Vlasov-Maxwell system can preserve local conservation laws, in addition to the symplectic structure, both of which are the intrinsic physical properties of infinite dimensional Hamiltonian systems in physics.

  13. Purely non-local Hamiltonian formalism, Kohno connections and ∨-systems

    SciTech Connect

    Arsie, Alessandro; Lorenzoni, Paolo

    2014-11-15

    In this paper, we extend purely non-local Hamiltonian formalism to a class of Riemannian F-manifolds, without assumptions on the semisimplicity of the product ○ or on the flatness of the connection ∇. In the flat case, we show that the recurrence relations for the principal hierarchy can be re-interpreted using a local and purely non-local Hamiltonian operators and in this case they split into two Lenard-Magri chains, one involving the even terms, the other involving the odd terms. Furthermore, we give an elementary proof that the Kohno property and the ∨-system condition are equivalent under suitable assumptions and we show how to associate a purely non-local Hamiltonian structure to any ∨-system, including degenerate ones.

  14. Electronic structure and nonlinear optical properties of the fullerenes C60 and C70: A valence-effective-Hamiltonian study

    NASA Astrophysics Data System (ADS)

    Shuai, Zhigang; Brédas, J. L.

    1992-12-01

    Based on the geometries optimized by the AM1 semiempirical technique (Austin Model 1 of Dewar et al.), we exploit the valence-effective-Hamiltonian (VEH) method to study the electronic structures of C60 and C70. The valence-electronic density of states (DOS) calculated is found to be in excellent agreement with the high-resolution energy-distribution curves obtained from synchrotron-photoemission experiments in terms of both positions and relative intensities of the peaks. The maximum difference in peak position between theory and experiment is 0.4 eV. This shows that the VEH method provides a very reasonable description of these two fullerenes. We then apply the VEH-SOS (sum-over-states) approach to study the nonlinear optical response of C60 and C70. We obtain that the off-resonance third-order susceptibility χ(3) is on the order of 10-12 esu. Our results are fully consistent with the electric-field-induced second-harmonic generation and third-harmonic-generation (THG) measurements by Wang and Cheng and the degenerate-four-wave-mixing measurements by Kafafi et al., but about three to four orders of magnitude lower than the data reported by Blau et al. and by Yang et et al. The static χ(3) values of C60 and C70 are compared to those of polyacetylene. We also investigate the dynamic nonlinear optical response by calculating the THG spectrum. We find that the lowest two-photon and three-photon resonances occur at almost the same frequency for C60, because of the symmetry of the molecule.

  15. Analysis of the nonlinear behavior of shear-Alfvén modes in tokamaks based on Hamiltonian mapping techniques

    SciTech Connect

    Briguglio, S. Vlad, G.; Fogaccia, G.; Di Troia, C.; Fusco, V.; Wang, X.; Zonca, F.

    2014-11-15

    We present a series of numerical simulation experiments set up to illustrate the fundamental physics processes underlying the nonlinear dynamics of Alfvénic modes resonantly excited by energetic particles in tokamak plasmas and of the ensuing energetic particle transports. These phenomena are investigated by following the evolution of a test particle population in the electromagnetic fields computed in self-consistent MHD-particle simulation performed by the HMGC code. Hamiltonian mapping techniques are used to extract and illustrate several features of wave-particle dynamics. The universal structure of resonant particle phase space near an isolated resonance is recovered and analyzed, showing that bounded orbits and untrapped trajectories, divided by the instantaneous separatrix, form phase space zonal structures, whose characteristic non-adiabatic evolution time is the same as the nonlinear time of the underlying fluctuations. Bounded orbits correspond to a net outward resonant particle flux, which produces a flattening and/or gradient inversion of the fast ion density profile around the peak of the linear wave-particle resonance. The connection of this phenomenon to the mode saturation is analyzed with reference to two different cases: a Toroidal Alfvén eigenmode in a low shear magnetic equilibrium and a weakly unstable energetic particle mode for stronger magnetic shear. It is shown that, in the former case, saturation is reached because of radial decoupling (resonant particle redistribution matching the mode radial width) and is characterized by a weak dependence of the mode amplitude on the growth rate. In the latter case, saturation is due to resonance detuning (resonant particle redistribution matching the resonance width) with a stronger dependence of the mode amplitude on the growth rate.

  16. Numerical integration of nearly-Hamiltonian systems. [Van der Pol oscillator and perturbed Keplerian motion

    NASA Technical Reports Server (NTRS)

    Bond, V. R.

    1978-01-01

    The reported investigation is concerned with the solution of systems of differential equations which are derived from a Hamiltonian function in the extended phase space. The problem selected involves a one-dimensional perturbed harmonic oscillator. The van der Pol equation considered has an exact asymptotic value for its amplitude. Comparisons are made between a numerical solution and a known analytical solution. In addition to the van der Pol problem, known solutions regarding the restricted problem of three bodies are used as examples for perturbed Keplerian motion. The extended phase space Hamiltonian discussed by Stiefel and Scheifele (1971) is considered. A description is presented of two canonical formulations of the perturbed harmonic oscillator.

  17. Periodic solutions of the N-vortex Hamiltonian system in planar domains

    NASA Astrophysics Data System (ADS)

    Bartsch, Thomas; Dai, Qianhui

    2016-02-01

    We investigate the existence of collision-free nonconstant periodic solutions of the N-vortex problem in domains Ω ⊂ C. These are solutions z (t) = (z1 (t) , … ,zN (t)) ∈ΩN of the first order Hamiltonian system

  18. Darboux integrability of 2-dimensional Hamiltonian systems with homogenous potentials of degree 3

    SciTech Connect

    Llibre, Jaume; Valls, Claudia

    2014-03-15

    We provide a characterization of all Hamiltonian systems of the form H=(p{sub 1}{sup 2}+p{sub 2}{sup 2})/2+V(q{sub 1},q{sub 2}), where V is a homogenous polynomial of degree 3 which are completely integrable with Darboux first integrals.

  19. Polynomial integrability of Hamiltonian systems with homogeneous potentials of degree -k

    NASA Astrophysics Data System (ADS)

    Oliveira, Regilene; Valls, Claudia

    2016-12-01

    In this paper we shall answer positively two open problems proposed by Llibre-Mahdi-Valls (2011) in [9]. More precisely, we characterize the polynomial integrability of Hamiltonian system with potentials given by the inverse of a homogeneous potential of degree k.

  20. Periodic motion in nonlinear systems

    NASA Technical Reports Server (NTRS)

    Williamson, D.

    1975-01-01

    In this paper it is shown how some basic ideas from system theory and differential geometry can be used to establish some new results on the existence of periodic motion in autonomous feedback systems. The conditions are expressed in terms of the frequency response characteristic of the open-loop system and certain general properties of the nonlinearities.

  1. Oscillations in nonlinear feedback systems.

    NASA Technical Reports Server (NTRS)

    Williamson, D.

    1973-01-01

    It is shown how some basic ideas from system theory and differential geometry can be used to establish new results concerning the existance of oscillations for autonomous feedback systems. The conditions obtained are expressed in terms of the frequency response characteristic of the open-loop system and certain general properties of the nonlinearity.

  2. An integrable system and associated integrable models as well as Hamiltonian structures

    NASA Astrophysics Data System (ADS)

    Tam, Hon-Wah; Zhang, Yufeng

    2012-10-01

    Starting from an existed Lie algebra introduces a new Lie algebra A1 = {e1, e2, e3} so that two isospectral Lax matrices are established. By employing the Tu scheme an integrable equation hierarchy denoted by IEH is obtained from which a few reduced evolution equations are presented. One of them is the mKdV equation. The elliptic variable solutions and three kinds of Darboux transformations for one coupled equation which is from the IEH are worked out, respectively. Finally, we take use of the Lie algebra A1 to generate eight higher-dimensional Lie algebras from which the linear integrable couplings, the nonlinear integrable couplings, and the bi-integrable couplings of the IEH are engendered, whose Hamiltonian structures are also obtained by the variational identity. Then further reduce one coupled integrable equation to get a nonlinear generalized mKdV equation.

  3. Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems

    NASA Astrophysics Data System (ADS)

    Abanin, Dmitry A.; De Roeck, Wojciech; Ho, Wen Wei; Huveneers, François

    2017-01-01

    We establish some general dynamical properties of quantum many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasiconserved extensive quantity H*, which plays the role of an effective static Hamiltonian. The dynamics of the system (e.g., evolution of any local observable) is well approximated by the evolution with the Hamiltonian H* up to time τ*, which is exponentially large in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where H* is ergodic, the driven system prethermalizes to a thermal state described by H* at intermediate times t ≲τ* , eventually heating up to an infinite-temperature state after times t ˜τ* . Our results indicate that rapidly driven many-body systems generically exhibit prethermalization and very slow heating. We briefly discuss implications for experiments which realize topological states by periodic driving.

  4. Asymptotic analysis of a class of three-degree-of-freedom Hamiltonian systems near stable equilibria

    NASA Astrophysics Data System (ADS)

    Wang, L.; Bosley, D. L.; Kevorkian, J.

    A conservative near-integrable Hamiltonian dynamical system is examined, which to leading order consists of three uncoupled harmonic oscillators with constant frequencies in the ratio 1:2:α for certain rational α. Formally, the problem considered can arise by perturbing any three-degree-of-freedom Hamiltonian near a stable equilibrium point, so that the Hamiltonian consists of a power series expansion in a small parameter, where successive terms are homogeneous polynomials of increasing degree in the coordinates and the momenta. The special case of two exact simultaneous resonances, one in the first perturbation term and one in the second, is examined and explicit asymptotic solutions are obtained. The solution procedure involves reducing the original Hamiltonian to two degrees of freedom using one integral of the motion; then transforming to standard form to find two additional adiabatic invariants by near-identity averaging canonical transformations. A specific example is studied numerically to verify the asymptotic validity of the results over long times.

  5. Trace Formula for Linear Hamiltonian Systems with its Applications to Elliptic Lagrangian Solutions

    NASA Astrophysics Data System (ADS)

    Hu, Xijun; Ou, Yuwei; Wang, Penghui

    2015-04-01

    In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm-Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem, which depends on the mass parameter and the eccentricity . Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagrangian solutions.

  6. Modeling Optical Spectra of Large Organic Systems Using Real-Time Propagation of Semiempirical Effective Hamiltonians.

    PubMed

    Ghosh, Soumen; Andersen, Amity; Gagliardi, Laura; Cramer, Christopher J; Govind, Niranjan

    2017-09-12

    We present an implementation of a time-dependent semiempirical method (INDO/S) in NWChem using real-time (RT) propagation to address, in principle, the entire spectrum of valence electronic excitations. Adopting this model, we study the UV/vis spectra of medium-sized systems such as P3B2 and f-coronene, and in addition much larger systems such as ubiquitin in the gas phase and the betanin chromophore in the presence of two explicit solvents (water and methanol). RT-INDO/S provides qualitatively and often quantitatively accurate results when compared with RT- TDDFT or experimental spectra. Even though we only consider the INDO/S Hamiltonian in this work, our implementation provides a framework for performing electron dynamics in large systems using semiempirical Hartree-Fock Hamiltonians in general.

  7. Forward Period Analysis Method of the Periodic Hamiltonian System

    PubMed Central

    Wang, Pengfei

    2016-01-01

    Using the forward period analysis (FPA), we obtain the period of a Morse oscillator and mathematical pendulum system, with the accuracy of 100 significant digits. From these results, the long-term [0, 1060] (time unit) solutions, ranging from the Planck time to the age of the universe, are computed reliably and quickly with a parallel multiple-precision Taylor series (PMT) scheme. The application of FPA to periodic systems can greatly reduce the computation time of long-term reliable simulations. This scheme provides an efficient way to generate reference solutions, against which long-term simulations using other schemes can be tested. PMID:27727295

  8. Forward Period Analysis Method of the Periodic Hamiltonian System.

    PubMed

    Wang, Pengfei

    2016-01-01

    Using the forward period analysis (FPA), we obtain the period of a Morse oscillator and mathematical pendulum system, with the accuracy of 100 significant digits. From these results, the long-term [0, 1060] (time unit) solutions, ranging from the Planck time to the age of the universe, are computed reliably and quickly with a parallel multiple-precision Taylor series (PMT) scheme. The application of FPA to periodic systems can greatly reduce the computation time of long-term reliable simulations. This scheme provides an efficient way to generate reference solutions, against which long-term simulations using other schemes can be tested.

  9. Symmetry of quantum phase space in a degenerate Hamiltonian system

    NASA Astrophysics Data System (ADS)

    Berman, G. P.; Demikhovskii, V. Ya.; Kamenev, D. I.

    2000-09-01

    The structure of the global "quantum phase space" is analyzed for the harmonic oscillator perturbed by a monochromatic wave in the limit when the perturbation amplitude is small. Usually, the phenomenon of quantum resonance was studied in nondegenerate [G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, Chur, 1985)] and degenerate [Demikhovskii, Kamenev, and Luna-Acosta, Phys. Rev. E 52, 3351 (1995)] classically chaotic systems only in the particular regions of the classical phase space, such as the center of the resonance or near the separatrix. The system under consideration is degenerate, and even an infinitely small perturbation generates in the classical phase space an infinite number of the resonant cells which are arranged in the pattern with the axial symmetry of the order 2μ (where μ is the resonance number). We show analytically that the Husimi functions of all Floquet states (the quantum phase space) have the same symmetry as the classical phase space. This correspondence is demonstrated numerically for the Husimi functions of the Floquet states corresponding to the motion near the elliptic stable points (centers of the classical resonance cells). The derived results are valid in the resonance approximation when the perturbation amplitude is small enough, and the stochastic layers in the classical phase space are exponentially thin. The developed approach can be used for studying a global symmetry of more complicated quantum systems with chaotic behavior.

  10. Pseudo PT-symmetry in time periodic non-Hermitian Hamiltonians systems

    NASA Astrophysics Data System (ADS)

    Maamache, Mustapha; Lamri, Sarra; Cherbal, Omar

    2017-03-01

    We investigate the concept of the pseudo-parity-time (pseudo- PT) symmetry in periodic quantum systems. This pseudo parity-time symmetry manifests itself dynamically in the framework of the non-unitary evolution (Floquet) operator U(τ) =e-iLτ, over a period τ, which shows that the stability of the dynamics occurs when the PT-symmetry (or pseudo- PT) of the time-independent non-Hermitian Hamiltonian L is unbroken i.e. its quasienergies En are real. Nevertheless, when the PT-symmetry of the non-Hermitian Hamiltonian L is broken, which corresponds to the complex conjugate quasienergies En, an instable dynamics arises. We investigate in greater detail a harmonic oscillator with imaginary time-dependent periodic driving term linear in x. The Floquet operator for the modulated system is pseudo- PT symmetric if the relative phase ϕ of the applied mode is not 0 or π.

  11. Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson.

    PubMed

    Miller, Michael I; Trouvé, Alain; Younes, Laurent

    2015-01-01

    The Computational Anatomy project is the morphome-scale study of shape and form, which we model as an orbit under diffeomorphic group action. Metric comparison calculates the geodesic length of the diffeomorphic flow connecting one form to another. Geodesic connection provides a positioning system for coordinatizing the forms and positioning their associated functional information. This article reviews progress since the Euler-Lagrange characterization of the geodesics a decade ago. Geodesic positioning is posed as a series of problems in Hamiltonian control, which emphasize the key reduction from the Eulerian momentum with dimension of the flow of the group, to the parametric coordinates appropriate to the dimension of the submanifolds being positioned. The Hamiltonian viewpoint provides important extensions of the core setting to new, object-informed positioning systems. Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series studies of growth and atrophy via shape splines.

  12. Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential

    NASA Astrophysics Data System (ADS)

    Martínez, Y. P.; Vidal, C.

    2016-12-01

    In this paper we study the global dynamics of the Hamiltonian systems x ˙ =Hy (x , y), y ˙ = -Hx (x , y), where the Hamiltonian function H has the particular form H (x , y) =y2 / 2 + P (x) / Q (x), P (x) , Q (x) ∈ R [ x ] are polynomials, in particular H is the sum of the kinetic and a rational potential energies. Firstly, we provide the normal forms by a suitable μ-symplectic change of variables. Then, the global topological classification of the phase portraits of these systems having canonical forms in the Poincaré disk in the cases where degree (P) = 0 , 1 , 2 and degree (Q) = 0 , 1 , 2 are studied as a function of the parameters that define each polynomial. We use a blow-up technique for finite equilibrium points and the Poincaré compactification for the infinite equilibrium points. Finally, we show some applications.

  13. Hamiltonian Description of Singular Lagrangian Systems with Spontaneously Broken Time Translation Symmetry

    NASA Astrophysics Data System (ADS)

    Zhao, Liu; Yu, Pengfei; Xu, Wei

    2013-02-01

    Shapere and Wilczek recently found some singular Lagrangian systems which spontaneously breaks time translation symmetry. The common feature of their models is that the energy functions are multi-valued in terms of the canonical phase space variables and the symmetry breaking ground states are all located at the brunching point singularities. By enlarging the phase space and making use of Dirac's theory on constrained Hamiltonian systems, we present the Hamiltonian description of some of the models discussed by Shapere and Wilczek and found that both the multi-valuedness and the brunching point singularities can be avoided, while the spontaneous breaking of time translation becomes more transparent. It is also shown that the breaking of time translation is always accompanied by the breaking of time reversal.

  14. Transient chaos in multidimensional Hamiltonian system with weak dissipation

    NASA Astrophysics Data System (ADS)

    Felk, E. V.; Savin, A. V.; Kuznetsov, A. P.

    2017-06-01

    The dynamics of two coupled twist maps with weak dissipation is studied. The calculation of Lyapunov exponents is used to analyze the structure of the action plane of the system. The chaotic transient dynamics is revealed for extremely small values of dissipation by calculation of finite-time Lyapunov exponents. The stagger-and-step method is used to obtain the chaotic saddle and it is found that it is similar to the Arnold web.

  15. Projection formalism for constrained dynamical systems: from Newtonian to Hamiltonian mechanics.

    PubMed

    Kneller, Gerald R

    2007-10-28

    The Hamiltonian of a holonomically constrained dynamical many-particle system in Cartesian coordinates has been recently derived for applications in statistical mechanics [G. R. Kneller, J. Chem. Phys. 125, 114107 (2006)]. Using the same projector formalism, we show here the equivalence of the corresponding equations of motion with those obtained from a Newtonian and a Lagrangian description. In the case of Newtonian mechanics, the general case of nonholonomic constraints is considered, too.

  16. Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces

    NASA Astrophysics Data System (ADS)

    Maciejewski, Andrzej J.; Szumiński, Wojciech; Przybylska, Maria

    2017-02-01

    We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations.

  17. On the Existence of Periodic Solutions for a Class of Symmetric Hamiltonian Systems.

    DTIC Science & Technology

    1986-01-01

    AMD-R167 487 ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR R CLRSS OF u1 SYNNETRIC HNMILTONI.. (U) NISCONSIN UNIY-MDISON IITHEMATICS RESERRCH CENTER P...1,,... 1.,.1...-,....-.I- - ; :. . : < : ’..p .. . . MRC Technical Summary Report #2901 ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF...MADISON , MATHEMATICS RESEARCH CENTER -%>."S.- ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF SYMMETRIC HAMILTONIAN SYSTEMS Paul R. Rabinowitz

  18. Hamiltonian mechanics and planar fishlike locomotion

    NASA Astrophysics Data System (ADS)

    Kelly, Scott; Xiong, Hailong; Burgoyne, Will

    2007-11-01

    A free deformable body interacting with a system of point vortices in the plane constitutes a Hamiltonian system. A free Joukowski foil with variable camber shedding point vortices in an ideal fluid according to a periodically applied Kutta condition provides a model for fishlike locomotion which bridges the gap between inviscid analytical models that sacrifice realism for tractability and viscous computational models inaccessible to tools from nonlinear control theory. We frame such a model in the context of Hamiltonian mechanics and describe its relevance both to the study of hydrodynamic interactions within schools of fish and to the realization of model-based control laws for biomimetic autonomous robotic vehicles.

  19. Periodic response of nonlinear systems

    NASA Technical Reports Server (NTRS)

    Nataraj, C.; Nelson, H. D.

    1988-01-01

    A procedure is developed to determine approximate periodic solutions of autonomous and non-autonomous systems. The trignometric collocation method (TCM) is formalized to allow for the analysis of relatively small order systems directly in physical coordinates. The TCM is extended to large order systems by utilizing modal analysis in a component mode synthesis strategy. The procedure was coded and verified by several check cases. Numerical results for two small order mechanical systems and one large order rotor dynamic system are presented. The method allows for the possibility of approximating periodic responses for large order forced and self-excited nonlinear systems.

  20. Control of Nonlinear Systems

    DTIC Science & Technology

    2004-01-01

    characteristics, and applied to presented small-gain theorems guaranteeing the lack of oscillatory or more complicated behavior in a large class of Lotka ... Volterra systems with predator-prey interactions as well as chemostats, which describe the interaction between microbial species which are competing

  1. Charge transfer in strongly correlated systems: An exact diagonalization approach to model Hamiltonians

    SciTech Connect

    Schöppach, Andreas; Gnandt, David; Koslowski, Thorsten

    2014-04-07

    We study charge transfer in bridged di- and triruthenium complexes from a theoretical and computational point of view. Ab initio computations are interpreted from the perspective of a simple empirical Hamiltonian, a chemically specific Mott-Hubbard model of the complexes' π electron systems. This Hamiltonian is coupled to classical harmonic oscillators mimicking a polarizable dielectric environment. The model can be solved without further approximations in a valence bond picture using the method of exact diagonalization and permits the computation of charge transfer reaction rates in the framework of Marcus' theory. In comparison to the exact solution, the Hartree-Fock mean field theory overestimates both the activation barrier and the magnitude of charge-transfer excitations significantly. For triruthenium complexes, we are able to directly access the interruthenium antiferromagnetic coupling strengths.

  2. Charge transfer in strongly correlated systems: an exact diagonalization approach to model Hamiltonians.

    PubMed

    Schöppach, Andreas; Gnandt, David; Koslowski, Thorsten

    2014-04-07

    We study charge transfer in bridged di- and triruthenium complexes from a theoretical and computational point of view. Ab initio computations are interpreted from the perspective of a simple empirical Hamiltonian, a chemically specific Mott-Hubbard model of the complexes' π electron systems. This Hamiltonian is coupled to classical harmonic oscillators mimicking a polarizable dielectric environment. The model can be solved without further approximations in a valence bond picture using the method of exact diagonalization and permits the computation of charge transfer reaction rates in the framework of Marcus' theory. In comparison to the exact solution, the Hartree-Fock mean field theory overestimates both the activation barrier and the magnitude of charge-transfer excitations significantly. For triruthenium complexes, we are able to directly access the interruthenium antiferromagnetic coupling strengths.

  3. Hamiltonian Description of Multi-fluid Streaming

    NASA Astrophysics Data System (ADS)

    Valls, C.; de La Llave, R.; Morrison, P. J.

    2001-10-01

    The general noncanonical Hamiltonian description of interpenetrating fluids coupled by electrostatic, gravitational, or other forces is presented. This formalism is used to describe equilibrium and nonlinear stability using techniques of Hamiltonian dynamics theory. For example, we study the stability of two warm counter-streaming electron beams in a neutralizing ion background. The normal modes are obtained from an energy functional by computing the lowest-order expression for the perturbed energy about an equilibrium, and transforming the corresponding system into action-angle variables. Higher-order terms in the Hamiltonian provide coupling between normal modes and can lead to instability because of the presence of negative energy modes (NEM's). (The signature of the NEM's is determined by the signature of the Hamiltonian, Moser's bracket definition, or the conventional plasma definition in terms of the dielectric function, all of which are shown to be equivalent.) The possible nonlinear behavior is discovered by constructing the Birkhoff normal form. Accounting for resonances, we transform away terms in the Hamiltonian to address the question of long-time stability for such systems.

  4. The Hamiltonian structure of a coupled system derived from a supersymmetric breaking of super Korteweg-de Vries equations

    SciTech Connect

    Restuccia, A.; Sotomayor, A.

    2013-11-15

    A supersymmetric breaking procedure for N= 1 super Korteweg-de Vries (KdV), using a Clifford algebra, is implemented. Dirac's method for the determination of constraints is used to obtain the Hamiltonian structure, via a Lagrangian, for the resulting solitonic system of coupled KdV type system. It is shown that the Hamiltonian obtained by this procedure is bounded from below and in that sense represents a model which is physically admissible.

  5. Statistically preferred basis of an open quantum system: its relation to the eigenbasis of a renormalized self-Hamiltonian.

    PubMed

    He, Lewei; Wang, Wen-Ge

    2014-02-01

    We study the problem of the basis of an open quantum system, under a quantum chaotic environment, which is preferred in view of its stationary reduced density matrix (RDM), that is, the basis in which the stationary RDM is diagonal. It is shown that, under an initial condition composed of sufficiently many energy eigenstates of the total system, such a basis is given by the eigenbasis of a renormalized self-Hamiltonian of the system, in the limit of large Hilbert space of the environment. Here, the renormalized self-Hamiltonian is given by the unperturbed self-Hamiltonian plus a certain average of the interaction Hamiltonian over the environmental degrees of freedom. Numerical simulations performed in two models, both with the kicked rotor as the environment, give results consistent with the above analytical predictions.

  6. The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants

    NASA Astrophysics Data System (ADS)

    Fuchssteiner, Benno; Oevel, Walter

    1982-03-01

    Using a bi-Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth- and seventh-order nonlinear partial differential equations; among them, the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C∞ vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ||t||→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.

  7. Hamiltonian model and dynamic analyses for a hydro-turbine governing system with fractional item and time-lag

    NASA Astrophysics Data System (ADS)

    Xu, Beibei; Chen, Diyi; Zhang, Hao; Wang, Feifei; Zhang, Xinguang; Wu, Yonghong

    2017-06-01

    This paper focus on a Hamiltonian mathematical modeling for a hydro-turbine governing system including fractional item and time-lag. With regards to hydraulic pressure servo system, a universal dynamical model is proposed, taking into account the viscoelastic properties and low-temperature impact toughness of constitutive materials as well as the occurrence of time-lag in the signal transmissions. The Hamiltonian model of the hydro-turbine governing system is presented using the method of orthogonal decomposition. Furthermore, a novel Hamiltonian function that provides more detailed energy information is presented, since the choice of the Hamiltonian function is the key issue by putting the whole dynamical system to the theory framework of the generalized Hamiltonian system. From the numerical experiments based on a real large hydropower station, we prove that the Hamiltonian function can describe the energy variation of the hydro-turbine suitably during operation. Moreover, the effect of the fractional α and the time-lag τ on the dynamic variables of the hydro-turbine governing system are explored and their change laws identified, respectively. The physical meaning between fractional calculus and time-lag are also discussed in nature. All of the above theories and numerical results are expected to provide a robust background for the safe operation and control of large hydropower stations.

  8. Nonlinear systems approach to control system design

    NASA Technical Reports Server (NTRS)

    Meyer, G.

    1984-01-01

    Consider some of the control system design methods for plants with nonlinear dynamics. If the nonlinearity is weak relative to the size of the operating region, then the linear methods apply directly. Fixed-gain design may be feasible even for significant nonlinearities. It may be possible to find a single gain which provides adequate control of the linear models at several perturbation points. If the nonlinearity is restricted to a sector, that fact may be used to obtain a fixed-gain controller. Otherwise, a gain may have to be associated with each perturbation point Pi. A gain schedule K(p(v)) is obtained by connecting the perturbation points by a function, say p(v), of the scheduling parameter v (i.e., speed). When the scheduling parameter must be multidimensional, this approach is difficult; the objective is to develop an easier procedure.

  9. Chaos, ergodicity, and the thermodynamics of lower-dimensional time-independent Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Kandrup, Henry E.; Sideris, Ioannis V.; Bohn, Courtlandt L.

    2002-01-01

    This paper uses the assumptions of ergodicity and a microcanonical distribution to compute estimates of the largest Lyapunov exponents in lower-dimensional Hamiltonian systems. That the resulting estimates are in reasonable agreement with the actual values computed numerically corroborates the intuition that chaos in such systems can be understood as arising generically from a parametric instability and that this instability may be modeled by a stochastic-oscillator equation [cf. Casetti, Clementi, and Pettini, Phys. Rev. E 54, 5969 (1996)], linearized perturbations of a chaotic orbit satisfying a harmonic-oscillator equation with a randomly varying frequency.

  10. The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

    SciTech Connect

    J. Squire, H. Qin and W.M. Tang

    2012-09-25

    We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. __________________________________________________

  11. Weak Ergodicity Breaking and Aging of Chaotic Transport in Hamiltonian Systems

    NASA Astrophysics Data System (ADS)

    Albers, Tony; Radons, Günter

    2014-10-01

    Momentum diffusion is a widespread phenomenon in generic Hamiltonian systems. We show for the prototypical standard map that this implies weak ergodicity breaking for the superdiffusive transport in coordinate direction with an averaging-dependent quadratic and cubic increase of the mean-squared displacement (MSD), respectively. This is explained via integrated Brownian motion, for which we derive aging time dependent expressions for the ensemble-averaged MSD, the distribution of time-averaged MSDs, and the ergodicity breaking parameter. Generalizations to other systems showing momentum diffusion are pointed out.

  12. Noise in Nonlinear Dynamical Systems

    NASA Astrophysics Data System (ADS)

    Moss, Frank; McClintock, P. V. E.

    2009-08-01

    List of contributors; Preface; Introduction to volume three; 1. The effects of coloured quadratic noise on a turbulent transition in liquid He II J. T. Tough; 2. Electrohydrodynamic instability of nematic liquid crystals: growth process and influence of noise S. Kai; 3. Suppression of electrohydrodynamic instabilities by external noise Helmut R. Brand; 4. Coloured noise in dye laser fluctuations R. Roy, A. W. Yu and S. Zhu; 5. Noisy dynamics in optically bistable systems E. Arimondo, D. Hennequin and P. Glorieux; 6. Use of an electronic model as a guideline in experiments on transient optical bistability W. Lange; 7. Computer experiments in nonlinear stochastic physics Riccardo Mannella; 8. Analogue simulations of stochastic processes by means of minimum component electronic devices Leone Fronzoni; 9. Analogue techniques for the study of problems in stochastic nonlinear dynamics P. V. E. McClintock and Frank Moss; Index.

  13. Research on Nonlinear Dynamical Systems.

    DTIC Science & Technology

    1983-01-10

    Professor J. P. LaSalle Grant DAAG29-79 C 0161 September 1, 1979 - September 24, 1982 Principal Investigators: H. T. Banks C. M. Dafermos J. K. Hale E...F. Infante J. P. LaSalle . J. Mallet-Paret Lefschetz Center for Dynamical Systems Division of Applied Mathematics D T I Brown University L emtc...publications LaSALLE , J.P. [94] Stability of nonautonomous systems, Journal of Nonlinear Analysis: Theory, Methods, and Applications, Vol.1, No.1

  14. Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula with applications to graphene-type systems

    NASA Astrophysics Data System (ADS)

    Vogl, M.; Pankratov, O.; Shallcross, S.

    2017-07-01

    We present a tractable and physically transparent semiclassical theory of matrix-valued Hamiltonians, i.e., those that describe quantum systems with internal degrees of freedoms, based on a generalization of the Gutzwiller trace formula for a n ×n dimensional Hamiltonian H (p ̂,q ̂) . The classical dynamics is governed by n Hamilton-Jacobi (HJ) equations that act in a phase space endowed with a classical Berry curvature encoding anholonomy in the parallel transport of the eigenvectors of H (p ,q ) ; these vectors describe the internal structure of the semiclassical particles. At the O (ℏ1) level and for nondegenerate HJ systems, this curvature results in an additional semiclassical phase composed of (i) a Berry phase and (ii) a dynamical phase resulting from the classical particles "moving through the Berry curvature". We show that the dynamical part of this semiclassical phase will, generally, be zero only for the case in which the Berry phase is topological (i.e., depends only on the winding number). We illustrate the method by calculating the Landau spectrum for monolayer graphene, the four-band model of AB bilayer graphene, and for a more complicated matrix Hamiltonian describing the silicene band structure. Finally, we apply our method to an inhomogeneous system consisting of a strain engineered one-dimensional moiré in bilayer graphene, finding localized states near the Dirac point that arise from electron trapping in a semiclassical moiré potential. The semiclassical density of states of these localized states we show to be in perfect agreement with an exact quantum mechanical calculation of the density of states.

  15. Persistence of regular motions for nearly integrable Hamiltonian systems in the thermodynamic limit

    NASA Astrophysics Data System (ADS)

    Carati, Andrea; Galgani, Luigi; Maiocchi, Alberto; Gangemi, Fabrizio; Gangemi, Roberto

    2016-11-01

    A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old Fermi-Pasta-Ulam problem, which gave origin to such discussions, and with the optical spectral lines, the existence of which was recently proven to be possible in classical models, just in virtue of such a persistence.

  16. On the structure of the Hamiltonian systems. The Fast Lyapunov Indicator: a new very sensitive tool

    NASA Astrophysics Data System (ADS)

    Froeschlè, C.; Lega, E.

    2000-10-01

    It is already known (Froeschlè, Lega and Gonczi 1997) that the Fast Lyapunov Indicator, i.e. the computation on a relatively short time of the largest Lyapunov indicator, allows one to discriminate between ordered and weak chaotic motion. We have found that, under certain conditions, the FLI also discriminates between resonant and non resonant orbits, not only for Hamiltonian systems with two degrees of freedom, but also for higher dimensional ones. This method not only allows one to display the resonant Arnold web, but also to detect the transition between Nekhoroshev's stable regime to Chirikov's diffusive one.

  17. Classification of hyperbolic singularities of rank zero of integrable Hamiltonian systems

    SciTech Connect

    Oshemkov, Andrey A

    2010-10-06

    A complete invariant is constructed that is a solution of the problem of semilocal classification of saddle singularities of integrable Hamiltonian systems. Namely, a certain combinatorial object (an f{sub n}-graph) is associated with every nondegenerate saddle singularity of rank zero; as a result, the problem of semilocal classification of saddle singularities of rank zero is reduced to the problem of enumeration of the f{sub n}-graphs. This enables us to describe a simple algorithm for obtaining the lists of saddle singularities of rank zero for a given number of degrees of freedom and a given complexity. Bibliography: 24 titles.

  18. One-Dimensional Self-Organization and Nonequilibrium Phase Transition in a Hamiltonian System

    NASA Astrophysics Data System (ADS)

    Wang, Jiao; Casati, Giulio

    2017-01-01

    Self-organization and nonequilibrium phase transitions are well known to occur in two- and three-dimensional dissipative systems. Here, instead, we provide numerical evidence that these phenomena also occur in a one-dimensional Hamiltonian system. To this end, we calculate the heat conductivity by coupling the two ends of our system to two heat baths at different temperatures. It is found that when the temperature difference is smaller than a critical value, the heat conductivity increases with the system size in power law with an exponent considerably smaller than 1. However, as the temperature difference exceeds the critical value, the system's behavior undergoes a transition and the heat conductivity tends to diverge linearly with the system size. Correspondingly, an ordered structure emerges. These findings suggest a new direction for exploring the transport problems in one dimension.

  19. Flamedoctor™: Nonlinear Burner Diagnostic System

    NASA Astrophysics Data System (ADS)

    Bailey, Ralph; Daw, Stuart; Finney, Charles; Flynn, Tom; Fuller, Tim

    2003-08-01

    Utility power plants are employing advanced control systems to improve performance over the load range. The performance of the boiler combustion system is critical to the overall performance. Flame Doctor™, which has been developed by McDermott Technology, Inc. and Oak Ridge National Laboratory under sponsorship of Electric Power Research Institute, performs diagnostics on an individual burner basis. The system consists of analogue-to-digital signal conversion and conditioning hardware, analysis software, and a graphical user interface. Time varying voltage signals from all of the burner flame scanners on a boiler are analyzed simultaneously. Nonlinear techniques such as symbolization and time asymmetry along with linear techniques such as power spectral analysis are used. The nonlinear techniques discriminate stability features in the combustion dynamics not possible with the linear techniques alone. The assessments for a variety of flame conditions are collected in a reference library. Libraries have been created for a number of flame scanners types. The Flame Doctor™ burner diagnostic system is described. Results from the first utility installation at Ameren UE Meramec power plant are shown. A live hook-up to the power plant is demonstrated. Flame Doctor™ is being offered commercially under alpha and beta demonstrations through the Electric Power Research Institute and Babcock & Wilcox.

  20. Dynamics of SU(1, 1) coherent states for the time-dependent quadratic Hamiltonian system

    NASA Astrophysics Data System (ADS)

    Choi, Jeong Ryeol

    2009-09-01

    The dynamics of SU(1, 1) coherent states introduced by Perelomov are investigated for the time-dependent quadratic Hamiltonian system. SU(1, 1) generators we employed are closely related to the invariant operator theory while those of the previous work of Gerry et al. [C.C. Gerry, P.K. Ma, E.R. Vrscay, Phys. Rev. A 39 (1989) 668] are associated to the simple harmonic oscillator. This is the main difference between the two approaches. The merit of the method used in this paper is that it admits wide sphere of analytical description for quantum features of time-dependent quadratic Hamiltonian system. Our development is applied to the Caldirola-Kanai oscillator and compared the corresponding results with those of the Gerry et al. after correcting some miscalculations of theirs. We showed that the results of our theory are in good agreement with the results of the corrected work of Gerry et al. even if the form of the SU(1, 1) generators we employed are somewhat different from those of their work. The nontrivial zero-point energy plays a dominant role in the very low energy limit (ξ→0) for the Caldirola-Kanai oscillator, leading the system to exhibit pure quantum effects as expected. On the other hand, it turn out for sufficiently high energy limit (ξ→1) that the characteristic feature of dissipating quantum energy become very much the same as that of the classical energy.

  1. A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

    NASA Astrophysics Data System (ADS)

    Villanueva, Jordi

    2017-04-01

    We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov's non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of O(ɛ ^{1/2}), where ɛ is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.

  2. Linear-optical simulation of the cooling of a cluster-state Hamiltonian system.

    PubMed

    Aguilar, G H; Kolb, T; Cavalcanti, D; Aolita, L; Chaves, R; Walborn, S P; Souto Ribeiro, P H

    2014-04-25

    A measurement-based quantum computer could consist of a local-gapped Hamiltonian system, whose thermal states-at sufficiently low temperature-are universal resources for the computation. Initialization of the computer would correspond to cooling the system. We perform an experimental quantum simulation of such a cooling process with entangled photons. We prepare three-qubit thermal cluster states exploiting the equivalence between local dephasing and thermalization for these states. This allows us to tune the system's temperature by changing the dephasing strength. We monitor the entanglement as the system cools down and observe the transitions from separability to bound entanglement, and then to free entanglement. We also analyze the performance of the system for measurement-based single-qubit state preparation. These studies constitute a basic characterization of experimental cluster-state computation under imperfect conditions.

  3. Dimensional Reduction for Filters of Nonlinear Systems with Time-Scale Separation

    DTIC Science & Technology

    2013-03-01

    Rapp, Edwin Kreuzer and N. Sri Namachchivaya, “Reduced Nor- mal Forms for Nonlinear Control of Underactuated Hoisting Systems ,” Archive of Applied Mechanics , Vol.82, 2012, pp. 297 - 315. 7 ... Mechanics , Vol. 78(6), 2011, pp. 61001-1 - 61001-10. 8. Lee DeVille, N. Sri Namachchivaya and Zoi Rapti, “Noisy Two Dimensional Non-Hamiltonian System ...AFRL-OSR-VA-TR-2013-0009 Dimensional Reduction for Filters of Nonlinear Systems with Time- Scale Separation Namachchivaya, N

  4. Reverse engineering of a nonlossy adiabatic Hamiltonian for non-Hermitian systems

    NASA Astrophysics Data System (ADS)

    Wu, Qi-Cheng; Chen, Ye-Hong; Huang, Bi-Hua; Xia, Yan; Song, Jie

    2016-11-01

    We generalize the quantum adiabatic theorem to the non-Hermitian system and build a strict adiabaticity condition to make the adiabatic evolution nonlossy when taking into account the effect of the adiabatic phase. According to the strict adiabaticity condition, the nonadiabatic couplings and the effect of the imaginary part of adiabatic phase should be eliminated as much as possible. Also, the non-Hermitian Hamiltonian reverse-engineering method is proposed for adiabatically driving an artificial quantum state. A concrete two-level system is adopted to show the usefulness of the reverse-engineering method. We obtain the desired target state by adjusting extra rotating magnetic fields at a predefined time. Furthermore, the numerical simulation shows that certain noise and dissipation in the systems are no longer undesirable but play a positive role in the scheme. Therefore, the scheme is quite useful for quantum information processing in some dissipative systems.

  5. Investigation of a Nonlinear Control System

    NASA Technical Reports Server (NTRS)

    Flugge-Lotz, I; Taylor, C F; Lindberg, H E

    1958-01-01

    A discontinuous variation of coefficients of the differential equation describing the linear control system before nonlinear elements are added is studied in detail. The nonlinear feedback is applied to a second-order system. Simulation techniques are used to study performance of the nonlinear control system and to compare it with the linear system for a wide variety of inputs. A detailed quantitative study of the influence of relay delays and of a transport delay is presented.

  6. Chaotic transport in Hamiltonian systems perturbed by a weak turbulent wave field

    SciTech Connect

    Abdullaev, S. S.

    2011-08-15

    Chaotic transport in a Hamiltonian system perturbed by a weak turbulent wave field is studied. It is assumed that a turbulent wave field has a wide spectrum containing up to thousands of modes whose phases are fluctuating in time with a finite correlation time. To integrate the Hamiltonian equations a fast symplectic mapping is derived. It has a large time-step equal to one full turn in angle variable. It is found that the chaotic transport across tori caused by the interactions of small-scale resonances have a fractal-like structure with the reduced or zero values of diffusion coefficients near low-order rational tori thereby forming transport barriers there. The density of rational tori is numerically calculated and its properties are investigated. It is shown that the transport barriers are formed in the gaps of the density of rational tori near the low-order rational tori. The dependencies of the depth and width of transport barriers on the wave field spectrum and the correlation time of fluctuating turbulent field (or the Kubo number) are studied. These numerical findings may have importance in understanding the mechanisms of transport barrier formation in fusion plasmas.

  7. Construction of Darboux coordinates and Poincaré-Birkhoff normal forms in noncanonical Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Junginger, Andrej; Main, Jörg; Wunner, Günter

    2017-06-01

    We demonstrate a general method to construct Darboux coordinates via normal form expansions in noncanonical Hamiltonian system obtained from e.g. a variational approach to quantum systems. The procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its Poincaré-Birkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamilton operator. The method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. Moreover, it directly allows to calculate classical and quantum reaction rates by applying transition state theory.

  8. SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

    NASA Astrophysics Data System (ADS)

    Choi, J. R.; Nahm, I. H.

    2007-01-01

    Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola-Kanai oscillator. The probability density of these coherent states for the Caldirola-Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed.

  9. Periodic orbits and nonintegrability of generalized classical Yang-Mills Hamiltonian systems

    SciTech Connect

    Jimenez-Lara, Lidia; Llibre, Jaume

    2011-03-15

    The averaging theory of first order is applied to study a generalized Yang-Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the nonintegrable classical Yang-Mills Hamiltonian systems, in the sense of Liouville-Arnold, which have the isolated periodic orbits found with averaging theory, cannot exist in any second first integral of class C{sup 1}. This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion.

  10. Reverse engineering of a Hamiltonian for a three-level system via the Rodrigues’ rotation formula

    NASA Astrophysics Data System (ADS)

    Kang, Yi-Hao; Huang, Bi-Hua; Lu, Pei-Min; Xia, Yan

    2017-02-01

    We propose a scheme to reversely construct a three-level Hamiltonian via the Rodrigues’ rotation formula and an auxiliary unitary transformation. The main goal of the scheme is designing feasible pulses to drive a three-level system to evolve rapidly from an arbitrary initial state to a desired final state. Numerical simulations demonstrate that the scheme is not only fast but also robust against the decoherence caused by fluctuations of control parameters and some dissipation factors. Besides, we apply the idea to implement a Hadamard gate in a three-level system, and the results show the present scheme is much faster compared with stimulated Raman adiabatic passage (STIRAP). Therefore, the scheme may be useful to find out an effective shortcut to the adiabatic passage in a three-level system.

  11. Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians

    SciTech Connect

    Buljubasich, Lisandro; Dente, Axel D.; Levstein, Patricia R.; Chattah, Ana K.; Pastawski, Horacio M.; Sánchez, Claudia M.

    2015-10-28

    We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.

  12. Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians

    NASA Astrophysics Data System (ADS)

    Buljubasich, Lisandro; Sánchez, Claudia M.; Dente, Axel D.; Levstein, Patricia R.; Chattah, Ana K.; Pastawski, Horacio M.

    2015-10-01

    We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.

  13. Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians.

    PubMed

    Buljubasich, Lisandro; Sánchez, Claudia M; Dente, Axel D; Levstein, Patricia R; Chattah, Ana K; Pastawski, Horacio M

    2015-10-28

    We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.

  14. Landau-Zener problem in a three-level neutrino system with nonlinear time dependence

    SciTech Connect

    Keraenen, P.; Maalampi, J.; Myyrylaeinen, M.; Riittinen, J.

    2007-02-01

    We consider the level-crossing problem in a three-level system with nonlinearly time-varying Hamiltonian (time-dependence t{sup -3}). We study the validity of the so-called independent crossing approximation in the Landau-Zener model by making comparison with results obtained numerically in the density matrix approach. We also demonstrate the failure of the so-called 'nearest zero' approximation of the Landau-Zener level-crossing probability integral.

  15. Path Integrals and Hamiltonians

    NASA Astrophysics Data System (ADS)

    Baaquie, Belal E.

    2014-03-01

    1. Synopsis; Part I. Fundamental Principles: 2. The mathematical structure of quantum mechanics; 3. Operators; 4. The Feynman path integral; 5. Hamiltonian mechanics; 6. Path integral quantization; Part II. Stochastic Processes: 7. Stochastic systems; Part III. Discrete Degrees of Freedom: 8. Ising model; 9. Ising model: magnetic field; 10. Fermions; Part IV. Quadratic Path Integrals: 11. Simple harmonic oscillators; 12. Gaussian path integrals; Part V. Action with Acceleration: 13. Acceleration Lagrangian; 14. Pseudo-Hermitian Euclidean Hamiltonian; 15. Non-Hermitian Hamiltonian: Jordan blocks; 16. The quartic potential: instantons; 17. Compact degrees of freedom; Index.

  16. Branched Hamiltonians and supersymmetry

    DOE PAGES

    Curtright, Thomas L.; Zachos, Cosmas K.

    2014-03-21

    Some examples of branched Hamiltonians are explored both classically and in the context of quantum mechanics, as recently advocated by Shapere and Wilczek. These are in fact cases of switchback potentials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian chaotic dynamical systems in a classical setting, and as encountered in analogous renormalization group flows for quantum theories which exhibit RG cycles. In conclusion, a basic two-worlds model, with a pair of Hamiltonian branches related by supersymmetry, is considered in detail.

  17. The Hamiltonian structure and Euler-Poincare formulation of the Vlasov-Maxwell and gyrokinetic systems

    SciTech Connect

    Squire, J.; Tang, W. M.; Qin, H.; Chandre, C.

    2013-02-15

    We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in H. Cendra et al., [J. Math. Phys. 39, 3138 (1998)]. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models, and Casimir type stability methods.

  18. Hamiltonian Structure of the Schrödinger Classical Dynamical System

    NASA Astrophysics Data System (ADS)

    Tessarotto, Massimo; Mond, Michael; Batic, Davide

    2016-09-01

    The connection between quantum mechanics and classical statistical mechanics has motivated in the past the representation of the Schrödinger quantum-wave equation in terms of "projections" onto the quantum configuration space of suitable phase-space asymptotic kinetic models. This feature has suggested the search of a possible exact super-dimensional classical dynamical system (CDS), denoted as Schrödinger CDS, which uniquely determines the time-evolution of the underlying quantum state describing a set of N like and mutually interacting quantum particles. In this paper the realization of the same CDS in terms of a coupled set of Hamiltonian systems is established. These are respectively associated with a quantum-hydrodynamic CDS advancing in time the quantum fluid velocity and a further one the RD-CDS, describing the relative dynamics with respect to the quantum fluid.

  19. A canonical form for nonlinear systems

    NASA Technical Reports Server (NTRS)

    Su, R.; Hunt, L. R.

    1986-01-01

    The concepts of transformation and canonical form have been used in analyzing linear systems. These ideas are extended to nonlinear systems. A coordinate system and a corresponding canonical form are developed for general nonlinear control systems. Their usefulness is demonstrated by showing that every feedback linearizable system becomes a system with only feedback paths in the canonical form. For control design involving a nonlinear system, one approach is to put the system in its canonical form and approximate by that part having only feedback paths.

  20. Numerical verification of the steepness of three and four degrees of freedom Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Schirinzi, Gabriella; Guzzo, Massimiliano

    2015-01-01

    We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev's theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4-L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degrees-of-freedom Hamiltonian system obtained from the Fermi-Pasta-Ulam problem by setting the number of particles equal to four.

  1. Strange attractor for the renormalization flow for invariant tori of Hamiltonian systems with two generic frequencies

    NASA Astrophysics Data System (ADS)

    Chandre, C.; Jauslin, H. R.

    2000-02-01

    We analyze the stability of invariant tori for Hamiltonian systems with two degrees of freedom by constructing a transformation that combines Kolmogorov-Arnold-Moser theory and renormalization-group techniques. This transformation is based on the continued fraction expansion of the frequency of the torus. We apply this transformation numerically for arbitrary frequencies that contain bounded entries in the continued fraction expansion. We give a global picture of renormalization flow for the stability of invariant tori, and we show that the properties of critical (and near critical) tori can be obtained by analyzing renormalization dynamics around a single hyperbolic strange attractor. We compute the fractal diagram, i.e., the critical coupling as a function of the frequencies, associated with a given one-parameter family.

  2. Maximized Gust Loads of a Closed-Loop, Nonlinear Aeroelastic System Using Nonlinear Systems Theory

    NASA Technical Reports Server (NTRS)

    Silva, Walter A.

    1999-01-01

    The problem of computing the maximized gust load for a nonlinear, closed-loop aeroelastic aircraft is discusses. The Volterra theory of nonlinear systems is applied in order to define a linearized system that provides a bounds on the response of the nonlinear system of interest. The method is applied to a simplified model of an Airbus A310.

  3. Hamiltonian Description of Convective-cell Generation

    SciTech Connect

    J.A. Krommes and R.A. Kolesnikov

    2004-03-11

    The nonlinear statistical growth rate eq for convective cells driven by drift-wave (DW) interactions is studied with the aid of a covariant Hamiltonian formalism for the gyrofluid nonlinearities. A statistical energy theorem is proven that relates eq to a second functional tensor derivative of the DW energy. This generalizes to a wide class of systems of coupled partial differential equations a previous result for scalar dynamics. Applications to (i) electrostatic ion-temperature-gradient-driven modes at small ion temperature, and (ii) weakly electromagnetic collisional DW's are noted.

  4. Hamiltonian description of the ideal fluid

    SciTech Connect

    Morrison, P.J.

    1994-01-01

    Fluid mechanics is examined from a Hamiltonian perspective. The Hamiltonian point of view provides a unifying framework; by understanding the Hamiltonian perspective, one knows in advance (within bounds) what answers to expect and what kinds of procedures can be performed. The material is organized into five lectures, on the following topics: rudiments of few-degree-of-freedom Hamiltonian systems illustrated by passive advection in two-dimensional fluids; functional differentiation, two action principles of mechanics, and the action principle and canonical Hamiltonian description of the ideal fluid; noncanonical Hamiltonian dynamics with examples; tutorial on Lie groups and algebras, reduction-realization, and Clebsch variables; and stability and Hamiltonian systems.

  5. Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters

    SciTech Connect

    Kandrup, H.E. ); Morrison, P.J. . Inst. for Fusion Studies)

    1992-11-01

    The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems like clusters of stars that are so dense that gravity must be described by the Einstein equation, is presented. In particular, it is demonstrated explicitly in the context of a 3 + 1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be viewed as a Hamiltonian system, where the dynamics is generated by a noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function f (a noncanonical variable) being the conserved ADM mass-energy H[sub ADM]. An explicit expression is derived for the energy [delta]([sup 2])H[sub ADM] associated with an arbitrary phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if [delta]([sup 2])H[sub ADM] is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems.

  6. Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters

    SciTech Connect

    Kandrup, H.E.; Morrison, P.J.

    1992-11-01

    The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems like clusters of stars that are so dense that gravity must be described by the Einstein equation, is presented. In particular, it is demonstrated explicitly in the context of a 3 + 1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be viewed as a Hamiltonian system, where the dynamics is generated by a noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function f (a noncanonical variable) being the conserved ADM mass-energy H{sub ADM}. An explicit expression is derived for the energy {delta}({sup 2})H{sub ADM} associated with an arbitrary phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if {delta}({sup 2})H{sub ADM} is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems.

  7. A Few Discrete Lattice Systems and Their Hamiltonian Structures, Conservation Laws

    NASA Astrophysics Data System (ADS)

    Guo, Xiu-Rong; Zhang, Yu-Feng; Zhang, Xiang-Zhi; Yue, Rong

    2017-04-01

    With the help of three shift operators and r-matrix theory, a few discrete lattice systems are obtained which can be reduced to the well-known Toda lattice equation with a constraint whose Hamiltonian structures are generated by Poisson tensors of some induced Lie-Poisson bracket. The recursion operators of these lattice systems are constructed starting from Lax representations. Finally, reducing the given shift operators to get a simpler one and its expanding shift operators, we produce a lattice system with three vector fields whose recursion operator is given. Furthermore, we reduce the lattice system with three vector fields to get a lattice system whose Lax pair and conservation laws are obtained, respectively. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province Hosted by China University of Mining and Technology (2014), the the Key Discipline Construction by China University of Mining and Technology under Grant No. XZD201602, the Shandong Provincial Natural Science Foundation, China under Grant Nos. ZR2016AM31, ZR2016AQ19, ZR2015EM042, the Development of Science and Technology Plan Projects of TaiAn City under Grant No. 2015NS1048, National Social Science Foundation of China under Grant No. 13BJY026, and A Project of Shandong Province Higher Educational Science and Technology Program under Grant No. J14LI58

  8. Nonlinear Dynamics of Parametrically Excited Gyroscopic Systems

    SciTech Connect

    Namachchivaya. N.S.

    2001-06-01

    The primary objective of this project is to determine how some of the powerful geometric methods of dynamical systems can be applied to study nonlinear gyroscopic systems. We proposed to develop techniques to predict local and global behavior and instability mechanisms and to analyze the interactions between noise, stability, and nonlinearities inherent in gyroscopic systems. In order to obtain these results we use the method of normal forms, global bifurcation techniques, and various other dynamical systems tools.

  9. Hamiltonian structure of classical N-body systems of finite-size particles subject to EM interactions

    NASA Astrophysics Data System (ADS)

    Cremaschini, C.; Tessarotto, M.

    2012-01-01

    An open issue in classical relativistic mechanics is the consistent treatment of the dynamics of classical N-body systems of mutually interacting particles. This refers, in particular, to charged particles subject to EM interactions, including both binary interactions and self-interactions ( EM-interacting N- body systems). The correct solution to the question represents an overriding prerequisite for the consistency between classical and quantum mechanics. In this paper it is shown that such a description can be consistently obtained in the context of classical electrodynamics, for the case of a N-body system of classical finite-size charged particles. A variational formulation of the problem is presented, based on the N -body hybrid synchronous Hamilton variational principle. Covariant Lagrangian and Hamiltonian equations of motion for the dynamics of the interacting N-body system are derived, which are proved to be delay-type ODEs. Then, a representation in both standard Lagrangian and Hamiltonian forms is proved to hold, the latter expressed by means of classical Poisson Brackets. The theory developed retains both the covariance with respect to the Lorentz group and the exact Hamiltonian structure of the problem, which is shown to be intrinsically non-local. Different applications of the theory are investigated. The first one concerns the development of a suitable Hamiltonian approximation of the exact equations that retains finite delay-time effects characteristic of the binary interactions and self-EM-interactions. Second, basic consequences concerning the validity of Dirac generator formalism are pointed out, with particular reference to the instant-form representation of Poincaré generators. Finally, a discussion is presented both on the validity and possible extension of the Dirac generator formalism as well as the failure of the so-called Currie "no-interaction" theorem for the non-local Hamiltonian system considered here.

  10. Poincaré-Treshchev Mechanism in Multi-scale, Nearly Integrable Hamiltonian Systems

    NASA Astrophysics Data System (ADS)

    Xu, Lu; Li, Yong; Yi, Yingfei

    2017-08-01

    This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53-83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419-1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré-Treshchev mechanism.

  11. On an application of the intermediate Hamiltonian method for molecular systems

    SciTech Connect

    Seto, R.; Stankevich, I.V.

    1999-04-01

    An application of the intermediate Hamiltonian method is reported in estimation of the lower bounds to the potential energy curve of the hydrogen molecule ion. An improvement of the method and its limitation are also discussed.

  12. Hamiltonian dynamics of thermostated systems: two-temperature heat-conducting phi4 chains.

    PubMed

    Hoover, Wm G; Hoover, Carol G

    2007-04-28

    We consider and compare four Hamiltonian formulations of thermostated mechanics, three of them kinetic, and the other one configurational. Though all four approaches "work" at equilibrium, their application to many-body nonequilibrium simulations can fail to provide a proper flow of heat. All the Hamiltonian formulations considered here are applied to the same prototypical two-temperature "phi4" model of a heat-conducting chain. This model incorporates nearest-neighbor Hooke's-Law interactions plus a quartic tethering potential. Physically correct results, obtained with the isokinetic Gaussian and Nose-Hoover thermostats, are compared with two other Hamiltonian results. The latter results, based on constrained Hamiltonian thermostats, fail to model correctly the flow of heat.

  13. Stochastic surrogate Hamiltonian

    NASA Astrophysics Data System (ADS)

    Katz, Gil; Gelman, David; Ratner, Mark A.; Kosloff, Ronnie

    2008-07-01

    The surrogate Hamiltonian is a general scheme to simulate the many body quantum dynamics composed of a primary system coupled to a bath. The method has been based on a representative bath Hamiltonian composed of two-level systems that is able to mimic the true system-bath dynamics up to a prespecified time. The original surrogate Hamiltonian method is limited to short time dynamics since the size of the Hilbert space required to obtain convergence grows exponentially with time. By randomly swapping bath modes with a secondary thermal reservoir, the method can simulate quantum dynamics of the primary system from short times to thermal equilibrium. By averaging a small number of realizations converged values of the system observables are obtained avoiding the exponential increase in resources. The method is demonstrated for the equilibration of a molecular oscillator with a thermal bath.

  14. Stochastic surrogate Hamiltonian

    SciTech Connect

    Katz, Gil; Kosloff, Ronnie; Gelman, David; Ratner, Mark A.

    2008-07-21

    The surrogate Hamiltonian is a general scheme to simulate the many body quantum dynamics composed of a primary system coupled to a bath. The method has been based on a representative bath Hamiltonian composed of two-level systems that is able to mimic the true system-bath dynamics up to a prespecified time. The original surrogate Hamiltonian method is limited to short time dynamics since the size of the Hilbert space required to obtain convergence grows exponentially with time. By randomly swapping bath modes with a secondary thermal reservoir, the method can simulate quantum dynamics of the primary system from short times to thermal equilibrium. By averaging a small number of realizations converged values of the system observables are obtained avoiding the exponential increase in resources. The method is demonstrated for the equilibration of a molecular oscillator with a thermal bath.

  15. Nonlinear waves in PT -symmetric systems

    NASA Astrophysics Data System (ADS)

    Konotop, Vladimir V.; Yang, Jianke; Zezyulin, Dmitry A.

    2016-07-01

    Recent progress on nonlinear properties of parity-time (PT )-symmetric systems is comprehensively reviewed in this article. PT symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying PT symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a PT -symmetric system. The natural inclusion of nonlinearity into these PT systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above PT -symmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear PT -symmetric systems arising from various physical disciplines are presented, nonlinear properties of these systems are thoroughly elucidated, and relevant experimental results are described. In addition, emerging applications of PT symmetry are pointed out.

  16. A Possible Method for Non-Hermitian and Non-PT-Symmetric Hamiltonian Systems

    PubMed Central

    Li, Jun-Qing; Miao, Yan-Gang; Xue, Zhao

    2014-01-01

    A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity. PMID:24896084

  17. A possible method for non-Hermitian and Non-PT-symmetric Hamiltonian systems.

    PubMed

    Li, Jun-Qing; Miao, Yan-Gang; Xue, Zhao

    2014-01-01

    A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.

  18. Shannon's theory in nonlinear systems

    NASA Astrophysics Data System (ADS)

    Killey, Robert I.; Behrens, Carsten

    2011-01-01

    An exponential growth in the capacity of optical networks has taken place over the last decade, but the extent to which future capacity growth can continue is limited by physical laws governing signal propagation through optical fibres. While the classic theory of communication developed by Claude Shannon allows the analytical calculation of information spectral density limits for linear channels with white additive Gaussian noise, the nonlinear nature of optical fibres makes these limits much more difficult to determine for long-haul optical transmission. Accurately predicting the ultimate limits has been the focus of much recent research. This paper describes the sources of linear and nonlinear signal impairments, reviews progress on extending Shannon's theory to the case of nonlinear signal propagation, and discusses new optical and electronic signal processing techniques that may be used to approach the Shannon limit in future networks.

  19. On optimal performance of nonlinear energy sinks in multiple-degree-of-freedom systems

    NASA Astrophysics Data System (ADS)

    Tripathi, Astitva; Grover, Piyush; Kalmár-Nagy, Tamás

    2017-02-01

    We study the problem of optimizing the performance of a nonlinear spring-mass-damper attached to a class of multiple-degree-of-freedom systems. We aim to maximize the rate of one-way energy transfer from primary system to the attachment, and focus on impulsive excitation of a two-degree-of-freedom primary system with an essentially nonlinear attachment. The nonlinear attachment is shown to be able to perform as a 'nonlinear energy sink' (NES) by taking away energy from the primary system irreversibly for some types of impulsive excitations. Using perturbation analysis and exploiting separation of time scales, we perform dimensionality reduction of this strongly nonlinear system. Our analysis shows that efficient energy transfer to nonlinear attachment in this system occurs for initial conditions close to homoclinic orbit of the slow time-scale undamped system, a phenomenon that has been previously observed for the case of single-degree-of-freedom primary systems. Analytical formulae for optimal parameters for given impulsive excitation input are derived. Generalization of this framework to systems with arbitrary number of degrees-of-freedom of the primary system is also discussed. The performance of both linear and nonlinear optimally tuned attachments is compared. While NES performance is sensitive to magnitude of the initial impulse, our results show that NES performance is more robust than linear tuned mass damper to several parametric perturbations. Hence, our work provides evidence that homoclinic orbits of the underlying Hamiltonian system play a crucial role in efficient nonlinear energy transfers, even in high dimensional systems, and gives new insight into robustness of systems with essential nonlinearity.

  20. Multistage slow relaxation in a Hamiltonian system: The Fermi-Pasta-Ulam model

    NASA Astrophysics Data System (ADS)

    Matsuyama, Hironori J.; Konishi, Tetsuro

    2015-08-01

    The relaxation process toward equipartition of energy among normal modes in a Hamiltonian system with many degrees of freedom, the Fermi-Pasta-Ulam (FPU) model is investigated numerically. We introduce a general indicator of relaxation σ which denotes the distance from equipartition state. In the time evolution of σ , some long-time interferences with relaxation, named "plateaus," are observed. In order to examine the details of the plateaus, relaxation time of σ and excitation time for each normal mode are measured as a function of the energy density ɛ0=E0/N . As a result, multistage relaxation is detected in the finite-size system. Moreover, by an analysis of the Lyapunov spectrum, the spectrum of mode energy occupancy, and the power spectrum of mode energy, we characterize the multistage slow relaxation, and some dynamical phases are extracted: quasiperiodic motion, stagnant motion (escaping from quasiperiodic motion), local chaos, and stronger chaos with nonthermal noise. We emphasize that the plateaus are robust against the arranging microscopic state. In other words, we can often observe plateaus and multistage slow relaxation in the FPU phase space. Slow relaxation is expected to remain or vanish in the thermodynamic limit depending on indicators.

  1. A canonical form for nonlinear systems

    NASA Technical Reports Server (NTRS)

    Su, R.; Hunt, L. R.

    1985-01-01

    The conceptions of transformation and canonical form have been much used to analyze the structure of linear systems. A coordinate system and a corresponding canonical form are developed for general nonlinear control systems. Their usefulness is demonstrated by showing that every feedback linearizable system becomes a system with only feedback paths in the canonical form.

  2. Stabilization of nonlinear systems using linear observers

    NASA Technical Reports Server (NTRS)

    Strane, R. E.; Vogt, W. G.

    1974-01-01

    It is shown that a linear observer can always be employed to stabilize a nonlinear system which contains a true Popov type nonlinearity in the closed interval from 0 to k, where k is finite, provided the nonlinear function and a completely observable output of the linear portion are available as inputs to the observer. Taking into consideration the case in which a completely observable output is not available from the linear portion, stabilization is shown to be possible if the original linear approximation of the system is asymptotically stable.

  3. A design methodology for nonlinear systems containing parameter uncertainty: Application to nonlinear controller design

    NASA Technical Reports Server (NTRS)

    Young, G.

    1982-01-01

    A design methodology capable of dealing with nonlinear systems, such as a controlled ecological life support system (CELSS), containing parameter uncertainty is discussed. The methodology was applied to the design of discrete time nonlinear controllers. The nonlinear controllers can be used to control either linear or nonlinear systems. Several controller strategies are presented to illustrate the design procedure.

  4. Nonlinear model updating applied to the IMAC XXXII Round Robin benchmark system

    NASA Astrophysics Data System (ADS)

    Kurt, Mehmet; Moore, Keegan J.; Eriten, Melih; McFarland, D. Michael; Bergman, Lawrence A.; Vakakis, Alexander F.

    2017-05-01

    We consider the application of a new nonlinear model updating strategy to a computational benchmark system. The approach relies on analyzing system response time series in the frequency-energy domain by constructing both Hamiltonian and forced and damped frequency-energy plots (FEPs). The system parameters are then characterized and updated by matching the backbone branches of the FEPs with the frequency-energy wavelet transforms of experimental and/or computational time series. The main advantage of this method is that no nonlinearity model is assumed a priori, and the system model is updated solely based on simulation and/or experimental measured time series. By matching the frequency-energy plots of the benchmark system and its reduced-order model, we show that we are able to retrieve the global strongly nonlinear dynamics in the frequency and energy ranges of interest, identify bifurcations, characterize local nonlinearities, and accurately reconstruct time series. We apply the proposed methodology to a benchmark problem, which was posed to the system identification community prior to the IMAC XXXII (2014) and XXXIII (2015) Conferences as a "Round Robin Exercise on Nonlinear System Identification". We show that we are able to identify the parameters of the non-linear element in the problem with a priori knowledge about its position.

  5. Patterns in a Nonlinear Optical System

    NASA Astrophysics Data System (ADS)

    Arecchi, F. T.; Ramazza, P. L.

    We discuss the general features of patten formation in nonlinear optics, regarding the system sizes along the coordinates longitudinal and transverse to the wavefront propagation as the crucial parameters in determining the possible dynamical behaviours. As a specific example of optical pattern forming system, we review the phenomena observed in a prototypical nonlinear interferometer formed by a Kerr-like medium with optical feedback. Particular attention is devoted to the role of nonlocal interactions in determining the pattern forming scenarios observed.

  6. The Nekhoroshev theorem and the observation of long-term diffusion in Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Guzzo, Massimiliano; Lega, Elena

    2016-11-01

    The long-term diffusion properties of the action variables in real analytic quasiintegrable Hamiltonian systems is a largely open problem. The Nekhoroshev theorem provides bounds to such a diffusion as well as a set of techniques, constituting its proof, which have been used to inspect also the instability of the action variables on times longer than the Nekhoroshev stability time. In particular, the separation of the motions in a superposition of a fast drift oscillation and an extremely slow diffusion along the resonances has been observed in several numerical experiments. Global diffusion, which occurs when the range of the slow diffusion largely exceeds the range of fast drift oscillations, needs times larger than the Nekhoroshev stability times to be observed, and despite the power of modern computers, it has been detected only in a small interval of the perturbation parameter, just below the critical threshold of application of the theorem. In this paper we show through an example how sharp this phenomenon is.

  7. Nonlinear resonance: Performance report, August 1, 1989--November 30, 1991

    SciTech Connect

    Kevorkian, J.

    1991-12-31

    This report discusses research concentrated on slowly varying nonlinear oscillatory systems. Some of the topics discussed are; adiabatic invariants and transient resonance in very slowly varying hamiltonians systems; sustained resonance in very slowly varying hamiltonian systems; free-electron lasers with very slow wiggler taper; and bursting oscillators. (LSP)

  8. Nonlinear resonance: Performance report, August 1, 1989--November 30, 1991

    SciTech Connect

    Kevorkian, J.

    1991-01-01

    This report discusses research concentrated on slowly varying nonlinear oscillatory systems. Some of the topics discussed are; adiabatic invariants and transient resonance in very slowly varying hamiltonians systems; sustained resonance in very slowly varying hamiltonian systems; free-electron lasers with very slow wiggler taper; and bursting oscillators. (LSP)

  9. Effective Hamiltonians of polymethineimine, polyazine and polyazoethene: A density matrix variation approach

    NASA Astrophysics Data System (ADS)

    Chen, GuanHua; Su, ZhongMin; Shen, ZhenWen; Yan, YiJing

    1998-08-01

    A new variation method is proposed to determine the effective Hamiltonians for conjugated π-electron systems. This method is based on the minimization of the difference between the ground state reduced single electron density matrix calculated from the effective Hamiltonian and its ab initio counterpart under a set of well-defined constraints. Applications are made to various oligomers of polymethineimine (PMI), polyazine (PAZ) and polyazoethene (PAE) at the Hartree-Fock level. Calculated are also the optical gaps of these oligomers. The effective Hamiltonians contain electron-electron Coulomb interactions and are suitable for the study of excited state dynamic processes such as nonlinear optical properties in π-conjugated systems.

  10. Experimental quantum Hamiltonian learning

    NASA Astrophysics Data System (ADS)

    Wang, Jianwei; Paesani, Stefano; Santagati, Raffaele; Knauer, Sebastian; Gentile, Antonio A.; Wiebe, Nathan; Petruzzella, Maurangelo; O'Brien, Jeremy L.; Rarity, John G.; Laing, Anthony; Thompson, Mark G.

    2017-06-01

    The efficient characterization of quantum systems, the verification of the operations of quantum devices and the validation of underpinning physical models, are central challenges for quantum technologies and fundamental physics. The computational cost of such studies could be improved by machine learning enhanced by quantum simulators. Here we interface two different quantum systems through a classical channel--a silicon-photonics quantum simulator and an electron spin in a diamond nitrogen-vacancy centre--and use the former to learn the Hamiltonian of the latter via Bayesian inference. We learn the salient Hamiltonian parameter with an uncertainty of approximately 10-5. Furthermore, an observed saturation in the learning algorithm suggests deficiencies in the underlying Hamiltonian model, which we exploit to further improve the model. We implement an interactive version of the protocol and experimentally show its ability to characterize the operation of the quantum photonic device.

  11. Dynamical systems approaches to nonlinear problems in systems and circuits

    SciTech Connect

    Salam, F.M.A.; Levi, M.L.

    1988-01-01

    Applications of dynamical-systems analysis to nonlinear circuits and physical systems are discussed in reviews and reports. Topics addressed include general analytical methods, general simulation methods, nonlinear circuits and systems in electrical engineering, control systems, solids and vibrations, and mechanical systems. Consideration is given to the applicability of the Mel'nikov method to highly dissipative systems, damping in nonlinear solid mechanics, a three-dimensional rotation instrument for displaying strange attractors, a chaotic saddle catastrophe in forced oscillators, soliton experiments in annular Josephson junctions, local bifurcation control, periodic and chaotic motions of a buckled beam experiencing parametric and external excitation, and robust nonlinear computed torque control for robot manipulators.

  12. The effect of system nonlinearities on system noise statistics

    NASA Technical Reports Server (NTRS)

    Robinson, L. H., Jr.

    1971-01-01

    The effects are studied of nonlinearities in a baseline communications system on the system noise amplitude statistics. So that a meaningful identification of system nonlinearities can be made, the baseline system is assumed to transmit a single biphase-modulated signal through a relay satellite to the receiving equipment. The significant nonlinearities thus identified include square-law or product devices (e.g., in the carrier reference recovery loops in the receivers), bandpass limiters, and traveling wave tube amplifiers.

  13. Automated reverse engineering of nonlinear dynamical systems.

    PubMed

    Bongard, Josh; Lipson, Hod

    2007-06-12

    Complex nonlinear dynamics arise in many fields of science and engineering, but uncovering the underlying differential equations directly from observations poses a challenging task. The ability to symbolically model complex networked systems is key to understanding them, an open problem in many disciplines. Here we introduce for the first time a method that can automatically generate symbolic equations for a nonlinear coupled dynamical system directly from time series data. This method is applicable to any system that can be described using sets of ordinary nonlinear differential equations, and assumes that the (possibly noisy) time series of all variables are observable. Previous automated symbolic modeling approaches of coupled physical systems produced linear models or required a nonlinear model to be provided manually. The advance presented here is made possible by allowing the method to model each (possibly coupled) variable separately, intelligently perturbing and destabilizing the system to extract its less observable characteristics, and automatically simplifying the equations during modeling. We demonstrate this method on four simulated and two real systems spanning mechanics, ecology, and systems biology. Unlike numerical models, symbolic models have explanatory value, suggesting that automated "reverse engineering" approaches for model-free symbolic nonlinear system identification may play an increasing role in our ability to understand progressively more complex systems in the future.

  14. ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS: On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

    NASA Astrophysics Data System (ADS)

    Eburilitu; Alatancang

    2010-03-01

    The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy's principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.

  15. Nonlinear dynamical system approaches towards neural prosthesis

    SciTech Connect

    Torikai, Hiroyuki; Hashimoto, Sho

    2011-04-19

    An asynchronous discrete-state spiking neurons is a wired system of shift registers that can mimic nonlinear dynamics of an ODE-based neuron model. The control parameter of the neuron is the wiring pattern among the registers and thus they are suitable for on-chip learning. In this paper an asynchronous discrete-state spiking neuron is introduced and its typical nonlinear phenomena are demonstrated. Also, a learning algorithm for a set of neurons is presented and it is demonstrated that the algorithm enables the set of neurons to reconstruct nonlinear dynamics of another set of neurons with unknown parameter values. The learning function is validated by FPGA experiments.

  16. SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

    NASA Astrophysics Data System (ADS)

    Choi, Jeong Ryeol; Yeon, Kyu Hwang

    2008-07-01

    The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schrödinger cat states defined as an eigenstate of hat{K}-2 are also studied. We applied our development to two-mode Caldirola-Kanai oscillator which is a typical example of the time-dependent quadratic Hamiltonian system. The time evolution of the quadrature distribution for the probability density in the coherent states are analyzed for the two-mode Caldirola-Kanai oscillator by plotting relevant figures.

  17. Nonlinear characteristics of an autoparametric vibration system

    NASA Astrophysics Data System (ADS)

    Yan, Zhimiao; Taha, Haithem E.; Tan, Ting

    2017-03-01

    The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.

  18. Nonlinear vibrating system identification via Hilbert decomposition

    NASA Astrophysics Data System (ADS)

    Feldman, Michael; Braun, Simon

    2017-02-01

    This paper deals with the identification of nonlinear vibration systems, based on measured signals for free and forced vibration regimes. Two categories of time domain signal are analyzed, one of a fast inter-modulation signal and a second as composed of several mono-components. To some extent, this attempts to imitate analytic studies of such systems, with its two major analysis groups - the perturbation and the harmonic balance methods. Two appropriate signal processing methods are then investigated, one based on demodulation and the other on signal decomposition. The Hilbert Transform (HT) has been shown to enable effective and simple methods of analysis. We show that precise identification of the nonlinear parameters can be obtained, contrary to other average HT based methods where only approximation parameters are obtained. The effectiveness of the proposed methods is demonstrated for the precise nonlinear system identification, using both the signal demodulation and the signal decomposition methods. Following the exposition of the tools used, both the signal demodulation as well as decomposition are applied to classical examples of nonlinear systems. Cases of nonlinear stiffness and damping forces are analyzed. These include, among other, an asymmetric Helmholtz oscillator, a backlash with nonlinear turbulent square friction, and a Duffing oscillator with dry friction.

  19. A generic formulation for emittance and lattice function evolution for non-Hamiltonian systems with stochastic effects

    SciTech Connect

    Berg, J. S.

    2015-05-03

    I describe a generic formulation for the evolution of emittances and lattice functions under arbitrary, possibly non-Hamiltonian, linear equations of motion. The average effect of stochastic processes, which would include ionization interactions and synchrotron radiation, is also included. I first compute the evolution of the covariance matrix, then the evolution of emittances and lattice functions from that. I examine the particular case of a cylindrically symmetric system, which is of particular interest for ionization cooling.

  20. Systems of Nonlinear Hyperbolic Partial Differential Equations

    DTIC Science & Technology

    1997-12-01

    McKinney) Travelling wave solutions of the modified Korteweg - deVries -Burgers Equation . J. Differential Equations , 116 (1995), 448-467. 4. (with D.G...SUBTITLE Systems of Nonlinear Hyperbolic Partial Differential Equations 6. AUTHOR’S) Michael Shearer PERFORMING ORGANIZATION NAMES(S) AND...DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) This project concerns properties of wave propagation in partial differential equations that are nonlinear

  1. Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems.

    PubMed

    Torres-Herrera, E J; Santos, Lea F

    2013-10-01

    We explore the role of the initial state on the onset of thermalization in isolated quantum many-body systems after a quench. The initial state is an eigenstate of an initial Hamiltonian H(I) and it evolves according to a different final Hamiltonian H(F). If the initial state has a chaotic structure with respect to H(F), i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens when H(I) is a full random matrix, because its states projected onto H(F), are fully delocalized. The results for the observables then agree with those obtained with thermal states at infinite temperature. However, finite real systems with few-body interactions, as the ones considered here, are deprived of fully extended eigenstates, even when described by a nonintegrable Hamiltonian. We examine how the initial state delocalizes as it gets closer to the middle of the spectrum of H(F), causing the observables to approach thermal averages, be the models integrable or chaotic. Our numerical studies are based on initial states with energies that cover the entire lower half of the spectrum of one-dimensional Heisenberg spin-1/2 systems.

  2. Hamiltonian and non-Hamiltonian perturbation theory for nearly periodic motion

    NASA Astrophysics Data System (ADS)

    Larsson, Jonas

    1986-02-01

    Kruskal's asymptotic theory of nearly period motion [M. Kruskal, J. Math. Phys. 4, 806 (1962)] (with applications to nonlinear oscillators, guiding center motion, etc.) is generalized and modified. A new more natural recursive formula, with considerable advantages in applications, determining the averaging transformations and the drift equations is derived. Also almost quasiperiodic motion is considered. For a Hamiltonian system, a manifestly Hamiltonian extension of Kruskal's theory is given by means of the phase-space Lagrangian formulation of Hamiltonian mechanics. By performing an averaging transformation on the phase-space Lagrangian for the system (L → L¯) and adding a total derivative dS/dτ, a nonoscillatory Lagrangian Λ=L¯+dS/dτ is obtained. The drift equations and the adiabatic invariant are now obtained from Λ. By truncating Λ to some finite order in the small parameter ɛ, manifestly Hamiltonian approximating systems are obtained. The utility of the method for treating the guiding-center motion is demonstrated in a separate paper.

  3. Connective stability of nonlinear matrix systems

    NASA Technical Reports Server (NTRS)

    Siljak, D. D.

    1974-01-01

    Consideration of stability under structural perturbations of free dynamic systems described by the differential equation dx/dt = A(t,x)x, where the matrix A(t,x) has time-varying nonlinear elements. The concept of 'connective stability' is introduced to study the structural properties of competitive-cooperative nonlinear matrix systems. It is shown that stability reliability in such systems is high and that they remain stable despite time-varying (including 'on-off') interaction among individual agents present in the system. The results obtained can be used to study stability aspects of mathematical models arising in as diverse fields as economics, biology, arms races, and transistor circuits.

  4. A formula relating sojourn times to the time of arrival in Hamiltonian dynamics

    NASA Astrophysics Data System (ADS)

    Gournay, A.; Tiedra de Aldecoa, R.

    2012-06-01

    We consider on a manifold M equipped with a Poisson bracket { ·, ·} a Hamiltonian H with complete flow and a family Φ ≡ (Φ1, …, Φd) of abstract position observables satisfying the condition {{Φj, H}, H} = 0 for each j. Under these assumptions, we prove a new formula relating sojourn times in dilated regions defined in terms of Φ to the time of arrival of classical orbits. The correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence. Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, central force systems, the Poincaré ball model, the wave equation, the nonlinear Schrödinger equation, the Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values.

  5. Damage detection in initially nonlinear systems

    SciTech Connect

    Bornn, Luke; Farrar, Charles; Park, Gyuhae

    2009-01-01

    The primary goal of Structural Health Monitoring (SHM) is to detect structural anomalies before they reach a critical level. Because of the potential life-safety and economic benefits, SHM has been widely studied over the past decade. In recent years there has been an effort to provide solid mathematical and physical underpinnings for these methods; however, most focus on systems that behave linearly in their undamaged state - a condition that often does not hold in complex 'real world' systems and systems for which monitoring begins mid-lifecycle. In this work, we highlight the inadequacy of linear-based methodology in handling initially nonlinear systems. We then show how the recently developed autoregressive support vector machine (AR-SVM) approach to time series modeling can be used for detecting damage in a system that exhibits initially nonlinear response. This process is applied to data acquired from a structure with induced nonlinearity tested in a laboratory environment.

  6. Augmented nonlinear differentiator design and application to nonlinear uncertain systems.

    PubMed

    Shao, Xingling; Liu, Jun; Li, Jie; Cao, Huiliang; Shen, Chong; Zhang, Xiaoming

    2017-03-01

    In this paper, an augmented nonlinear differentiator (AND) based on sigmoid function is developed to calculate the noise-less time derivative under noisy measurement condition. The essential philosophy of proposed AND in achieving high attenuation of noise effect is established by expanding the signal dynamics with extra state variable representing the integrated noisy measurement, then with the integral of measurement as input, the augmented differentiator is formulated to improve the estimation quality. The prominent advantages of the present differentiation technique are: (i) better noise suppression ability can be achieved without appreciable delay; (ii) the improved methodology can be readily extended to construct augmented high-order differentiator to obtain multiple derivatives. In addition, the convergence property and robustness performance against noises are investigated via singular perturbation theory and describing function method, respectively. Also, comparison with several classical differentiators is given to illustrate the superiority of AND in noise suppression. Finally, the robust control problems of nonlinear uncertain systems, including a numerical example and a mass spring system, are addressed to demonstrate the effectiveness of AND in precisely estimating the disturbance and providing the unavailable differential estimate to implement output feedback based controller.

  7. An eight-dimensional quantum mechanical Hamiltonian for X + YCZ3 system and its applications to H + CH4 reaction

    NASA Astrophysics Data System (ADS)

    Liu, Rui; Xiong, Hongwei; Yang, Minghui

    2012-11-01

    An eight-dimensional quantum mechanical Hamiltonian has been proposed based on Palma and Clary's model in which the non-reacting CZ3 group keeps a C3v symmetry in the X + YCZ3 ↔ XY + CZ3 reaction J. Palma and D. C. Clary [J. Chem. Phys. 112, 1859 (2000), 10.1063/1.480749]. By transforming the original Cartesian coordinate system (x, s) into a scaled polar coordinate system (q, γ), the vibrational Hamiltonian of CZ3 group is expressed in a simple form with a clear physical picture. This Hamiltonian is used to investigate the H + CH4 → H2 + CH3 reaction on the Jordan-Gilbert potential energy surface. The total reaction probabilities are calculated for the initial ground state, and umbrella, bending, symmetric, and asymmetric stretching excited states of CH4 with total angular momentum J = 0. The integral cross sections for the reaction are also studied for these initial vibrational states with a centrifugal-sudden approximation. The total integral cross sections for the asymmetric stretching vibrational excited state are in good agreement with the experimental observations. The results also showed the difference of dynamical behavior between reactions from symmetric and asymmetric stretching excited states. The thermal rate constants are calculated for the temperature range T = 250-2000 K and compared with the experimental and other theoretical results.

  8. Parametric Identification of Nonlinear Dynamical Systems

    NASA Technical Reports Server (NTRS)

    Feeny, Brian

    2002-01-01

    In this project, we looked at the application of harmonic balancing as a tool for identifying parameters (HBID) in a nonlinear dynamical systems with chaotic responses. The main idea is to balance the harmonics of periodic orbits extracted from measurements of each coordinate during a chaotic response. The periodic orbits are taken to be approximate solutions to the differential equations that model the system, the form of the differential equations being known, but with unknown parameters to be identified. Below we summarize the main points addressed in this work. The details of the work are attached as drafts of papers, and a thesis, in the appendix. Our study involved the following three parts: (1) Application of the harmonic balance to a simulation case in which the differential equation model has known form for its nonlinear terms, in contrast to a differential equation model which has either power series or interpolating functions to represent the nonlinear terms. We chose a pendulum, which has sinusoidal nonlinearities; (2) Application of the harmonic balance to an experimental system with known nonlinear forms. We chose a double pendulum, for which chaotic response were easily generated. Thus we confronted a two-degree-of-freedom system, which brought forth challenging issues; (3) A study of alternative reconstruction methods. The reconstruction of the phase space is necessary for the extraction of periodic orbits from the chaotic responses, which is needed in this work. Also, characterization of a nonlinear system is done in the reconstructed phase space. Such characterizations are needed to compare models with experiments. Finally, some nonlinear prediction methods can be applied in the reconstructed phase space. We developed two reconstruction methods that may be considered if the common method (method of delays) is not applicable.

  9. Discrete time learning control in nonlinear systems

    NASA Technical Reports Server (NTRS)

    Longman, Richard W.; Chang, Chi-Kuang; Phan, Minh

    1992-01-01

    In this paper digital learning control methods are developed primarily for use in single-input, single-output nonlinear dynamic systems. Conditions for convergence of the basic form of learning control based on integral control concepts are given, and shown to be satisfied by a large class of nonlinear problems. It is shown that it is not the gross nonlinearities of the differential equations that matter in the convergence, but rather the much smaller nonlinearities that can manifest themselves during the short time interval of one sample time. New algorithms are developed that eliminate restrictions on the size of the learning gain, and on knowledge of the appropriate sign of the learning gain, for convergence to zero error in tracking a feasible desired output trajectory. It is shown that one of the new algorithms can give guaranteed convergence in the presence of actuator saturation constraints, and indicate when the requested trajectory is beyond the actuator capabilities.

  10. Asymmetric wave propagation in nonlinear systems.

    PubMed

    Lepri, Stefano; Casati, Giulio

    2011-04-22

    A mechanism for asymmetric (nonreciprocal) wave transmission is presented. As a reference system, we consider a layered nonlinear, nonmirror-symmetric model described by the one-dimensional discrete nonlinear Schrödinger equation with spatially varying coefficients embedded in an otherwise linear lattice. We construct a class of exact extended solutions such that waves with the same frequency and incident amplitude impinging from left and right directions have very different transmission coefficients. This effect arises already for the simplest case of two nonlinear layers and is associated with the shift of nonlinear resonances. Increasing the number of layers considerably increases the complexity of the family of solutions. Finally, numerical simulations of asymmetric wave packet transmission are presented which beautifully display the rectifying effect.

  11. On Stabilization of Nonautonomous Nonlinear Systems

    SciTech Connect

    Bogdanov, A. Yu.

    2008-10-30

    The procedures to obtain the sufficient conditions of asymptotic stability for nonlinear nonstationary continuous-time systems are discussed. We consider different types of the following general controlled system: x = X(t,x,u) = F(t,x)+B(t,x)u, x(t{sub 0}) = x{sub 0}. (*) The basis of investigation is limiting equations, limiting Lyapunov functions, etc. The improved concept of observability of the pair of functional matrices is presented. By these results the problem of synthesis of asymptotically stable control nonlinear nonautonomous systems (with linear parts) involving the quadratic time-dependent Lyapunov functions is solved as well as stabilizing a given unstable system with nonlinear control law.

  12. Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling.

    PubMed

    Sigalov, G; Gendelman, O V; AL-Shudeifat, M A; Manevitch, L I; Vakakis, A F; Bergman, L A

    2012-03-01

    We show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits non-conventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the non-conventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.

  13. Controllability of Nonlinear Fractional Delay Dynamical Systems

    NASA Astrophysics Data System (ADS)

    Nirmala, R. Joice; Balachandran, K.; Rodríguez-Germa, L.; Trujillo, J. J.

    2016-02-01

    This paper is concerned with controllability of nonlinear fractional delay dynamical systems with delay in state variables. The solution representations of fractional delay differential equations have been established by using the Laplace transform technique and the Mittag-Leffler function. Necessary and sufficient conditions for the controllability criteria of linear fractional delay systems are established. Further sufficient condition for the controllability of nonlinear fractional delay dynamical system are obtained by using the fixed point argument. Examples and numerical simulation are presented to illustrate the results.

  14. Ontology of Earth's nonlinear dynamic complex systems

    NASA Astrophysics Data System (ADS)

    Babaie, Hassan; Davarpanah, Armita

    2017-04-01

    As a complex system, Earth and its major integrated and dynamically interacting subsystems (e.g., hydrosphere, atmosphere) display nonlinear behavior in response to internal and external influences. The Earth Nonlinear Dynamic Complex Systems (ENDCS) ontology formally represents the semantics of the knowledge about the nonlinear system element (agent) behavior, function, and structure, inter-agent and agent-environment feedback loops, and the emergent collective properties of the whole complex system as the result of interaction of the agents with other agents and their environment. It also models nonlinear concepts such as aperiodic, random chaotic behavior, sensitivity to initial conditions, bifurcation of dynamic processes, levels of organization, self-organization, aggregated and isolated functionality, and emergence of collective complex behavior at the system level. By incorporating several existing ontologies, the ENDCS ontology represents the dynamic system variables and the rules of transformation of their state, emergent state, and other features of complex systems such as the trajectories in state (phase) space (attractor and strange attractor), basins of attractions, basin divide (separatrix), fractal dimension, and system's interface to its environment. The ontology also defines different object properties that change the system behavior, function, and structure and trigger instability. ENDCS will help to integrate the data and knowledge related to the five complex subsystems of Earth by annotating common data types, unifying the semantics of shared terminology, and facilitating interoperability among different fields of Earth science.

  15. System interaction with linear and nonlinear characteristics

    SciTech Connect

    Lin, C.W. ); Tseng, W.S. )

    1991-01-01

    This book is covered under some of the following topics: seismic margins in piping systems, vibrational power flow in a cylindrical shell, inelastic pipework dynamics and aseismic design, an efficient method for dynamic analysis of a linearly elastic piping system with nonlinear supports.

  16. Zero curvature representation, bi-Hamiltonian structure, and an integrable hierarchy for the Zakharov-Ito system

    NASA Astrophysics Data System (ADS)

    Baxter, Mathew; Choudhury, S. Roy; Van Gorder, Robert A.

    2015-06-01

    In the present paper, we present an integrable hierarchy for the Zakharov-Ito system. We first construct the Lenard recursion sequence and zero curvature representation for the Zakharov-Ito system, following Cao's method as significantly generalized by other authors. We then construct the bi-Hamiltonian structures employing variational trace identities but woven together with the Lenard recursion sequences. From this, we are in a position to construct an integrable hierarchy of equations from the Zakharov-Ito system, and we obtain the recursion operator and Poisson brackets for constructing this hierarchy. Finally, we demonstrate that the obtained hierarchy is indeed Liouville integrable.

  17. Nonlinear coupling in the human motor system

    PubMed Central

    Chen, C.C.; Kilner, J.M.; Friston, K.J.; Kiebel, S. J.; Jolly, R.K.; Ward, N. S.

    2010-01-01

    The synchronous discharge of neuronal assemblies is thought to facilitate communication between areas within distributed networks in the human brain. This oscillatory activity is especially interesting, given the pathological modulation of specific frequencies in diseases affecting the motor system. Many studies investigating oscillatory activity have focussed on same frequency, or linear, coupling between areas of a network. In this study, our aim was to establish a functional architecture in the human motor system responsible for induced responses as measured in normal subjects with magnetoencephalography. Specifically, we looked for evidence for additional nonlinear (between-frequency) coupling among neuronal sources and, in particular, whether nonlinearities were found predominantly in connections within areas (intrinsic), between areas (extrinsic) or both. We modelled the event-related modulation of spectral responses during a simple hand-grip using dynamic casual modelling. We compared models with and without nonlinear connections under conditions of symmetric and asymmetric interhemispheric connectivity. Bayesian model comparison suggested that the task-dependent motor network was asymmetric during right hand movements. Furthermore, it revealed very strong evidence for nonlinear coupling between sources in this distributed network, but interactions among frequencies within a source appeared linear in nature. Our results provide empirical evidence for nonlinear coupling among distributed neuronal sources in the motor system and that these play an important role in modulating spectral responses under normal conditions. PMID:20573886

  18. System characterization in nonlinear random vibration

    SciTech Connect

    Paez, T.L.; Gregory, D.L.

    1986-01-01

    Linear structural models are frequently used for structural system characterization and analysis. In most situations they can provide satisfactory results, but under some circumstances they are insufficient for system definition. The present investigation proposes a model for nonlinear structure characterization, and demonstrates how the functions describing the model can be identified using a random vibration experiment. Further, it is shown that the model is sufficient to completely characterize the stationary random vibration response of a structure that has a harmonic frequency generating form of nonlinearity. An analytical example is presented to demonstrate the plausibility of the model.

  19. Chaotic Oscillations in Weakly Nonlinear Systems

    NASA Astrophysics Data System (ADS)

    Belogortsev, Andrey B.

    1995-01-01

    The weakly nonlinear oscillator is a classical model widely used for studying various nonlinear phenomena in such fields as physics, mechanics, biology, and electrical engineering. This work is devoted to the study of the properties of weakly nonlinear systems, which result in the appearance of their chaotic behavior. The analysis is concentrated on three classical types of weakly nonlinear systems: the Duffing oscillator, the van der Pol oscillator, and the relaxation oscillator. The method of averaging is applied to the original equations of motion of these systems to obtain the averaged equations, which serve as the basic mathematical models in this work. The secondary averaging method is applied to the Duffing and van der Pol oscillators, driven by a quasiperiodic force, and an analysis of their properties is performed. Analytical expressions for the response curves and bifurcation conditions of various types in these systems have been obtained for the first time. The theoretical results have been compared with numerical ones, which agree closely. An approach using a discrete mapping has also been applied to the quasiperiodically forced Duffing and van der Pol oscillators. Corresponding maps have been derived and analyzed for the first time. The analytical results obtained for the response curves of the oscillators and bifurcation conditions of the quasiperiodic solutions are in good agreement with the results obtained using the secondary averaging technique and with numerical results. The mechanisms for the appearance of chaotic motion in weakly nonlinear oscillators with different types of hysteresis (due to nonisochronism and due to a relaxation element) have been analysed and discussed. The bifurcation portraits of the weakly nonlinear oscillators have been obtained numerically and the general characteristics of the transition from regular to chaotic motion in such systems have been analyzed. The theoretical results are in good agreement with the numerical

  20. State Identification in Nonlinear Systems

    SciTech Connect

    Holloway, James Paul

    2005-02-06

    A state estimation method based on finding a system state that causes a model to match a set of system measurements is regularized by requiring that sudden changes in system state be avoided. The required optimization is accomplished by a pattern search algorithm. The method does not require derivative information or linearization of the model. Is has been applied to a 10 dimensional model of a fast reactor system.

  1. Orbit structure of Hamiltonian systems arising from Lie transformation group actions

    NASA Astrophysics Data System (ADS)

    Garzia, M. R.; Loparo, K. A.; Martin, C. F.

    This paper associates the Riccati group and its group action on linear-quadratic optimal control problems to the action of a Lie transformation group on a set of Hamiltonian matrices. In this Lie theoretic setting results are presented concerning the associated orbit structure and the structure of the group itself. These results are of importance in understanding the solution structure of matrix Riccati differential equations, and thus also of importance in linear-quadratic optimal control.

  2. Orbit structure of Hamiltonian systems arising from Lie transformation group actions

    NASA Technical Reports Server (NTRS)

    Garzia, M. R.; Loparo, K. A.; Martin, C. F.

    1983-01-01

    This paper associates the Riccati group and its group action on linear-quadratic optimal control problems to the action of a Lie transformation group on a set of Hamiltonian matrices. In this Lie theoretic setting results are presented concerning the associated orbit structure and the structure of the group itself. These results are of importance in understanding the solution structure of matrix Riccati differential equations, and thus also of importance in linear-quadratic optimal control.

  3. Orbit structure of Hamiltonian systems arising from Lie transformation group actions

    NASA Technical Reports Server (NTRS)

    Garzia, M. R.; Loparo, K. A.; Martin, C. F.

    1983-01-01

    This paper associates the Riccati group and its group action on linear-quadratic optimal control problems to the action of a Lie transformation group on a set of Hamiltonian matrices. In this Lie theoretic setting results are presented concerning the associated orbit structure and the structure of the group itself. These results are of importance in understanding the solution structure of matrix Riccati differential equations, and thus also of importance in linear-quadratic optimal control.

  4. Geometric Hamiltonian structures and perturbation theory

    SciTech Connect

    Omohundro, S.

    1984-08-01

    We have been engaged in a program of investigating the Hamiltonian structure of the various perturbation theories used in practice. We describe the geometry of a Hamiltonian structure for non-singular perturbation theory applied to Hamiltonian systems on symplectic manifolds and the connection with singular perturbation techniques based on the method of averaging.

  5. Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters

    SciTech Connect

    Kandrup, H.E. ); Morrison, P.J. )

    1993-07-01

    The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems, is presented. It is demonstrated explicitly in the context of a 3+1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function f (a noncanonical variable) being the conserved ADM mass-energy H[sub ADM]. This facilitates a geometric understanding of the evolution of f in an infinite-dimensional phase space, providing a natural interpretation of the constraints associated with conservation of phase space. This geometric interpretation also facilitates the derivation of improved criteria for linear stability by focusing on dynamically accessible perturbation [delta]f which satisfy all the constraints of phase space conservation. An explicit expression is derived for the energy [delta][sup (2)]H[sub ADM] associated with an arbitrary spherical phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if [delta][sup (2)]H[sub ADM] is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems. Intuition derived from simple finite models clarifies several features of the Vlasov-Einstein system; for example, how, negative energy modes preclude necessary and sufficient conditions for stability and why, unlike the Newtonian case, the existence of negative energy perturbations for some static, isotropic equilibrium apparently signals the onset of a linear instability. An Appendix exhibits the construction of a completely covariant bracket which generates the Vlasov-Einstein system for arbitrary configurations in a form independent of any assumed 3+1 splitting. 87 refs.

  6. Evolutionary quantitative genetics of nonlinear developmental systems.

    PubMed

    Morrissey, Michael B

    2015-08-01

    In quantitative genetics, the effects of developmental relationships among traits on microevolution are generally represented by the contribution of pleiotropy to additive genetic covariances. Pleiotropic additive genetic covariances arise only from the average effects of alleles on multiple traits, and therefore the evolutionary importance of nonlinearities in development is generally neglected in quantitative genetic views on evolution. However, nonlinearities in relationships among traits at the level of whole organisms are undeniably important to biology in general, and therefore critical to understanding evolution. I outline a system for characterizing key quantitative parameters in nonlinear developmental systems, which yields expressions for quantities such as trait means and phenotypic and genetic covariance matrices. I then develop a system for quantitative prediction of evolution in nonlinear developmental systems. I apply the system to generating a new hypothesis for why direct stabilizing selection is rarely observed. Other uses will include separation of purely correlative from direct and indirect causal effects in studying mechanisms of selection, generation of predictions of medium-term evolutionary trajectories rather than immediate predictions of evolutionary change over single generation time-steps, and the development of efficient and biologically motivated models for separating additive from epistatic genetic variances and covariances.

  7. Pseudo-invariants theory and real phases for systems with non-Hermitian time-dependent Hamiltonians

    NASA Astrophysics Data System (ADS)

    Maamache, Mustapha; Kaltoum Djeghiour, Oum; Mana, Naima; Koussa, Walid

    2017-09-01

    In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. Explicitly time-dependent pseudo-Hermitian invariants theory, with a time-dependent metric, is developed. We derive a simple relation between the eigenstates of this pseudo-Hermitian invariant and the solutions of the Schrödinger equation. A physical system is treated in detail: the time-dependent Swanson model, where an explicitly time-dependent pseudo-Hermitian invariant is derived as well as their eigenvalues and eigenstates.

  8. Quantum manifestations of classical stochasticity. I. Energetics of some nonlinear systems

    NASA Astrophysics Data System (ADS)

    Weissman, Yitzhak; Jortner, Joshua

    1982-08-01

    In this paper we present the results of a semiclassical investigation and a quantum mechanical study of the bound energy spectrum of the Henon-Heiles Hamiltonian (HHH) and of the Barbanis Hamiltonian (BH). We have derived a simple semiclassical formula for the energy levels E, and for their sensitivity dE/dɛ with respect to the strength ɛ of the nonlinear coupling for the HHH, and established general relations between E and its derivatives dnE/dɛn (n⩾1). Numerical quantum mechanical computations of the energy levels were conducted for the HHH and for the BH. The nonlinear coupling constant was adjusted so that for the HHH there will be ˜150 states up to the classical critical energy Ec and ˜300 states up to the dissociation energy ED. The E values were obtained by direct diagonalization using a basis containing 760 states, while the values of dE/dɛ were computed utilizing the Hellmann-Feynman theorem. Good agreement between the semiclassical and the quantum mechanical spectra was observerd well above Ec. These results raise the distict possibility that the semiclassical approxmation for these nonlinear systems does not break down in the vicinity of Ec and that the bound level structure does not provide a manifestation of the classical transition from quasiperiodic to chaotic motion.

  9. Bounds on nonlinear motion for a finite time

    SciTech Connect

    Warnock, R.L.; Ruth, R.D.

    1989-06-01

    Recent improvements in numerical methods to compute canonical transformations make it feasible to set interesting bounds on the motion of nonlinear Hamiltonian systems over a finite interval of time. 7 refs.

  10. Optimized spectral estimation for nonlinear synchronizing systems

    NASA Astrophysics Data System (ADS)

    Sommerlade, Linda; Mader, Malenka; Mader, Wolfgang; Timmer, Jens; Thiel, Marco; Grebogi, Celso; Schelter, Björn

    2014-03-01

    In many fields of research nonlinear dynamical systems are investigated. When more than one process is measured, besides the distinct properties of the individual processes, their interactions are of interest. Often linear methods such as coherence are used for the analysis. The estimation of coherence can lead to false conclusions when applied without fulfilling several key assumptions. We introduce a data driven method to optimize the choice of the parameters for spectral estimation. Its applicability is demonstrated based on analytical calculations and exemplified in a simulation study. We complete our investigation with an application to nonlinear tremor signals in Parkinson's disease. In particular, we analyze electroencephalogram and electromyogram data.

  11. Optimized spectral estimation for nonlinear synchronizing systems.

    PubMed

    Sommerlade, Linda; Mader, Malenka; Mader, Wolfgang; Timmer, Jens; Thiel, Marco; Grebogi, Celso; Schelter, Björn

    2014-03-01

    In many fields of research nonlinear dynamical systems are investigated. When more than one process is measured, besides the distinct properties of the individual processes, their interactions are of interest. Often linear methods such as coherence are used for the analysis. The estimation of coherence can lead to false conclusions when applied without fulfilling several key assumptions. We introduce a data driven method to optimize the choice of the parameters for spectral estimation. Its applicability is demonstrated based on analytical calculations and exemplified in a simulation study. We complete our investigation with an application to nonlinear tremor signals in Parkinson's disease. In particular, we analyze electroencephalogram and electromyogram data.

  12. Stochastic volatility models at ρ=±1 as second class constrained Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Contreras G., Mauricio

    2014-07-01

    systems (Dirac, 1958, 1967) must be employed, and Dirac's analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian path-integral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path-integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semi-classical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model.

  13. Optimal control of open quantum systems: a combined surrogate hamiltonian optimal control theory approach applied to photochemistry on surfaces.

    PubMed

    Asplund, Erik; Klüner, Thorsten

    2012-03-28

    In this paper, control of open quantum systems with emphasis on the control of surface photochemical reactions is presented. A quantum system in a condensed phase undergoes strong dissipative processes. From a theoretical viewpoint, it is important to model such processes in a rigorous way. In this work, the description of open quantum systems is realized within the surrogate hamiltonian approach [R. Baer and R. Kosloff, J. Chem. Phys. 106, 8862 (1997)]. An efficient and accurate method to find control fields is optimal control theory (OCT) [W. Zhu, J. Botina, and H. Rabitz, J. Chem. Phys. 108, 1953 (1998); Y. Ohtsuki, G. Turinici, and H. Rabitz, J. Chem. Phys. 120, 5509 (2004)]. To gain control of open quantum systems, the surrogate hamiltonian approach and OCT, with time-dependent targets, are combined. Three open quantum systems are investigated by the combined method, a harmonic oscillator immersed in an ohmic bath, CO adsorbed on a platinum surface, and NO adsorbed on a nickel oxide surface. Throughout this paper, atomic units, i.e., ℏ = m(e) = e = a(0) = 1, have been used unless otherwise stated.

  14. Optimal control of open quantum systems: A combined surrogate Hamiltonian optimal control theory approach applied to photochemistry on surfaces

    SciTech Connect

    Asplund, Erik; Kluener, Thorsten

    2012-03-28

    In this paper, control of open quantum systems with emphasis on the control of surface photochemical reactions is presented. A quantum system in a condensed phase undergoes strong dissipative processes. From a theoretical viewpoint, it is important to model such processes in a rigorous way. In this work, the description of open quantum systems is realized within the surrogate Hamiltonian approach [R. Baer and R. Kosloff, J. Chem. Phys. 106, 8862 (1997)]. An efficient and accurate method to find control fields is optimal control theory (OCT) [W. Zhu, J. Botina, and H. Rabitz, J. Chem. Phys. 108, 1953 (1998); Y. Ohtsuki, G. Turinici, and H. Rabitz, J. Chem. Phys. 120, 5509 (2004)]. To gain control of open quantum systems, the surrogate Hamiltonian approach and OCT, with time-dependent targets, are combined. Three open quantum systems are investigated by the combined method, a harmonic oscillator immersed in an ohmic bath, CO adsorbed on a platinum surface, and NO adsorbed on a nickel oxide surface. Throughout this paper, atomic units, i.e., ({Dirac_h}/2{pi})=m{sub e}=e=a{sub 0}= 1, have been used unless otherwise stated.

  15. Indirect learning control for nonlinear dynamical systems

    NASA Technical Reports Server (NTRS)

    Ryu, Yeong Soon; Longman, Richard W.

    1993-01-01

    In a previous paper, learning control algorithms were developed based on adaptive control ideas for linear time variant systems. The learning control methods were shown to have certain advantages over their adaptive control counterparts, such as the ability to produce zero tracking error in time varying systems, and the ability to eliminate repetitive disturbances. In recent years, certain adaptive control algorithms have been developed for multi-body dynamic systems such as robots, with global guaranteed convergence to zero tracking error for the nonlinear system euations. In this paper we study the relationship between such adaptive control methods designed for this specific class of nonlinear systems, and the learning control problem for such systems, seeking to converge to zero tracking error in following a specific command repeatedly, starting from the same initial conditions each time. The extension of these methods from the adaptive control problem to the learning control problem is seen to be trivial. The advantages and disadvantages of using learning control based on such adaptive control concepts for nonlinear systems, and the use of other currently available learning control algorithms are discussed.

  16. (Investigation of transitions from order to chaos in dynamical systems)

    SciTech Connect

    Not Available

    1990-01-01

    This report discusses: torus structure in higher dimensional hamiltonian systems; particle heating and stochastic web diffusion; scaling behavior of coupled conservative nonlinear systems; box counting algorithm and dimensional analysis of a pulsar; and universality of coupled nonlinear systems. (LSP)

  17. Low-energy effective Hamiltonians for correlated electron systems beyond density functional theory

    NASA Astrophysics Data System (ADS)

    Hirayama, Motoaki; Miyake, Takashi; Imada, Masatoshi; Biermann, Silke

    2017-08-01

    We propose a refined scheme of deriving an effective low-energy Hamiltonian for materials with strong electronic Coulomb correlations beyond density functional theory (DFT). By tracing out the electronic states away from the target degrees of freedom in a controlled way by a perturbative scheme, we construct an effective Hamiltonian for a restricted low-energy target space incorporating the effects of high-energy degrees of freedom in an effective manner. The resulting effective Hamiltonian can afterwards be solved by accurate many-body solvers. We improve this "multiscale ab initio scheme for correlated electrons" (MACE) primarily in two directions by elaborating and combining two frameworks developed by Hirayama et al. [M. Hirayama, T. Miyake, and M. Imada, Phys. Rev. B 87, 195144 (2013), 10.1103/PhysRevB.87.195144] and Casula et al. [M. Casula, P. Werner, L. Vaugier, F. Aryasetiawan, T. Miyake, A. J. Millis, and S. Biermann, Phys. Rev. Lett. 109, 126408 (2012), 10.1103/PhysRevLett.109.126408]: (1) Double counting of electronic correlations between the DFT and the low-energy solver is avoided by using the constrained G W scheme; and (2) the frequency dependent interactions emerging from the partial trace summation are successfully separated into a nonlocal part that is treated following ideas by Hirayama et al. and a local part treated nonperturbatively in the spirit of Casula et al. and are incorporated into the renormalization of the low-energy dispersion. The scheme is favorably tested on the example of SrVO3.

  18. Nonlinear amplitude approximation for bilinear systems

    NASA Astrophysics Data System (ADS)

    Jung, Chulwoo; D'Souza, Kiran; Epureanu, Bogdan I.

    2014-06-01

    An efficient method to predict vibration amplitudes at the resonant frequencies of dynamical systems with piecewise-linear nonlinearity is developed. This technique is referred to as bilinear amplitude approximation (BAA). BAA constructs a single vibration cycle at each resonant frequency to approximate the periodic steady-state response of the system. It is postulated that the steady-state response is piece-wise linear and can be approximated by analyzing the response over two time intervals during which the system behaves linearly. Overall the dynamics is nonlinear, but the system is in a distinct linear state during each of the two time intervals. Thus, the approximated vibration cycle is constructed using linear analyses. The equation of motion for analyzing the vibration of each state is projected along the overlapping space spanned by the linear mode shapes active in each of the states. This overlapping space is where the vibratory energy is transferred from one state to the other when the system switches from one state to the other. The overlapping space can be obtained using singular value decomposition. The space where the energy is transferred is used together with transition conditions of displacement and velocity compatibility to construct a single vibration cycle and to compute the amplitude of the dynamics. Since the BAA method does not require numerical integration of nonlinear models, computational costs are very low. In this paper, the BAA method is first applied to a single-degree-of-freedom system. Then, a three-degree-of-freedom system is introduced to demonstrate a more general application of BAA. Finally, the BAA method is applied to a full bladed disk with a crack. Results comparing numerical solutions from full-order nonlinear analysis and results obtained using BAA are presented for all systems.

  19. Research on Nonlinear Dynamical Systems.

    DTIC Science & Technology

    1976-10-19

    LaSalle , J .P ., “Stability theory and invariance principles ” , Dynamical Systems, An International Symposium, Vol.1, pp. 2 11—222 , Academic Press...1974 — 31 November 1975 Principal Investigator: Professor J. P. LaSalle Grant DAA G 29/76/G/0052 1 December 1975 - 31 August 1976 Principal...Investigator: Professor 3. P. LaSalle L.fsch.ts Cente r for. Dynamical Syst.m. Division of Appli.d Mathematics Brown Univ.r sity Providena., Rhod. ~~~~~ 02912 D

  20. Spectral decomposition of nonlinear systems with memory

    NASA Astrophysics Data System (ADS)

    Svenkeson, Adam; Glaz, Bryan; Stanton, Samuel; West, Bruce J.

    2016-02-01

    We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.

  1. On stabilisability of nonlinear systems on time scales

    NASA Astrophysics Data System (ADS)

    Bartosiewicz, Zbigniew; Piotrowska, Ewa

    2013-01-01

    In this article, stabilisability of nonlinear finite-dimensional control systems on arbitrary time scales is studied. The classical results on stabilisation of nonlinear continuous-time and discrete-time systems are extended to systems on arbitrary time scales with bounded graininess function. It is shown that uniform exponential stability of the linear approximation of a nonlinear system implies uniform exponential stability of the nonlinear system. Then this result is used to show a similar implication for uniform exponential stabilisability.

  2. Nonlinear plants, factorizations and stable feedback systems

    NASA Technical Reports Server (NTRS)

    Desoer, Charles A.; Kabuli, M. Guntekin

    1987-01-01

    For nonlinear plants represented by causal maps defined over extended spaces, right factorization and normalized right-coprime factorization concepts are discussed in terms of well-posed stable feedback systems. This setup covers continuous-time, discrete-time, time-invariant or time-varying input-output maps. The nonlinear maps are factored in terms of causal bounded-input bounded-output stable maps. In factored form, all instabilities of the original map are represented by the inverse of a causal stable `denominator' map. The existence of maps with right factorizations and normalized right-coprime factorizations is shown using a well-posed stable unity-feedback system. In the case where one of the subsystems has a normalized right-coprime factorization, the stability of the feedback system is equivalent to the stability of the pseudostate map.

  3. Consensus tracking for multiagent systems with nonlinear dynamics.

    PubMed

    Dong, Runsha

    2014-01-01

    This paper concerns the problem of consensus tracking for multiagent systems with a dynamical leader. In particular, it proposes the corresponding explicit control laws for multiple first-order nonlinear systems, second-order nonlinear systems, and quite general nonlinear systems based on the leader-follower and the tree shaped network topologies. Several numerical simulations are given to verify the theoretical results.

  4. Accidental degeneracies in nonlinear quantum deformed systems

    NASA Astrophysics Data System (ADS)

    Aleixo, A. N. F.; Balantekin, A. B.

    2011-09-01

    We construct a multi-parameter nonlinear deformed algebra for quantum confined systems that includes many other deformed models as particular cases. We demonstrate that such systems exhibit the property of accidental pairwise energy level degeneracies. We also study, as a special case of our multi-parameter deformation formalism, the extension of the Tamm-Dancoff cutoff deformed oscillator and the occurrence of accidental pairwise degeneracy in the energy levels of the deformed system. As an application, we discuss the case of a trigonometric Rosen-Morse potential, which is successfully used in models for quantum confined systems, ranging from electrons in quantum dots to quarks in hadrons.

  5. Nonlinear dynamic macromodeling techniques for audio systems

    NASA Astrophysics Data System (ADS)

    Ogrodzki, Jan; Bieńkowski, Piotr

    2015-09-01

    This paper develops a modelling method and a models identification technique for the nonlinear dynamic audio systems. Identification is performed by means of a behavioral approach based on a polynomial approximation. This approach makes use of Discrete Fourier Transform and Harmonic Balance Method. A model of an audio system is first created and identified and then it is simulated in real time using an algorithm of low computational complexity. The algorithm consists in real time emulation of the system response rather than in simulation of the system itself. The proposed software is written in Python language using object oriented programming techniques. The code is optimized for a multithreads environment.

  6. Low level monitoring and control of nonlinear systems

    SciTech Connect

    Cover, A.; Reneke, J.; Lenhart, S.; Protopopescu, V.

    1994-12-31

    In this paper, we propose a nonparametric method for monitoring and controlling nonlinear systems whose dynamics is, in general, unknown or only partially known. Our nonparametric method is based on the stochastic linearization of the underlying (unknown) nonlinear system.

  7. Model reduction of systems with localized nonlinearities.

    SciTech Connect

    Segalman, Daniel Joseph

    2006-03-01

    An LDRD funded approach to development of reduced order models for systems with local nonlinearities is presented. This method is particularly useful for problems of structural dynamics, but has potential application in other fields. The key elements of this approach are (1) employment of eigen modes of a reference linear system, (2) incorporation of basis functions with an appropriate discontinuity at the location of the nonlinearity. Galerkin solution using the above combination of basis functions appears to capture the dynamics of the system with a small basis set. For problems involving small amplitude dynamics, the addition of discontinuous (joint) modes appears to capture the nonlinear mechanics correctly while preserving the modal form of the predictions. For problems involving large amplitude dynamics of realistic joint models (macro-slip), the use of appropriate joint modes along with sufficient basis eigen modes to capture the frequencies of the system greatly enhances convergence, though the modal nature the result is lost. Also observed is that when joint modes are used in conjunction with a small number of elastic eigen modes in problems of macro-slip of realistic joint models, the resulting predictions are very similar to those of the full solution when seen through a low pass filter. This has significance both in terms of greatly reducing the number of degrees of freedom of the problem and in terms of facilitating the use of much larger time steps.

  8. Robust H ∞ control of a nonlinear uncertain system via a stable nonlinear output feedback controller

    NASA Astrophysics Data System (ADS)

    Harno, Hendra G.; Petersen, Ian R.

    2011-04-01

    A new approach to solving a nonlinear robust H ∞ control problem using a stable nonlinear output feedback controller is presented in this article. The class of nonlinear uncertain systems being considered is characterised in terms of integral quadratic constraints and global Lipschitz conditions describing the admissible uncertainties and nonlinearities, respectively. The nonlinear controller is able to exploit the plant nonlinearities through the inclusion of a copy of the known plant nonlinearities in the controller. The H ∞ control objective is to obtain an absolutely stable closed-loop system with a specified disturbance attenuation level. The solution to this control problem involves stabilising solutions to parametrised algebraic Riccati equations. We apply a differential evolution algorithm to solve a non-convex nonlinear optimisation problem arising in the controller synthesis.

  9. Singularity perturbed zero dynamics of nonlinear systems

    NASA Technical Reports Server (NTRS)

    Isidori, A.; Sastry, S. S.; Kokotovic, P. V.; Byrnes, C. I.

    1992-01-01

    Stability properties of zero dynamics are among the crucial input-output properties of both linear and nonlinear systems. Unstable, or 'nonminimum phase', zero dynamics are a major obstacle to input-output linearization and high-gain designs. An analysis of the effects of regular perturbations in system equations on zero dynamics shows that whenever a perturbation decreases the system's relative degree, it manifests itself as a singular perturbation of zero dynamics. Conditions are given under which the zero dynamics evolve in two timescales characteristic of a standard singular perturbation form that allows a separate analysis of slow and fast parts of the zero dynamics.

  10. Approximations of nonlinear systems having outputs

    NASA Technical Reports Server (NTRS)

    Hunt, L. R.; Su, R.

    1985-01-01

    For a nonlinear system with output derivative x = f(x) and y = h(x), two types of linearizations about a point x(0) in state space are considered. One is the usual Taylor series approximation, and the other is defined by linearizing the appropriate Lie derivatives of the output with respect to f about x(0). The latter is called the obvservation model and appears to be quite natural for observation. It is noted that there is a coordinate system in which these two kinds of linearizations agree. In this coordinate system, a technique to construct an observer is introduced.

  11. An electromechanical Ising Hamiltonian

    PubMed Central

    Mahboob, Imran; Okamoto, Hajime; Yamaguchi, Hiroshi

    2016-01-01

    Solving intractable mathematical problems in simulators composed of atoms, ions, photons, or electrons has recently emerged as a subject of intense interest. We extend this concept to phonons that are localized in spectrally pure resonances in an electromechanical system that enables their interactions to be exquisitely fashioned via electrical means. We harness this platform to emulate the Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from the parametric resonances of multiple modes. The coupling between the mechanical spins is created by generating two-mode squeezed states, which impart correlations between modes that can imitate a random, ferromagnetic state or an antiferromagnetic state on demand. These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling. PMID:28861469

  12. [Investigation of transitions from order to chaos in dynamical systems]. Annual progress report

    SciTech Connect

    Not Available

    1990-12-31

    This report discusses: torus structure in higher dimensional hamiltonian systems; particle heating and stochastic web diffusion; scaling behavior of coupled conservative nonlinear systems; box counting algorithm and dimensional analysis of a pulsar; and universality of coupled nonlinear systems. (LSP)

  13. Chaotic Hamiltonian Dynamics.

    NASA Astrophysics Data System (ADS)

    Bialek, James Mark

    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 Fractal 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. Predictions of Universal Behavior have been based on the study of algebraic mappings. The problem we examine has a Hamiltonian given by H = p^2 over {2} - lambda over{2pi}sin(2pi x)sin(2pit). This Hamiltonian describes the motion of a compass needle in a sinusoidally varying magnetic field or, equally well, the one dimensional motion of a particle in a standing wave potential. By treating the magnitude(lambda ) of the time dependent potential as a parameter and by examining the trajectories of the system in a Poincare surface of section, the resulting differential

  14. Controllability of non-linear biochemical systems.

    PubMed

    Ervadi-Radhakrishnan, Anandhi; Voit, Eberhard O

    2005-07-01

    Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.

  15. Non-linear dynamic compensation system

    NASA Technical Reports Server (NTRS)

    Lin, Yu-Hwan (Inventor); Lurie, Boris J. (Inventor)

    1992-01-01

    A non-linear dynamic compensation subsystem is added in the feedback loop of a high precision optical mirror positioning control system to smoothly alter the control system response bandwidth from a relatively wide response bandwidth optimized for speed of control system response to a bandwidth sufficiently narrow to reduce position errors resulting from the quantization noise inherent in the inductosyn used to measure mirror position. The non-linear dynamic compensation system includes a limiter for limiting the error signal within preselected limits, a compensator for modifying the limiter output to achieve the reduced bandwidth response, and an adder for combining the modified error signal with the difference between the limited and unlimited error signals. The adder output is applied to control system motor so that the system response is optimized for accuracy when the error signal is within the preselected limits, optimized for speed of response when the error signal is substantially beyond the preselected limits and smoothly varied therebetween as the error signal approaches the preselected limits.

  16. Control of nonlinear time-varying systems

    NASA Technical Reports Server (NTRS)

    Hunt, L. R.; Su, R.

    1981-01-01

    Necessary and sufficient conditions are given for a time-varying nonlinear system of specific form to be transformed into a time-invariant controllable linear system. Since the present work will be in a neighborhood of the origin, it is unnecessary to name specific sets and it is assumed that all assumptions, conditions and results hold in an open set in the appropriate Euclidean space that contains the origin. This theory can be combined with the global inverse function theorems to produce global results.

  17. NONLINEAR TIDES IN CLOSE BINARY SYSTEMS

    SciTech Connect

    Weinberg, Nevin N.; Arras, Phil; Quataert, Eliot; Burkart, Josh

    2012-06-01

    We study the excitation and damping of tides in close binary systems, accounting for the leading-order nonlinear corrections to linear tidal theory. These nonlinear corrections include two distinct physical effects: three-mode nonlinear interactions, i.e., the redistribution of energy among stellar modes of oscillation, and nonlinear excitation of stellar normal modes by the time-varying gravitational potential of the companion. This paper, the first in a series, presents the formalism for studying nonlinear tides and studies the nonlinear stability of the linear tidal flow. Although the formalism we present is applicable to binaries containing stars, planets, and/or compact objects, we focus on non-rotating solar-type stars with stellar or planetary companions. Our primary results include the following: (1) The linear tidal solution almost universally used in studies of binary evolution is unstable over much of the parameter space in which it is employed. More specifically, resonantly excited internal gravity waves in solar-type stars are nonlinearly unstable to parametric resonance for companion masses M' {approx}> 10-100 M{sub Circled-Plus} at orbital periods P Almost-Equal-To 1-10 days. The nearly static 'equilibrium' tidal distortion is, however, stable to parametric resonance except for solar binaries with P {approx}< 2-5 days. (2) For companion masses larger than a few Jupiter masses, the dynamical tide causes short length scale waves to grow so rapidly that they must be treated as traveling waves, rather than standing waves. (3) We show that the global three-wave treatment of parametric instability typically used in the astrophysics literature does not yield the fastest-growing daughter modes or instability threshold in many cases. We find a form of parametric instability in which a single parent wave excites a very large number of daughter waves (N Almost-Equal-To 10{sup 3}[P/10 days] for a solar-type star) and drives them as a single coherent unit with

  18. SU(1,1) Lie Algebra Applied to the Time-Dependent Quadratic Hamiltonian System Perturbed by a Singularity.

    NASA Astrophysics Data System (ADS)

    Choi, Jeong Ryeol; Choi, Seong Soo

    We realized SU(1,1) Lie algebra in terms of the appropriate SU(1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU(1,1) Lie algebra. Various expectation values in two kinds of the generalized SU(1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola-Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU(1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU(1,1) coherent states for the CKOPS are very similar to the classical trajectory.

  19. Integrated analysis and design for controlled nonlinear multibody systems

    NASA Technical Reports Server (NTRS)

    Wu, Shih-Chin; Juang, Jer-Nan; Belvin, W. Keith; Woodard, Stanley E.

    1991-01-01

    A hybrid approach for integrated analysis and design of nonlinear multibody systems is proposed. Based on the approach, a general-purpose design tool is developed for nonlinear dynamic systems subjected to nonlinear design constraints. For analysis purposes, second-order nonlinear equations of motion of the system are automatically generated using general-purpose multibody formulations. Once the equations are solved, they are written in first-order form to take advantage of the first-order formulations of design sensitivity analysis for dynamic systems. A nonlinear programming technique is then used to optimize the nonlinear systems, such that a nonlinear cost function is minimized and performance constraints are satisfied. The approach proved to be very general and useful for design and analysis of controlled multibody systems. The approach is applied to the design of passive dynamic controllers for slewing control of multibody systems.

  20. Deterministic and stochastic responses of nonlinear systems

    NASA Astrophysics Data System (ADS)

    Abou-Rayan, Ashraf Mohamed

    The responses of nonlinear systems to both deterministic and stochastic excitations are discussed. For a single degree of freedom system, the response of a simply supported buckled beam to parametric excitations is investigated. Two types of excitations are examined: deterministic and random. For the nonlinear response to a harmonic axial load, the method of multiple scales is used to determine to second order the amplitude and phase modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large amplitude responses are investigated by using simulations on a digital computer and are compared with results obtained using an analog computer. For the stochastic response to a wide-band random excitation, the Gaussian and non-Gaussian closure schemes are used to determine the response statistics. The results are compared with those obtained from real-time analysis (analog-computer simulation). The normality assumption is examined. A comparison between the responses to deterministic and random excitation is presented. For two degree of freedom systems, two methods are used to study the response under the action of broad-band random excitations. The first method is applicable to systems having cubic nonlinearities. It involves an averaging approach to reduce the number of moment equations for the non-Gaussian closure scheme from 69 to 14 equations. The results are compared with those obtained from numerical integrations of the moment equations and the exact stationary solution of the Fokker-Planck-Komologorov equation. The second method is applicable to systems having quadratic and cubic nonlinearities. Stationary solutions of the moment equations are determined and their stability is ascertained by examining the

  1. Constants of motion for constrained Hamiltonian systems: A particle around a charged rotating black hole

    SciTech Connect

    Igata, Takahisa; Ishihara, Hideki; Koike, Tatsuhiko

    2011-03-15

    We discuss constants of motion of a particle under an external field in a curved spacetime, taking into account the Hamiltonian constraint, which arises from the reparametrization invariance of the particle orbit. As the necessary and sufficient condition for the existence of a constant of motion, we obtain a set of equations with a hierarchical structure, which is understood as a generalization of the Killing tensor equation. It is also a generalization of the conventional argument in that it includes the case when the conservation condition holds only on the constraint surface in the phase space. In that case, it is shown that the constant of motion is associated with a conformal Killing tensor. We apply the hierarchical equations and find constants of motion in the case of a charged particle in an electromagnetic field in black hole spacetimes. We also demonstrate that gravitational and electromagnetic fields exist in which a charged particle has a constant of motion associated with a conformal Killing tensor.

  2. Hamiltonian formulation of general relativity.

    NASA Astrophysics Data System (ADS)

    Teitelboim, Claudio

    The following sections are included: * INTRODUCTION * HAMILTONIAN FORMULATION OF GAUGE THEORIES (PRE-BRST) * BRST HAMILTONIAN FORMULATION OF GAUGE THEORIES * DYNAMICS OF GRAVITATIONAL FIELD * DOES THE HAMILTONIAN VANISH? GENERAL COVARIANCE AS AN "ORDINARY" GAUGE INVARIANCE * GENERALLY COVARIANT SYSTEMS * TIME AS A CANONICAL VARIABLE. ZERO HAMILTONIAN * Parametrized Systems * Zero Hamiltonian * Parametrization and Explicit Time Dependence * TIME REPARAMETRIZATION INVARIANCE * Form of Gauge Transformations * Must the Hamiltonian be Zero for a Generally Covariant System? * Simple Example of a Generally Covariant System with a Nonzero Hamiltonian * "TRUE DYNAMICS" VERSUS GAUGE TRANSFORMATIONS * Interpretation of the Formalism * Reduced Phase Space * MUST TIME FLOW? * GAUGE INDEPENDENCE OF PATH INTEGRAL FOR A PARAMETRIZED SYSTEM ILLUSTRATED. EQUIVALENCE OF THE GAUGES t = τ AND t = 0 * Reduced Phase Space Transition Amplitude as a Reduced Phase Space Path Integral * Canonical Gauge Conditions * Gauge t = 0 * Gauge t α τ * BRST CHARGE OF GRAVITATIONAL FIELD * ELEMENTS OF BRST THEORY * THE GHOST, YOU'VE COME A LONG WAY BABY * Introduction * Quantum mechanics, the art of finding and combining simple elementary processes * Ghosts necessary to keep elementary processes simple * BRST symmetry: ghosts and matter become different components of single geometrical object * BRST SYMMETRY IN CLASSICAL MECHANICS * Ghosts have role in classical mechanics * Gauge invariance and constraints * Classical mechanics over Grassmann algebra necessary * Higher order structure functions * Rank defined. Open algebras * Ghosts. Ghost number. BRST generator as generating function for structure functions * A belianization of constraints. Existence of Ω * Uniqueness of Ω * Classical BRST cohomology * QUANTUM BRST THEORY * States and operators * Ghost number * BRST invariant states * Quantum BRST cohomology * Equivalence of the BRST physical subspace with the conventional gauge

  3. Global state feedback stabilisation of nonlinear systems with high-order and low-order nonlinearities

    NASA Astrophysics Data System (ADS)

    Zhang, Xing-Hui; Xie, Xue-Jun

    2014-03-01

    This paper studies the state feedback control problem for a class of nonlinear systems with high-order and low-order nonlinearities. The introduction of the sign function together with the method of adding a power integrator and Lyapunov stability theorem makes the closed-loop system globally asymptotically stable. Exploiting the idea of how to deal with growth nonlinearities with both high order and low order being relaxed to some intervals is the focus of this work.

  4. Hamiltonian decomposition for bulk and surface states.

    PubMed

    Sasaki, Ken-Ichi; Shimomura, Yuji; Takane, Yositake; Wakabayashi, Katsunori

    2009-04-10

    We demonstrate that a tight-binding Hamiltonian with nearest- and next-nearest-neighbor hopping integrals can be decomposed into bulk and boundary parts for honeycomb lattice systems. The Hamiltonian decomposition reveals that next-nearest-neighbor hopping causes sizable changes in the energy spectrum of surface states even if the correction to the energy spectrum of bulk states is negligible. By applying the Hamiltonian decomposition to edge states in graphene systems, we show that the next-nearest-neighbor hopping stabilizes the edge states. The application of Hamiltonian decomposition to a general lattice system is discussed.

  5. Observability and Information Structure of Nonlinear Systems,

    DTIC Science & Technology

    1985-10-01

    defined by Shannon and used as a measure of mut.:al infor-mation between event x. and y4. If p(x.l IY.) I I(x., y.) xil -in (1/p(x.)) =- JInp (x.) (2...entropy H(x,y) in a similar way as H(x,y) = - fx,yp(xiy)lnp(x,y)cdlY, = -E[ JInp (x,y)]. (3-13) With the above definitions, mutual information between x...Observabiity of Nonlinear Systems, Eng. Cybernetics, Volume 1, pp 338-345, 1972. 18. Sen , P., Chidambara, M.R., Observability of a Class of Nonli-.ear

  6. Feedback nonlinear discrete-time systems

    NASA Astrophysics Data System (ADS)

    Yu, Miao; Wang, Jiasen; Qi, Donglian

    2014-11-01

    In this paper, we design an adaptive iterative learning control method for a class of high-order nonlinear output feedback discrete-time systems with random initial conditions and iteration-varying desired trajectories. An n-step ahead predictor approach is employed to estimate future outputs. The discrete Nussbaum gain method is incorporated into the control design to deal with unknown control directions. The proposed control algorithm ensures that the tracking error converges to zero asymptotically along the iterative learning axis except for the beginning outputs affected by random initial conditions. A numerical simulation is carried out to demonstrate the efficacy of the presented control laws.

  7. Particle systems and nonlinear Landau damping

    SciTech Connect

    Villani, Cédric

    2014-03-15

    Some works dealing with the long-time behavior of interacting particle systems are reviewed and put into perspective, with focus on the classical Kolmogorov–Arnold–Moser theory and recent results of Landau damping in the nonlinear perturbative regime, obtained in collaboration with Clément Mouhot. Analogies are discussed, as well as new qualitative insights in the theory. Finally, the connection with a more recent work on the inviscid Landau damping near the Couette shear flow, by Bedrossian and Masmoudi, is briefly discussed.

  8. Design of suboptimal regulators for nonlinear systems

    NASA Technical Reports Server (NTRS)

    Balaram, J.; Saridis, G. N.

    1985-01-01

    An optimal feedback control law is preferred for the regulation of a deterministic nonlinear system. In this paper, a practical, iterative design method leading to a sequence of suboptimal control laws with successively improved performance is presented. The design method requires the determination of an upper bound to the performance of each successive control law. This is obtained by solving a partial differential inequality by means of a linear programming technique. Robustness properties and the application of the design method to the control of a robot manipulator arm are also presented.

  9. Particle systems and nonlinear Landau dampinga)

    NASA Astrophysics Data System (ADS)

    Villani, Cédric

    2014-03-01

    Some works dealing with the long-time behavior of interacting particle systems are reviewed and put into perspective, with focus on the classical Kolmogorov-Arnold-Moser theory and recent results of Landau damping in the nonlinear perturbative regime, obtained in collaboration with Clément Mouhot. Analogies are discussed, as well as new qualitative insights in the theory. Finally, the connection with a more recent work on the inviscid Landau damping near the Couette shear flow, by Bedrossian and Masmoudi, is briefly discussed.

  10. Lie superbialgebra structures on the Lie superalgebra (C3+A) and deformation of related integrable Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Eghbali, A.; Rezaei-Aghdam, A.

    2017-06-01

    Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra (C3+A ) are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra (C3+A ) , the Lie superbialgebra structures on the Lie superalgebra (C3+A ) are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras (C3+A ) are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra Uλ (Cp=1 2 ,𝜖⊕A1 ,1 )(C3+A).

  11. Output feedback adaptive fuzzy control of uncertain MIMO nonlinear systems with unknown input nonlinearities.

    PubMed

    Shahnazi, Reza

    2015-01-01

    An adaptive fuzzy output feedback controller is proposed for a class of uncertain MIMO nonlinear systems with unknown input nonlinearities. The input nonlinearities can be backlash-like hysteresis or dead-zone. Besides, the gains of unknown input nonlinearities are unknown nonlinear functions. Based on universal approximation theorem, the unknown nonlinear functions are approximated by fuzzy systems. The proposed method does not need the availability of the states and an observer based on strictly positive real (SPR) theory is designed to estimate the states. An adaptive robust structure is used to cope with fuzzy approximation error and external disturbances. The semi-global asymptotic stability of the closed-loop system is guaranteed via Lyapunov approach. The applicability of the proposed method is also shown via simulations. Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.

  12. The construction of arbitrary order ERKN methods based on group theory for solving oscillatory Hamiltonian systems with applications

    NASA Astrophysics Data System (ADS)

    Mei, Lijie; Wu, Xinyuan

    2016-10-01

    In general, extended Runge-Kutta-Nyström (ERKN) methods are more effective than traditional Runge-Kutta-Nyström (RKN) methods in dealing with oscillatory Hamiltonian systems. However, the theoretical analysis for ERKN methods, such as the order conditions, the symplectic conditions and the symmetric conditions, becomes much more complicated than that for RKN methods. Therefore, it is a bottleneck to construct high-order ERKN methods efficiently. In this paper, we first establish the ERKN group Ω for ERKN methods and the RKN group G for RKN methods, respectively. We then rigorously show that ERKN methods are a natural extension of RKN methods, that is, there exists an epimorphism η of the ERKN group Ω onto the RKN group G. This epimorphism gives a global insight into the structure of the ERKN group by the analysis of its kernel and the corresponding RKN group G. Meanwhile, we establish a particular mapping φ of G into Ω so that each image element is an ideal representative element of the congruence class in Ω. Furthermore, an elementary theoretical analysis shows that this map φ can preserve many structure-preserving properties, such as the order, the symmetry and the symplecticity. From the epimorphism η together with its section φ, we may gain knowledge about the structure of the ERKN group Ω via the RKN group G. In light of the theoretical analysis of this paper, we obtain high-order structure-preserving ERKN methods in an effective way for solving oscillatory Hamiltonian systems. Numerical experiments are carried out and the results are very promising, which strongly support our theoretical analysis presented in this paper.

  13. Parameter identification for nonlinear aerodynamic systems

    NASA Technical Reports Server (NTRS)

    Pearson, Allan E.

    1990-01-01

    Parameter identification for nonlinear aerodynamic systems is examined. It is presumed that the underlying model can be arranged into an input/output (I/O) differential operator equation of a generic form. The algorithm estimation is especially efficient since the equation error can be integrated exactly given any I/O pair to obtain an algebraic function of the parameters. The algorithm for parameter identification was extended to the order determination problem for linear differential system. The degeneracy in a least squares estimate caused by feedback was addressed. A method of frequency analysis for determining the transfer function G(j omega) from transient I/O data was formulated using complex valued Fourier based modulating functions in contrast with the trigonometric modulating functions for the parameter estimation problem. A simulation result of applying the algorithm is given under noise-free conditions for a system with a low pass transfer function.

  14. Impulse position control algorithms for nonlinear systems

    SciTech Connect

    Sesekin, A. N.; Nepp, A. N.

    2015-11-30

    The article is devoted to the formalization and description of impulse-sliding regime in nonlinear dynamical systems that arise in the application of impulse position controls of a special kind. The concept of trajectory impulse-sliding regime formalized as some limiting network element Euler polygons generated by a discrete approximation of the impulse position control This paper differs from the previously published papers in that it uses a definition of solutions of systems with impulse controls, it based on the closure of the set of smooth solutions in the space of functions of bounded variation. The need for the study of such regimes is the fact that they often arise when parry disturbances acting on technical or economic control system.

  15. Impulse position control algorithms for nonlinear systems

    NASA Astrophysics Data System (ADS)

    Sesekin, A. N.; Nepp, A. N.

    2015-11-01

    The article is devoted to the formalization and description of impulse-sliding regime in nonlinear dynamical systems that arise in the application of impulse position controls of a special kind. The concept of trajectory impulse-sliding regime formalized as some limiting network element Euler polygons generated by a discrete approximation of the impulse position control This paper differs from the previously published papers in that it uses a definition of solutions of systems with impulse controls, it based on the closure of the set of smooth solutions in the space of functions of bounded variation. The need for the study of such regimes is the fact that they often arise when parry disturbances acting on technical or economic control system.

  16. Hamiltonian thermostats fail to promote heat flow

    NASA Astrophysics Data System (ADS)

    Hoover, Wm. G.; Hoover, Carol G.

    2013-12-01

    Hamiltonian mechanics can be used to constrain temperature simultaneously with energy. We illustrate the interesting situations that develop when two different temperatures are imposed within a composite Hamiltonian system. The model systems we treat are ϕ4 chains, with quartic tethers and quadratic nearest-neighbor Hooke's-law interactions. This model is known to satisfy Fourier's law. Our prototypical problem sandwiches a Newtonian subsystem between hot and cold Hamiltonian reservoir regions. We have characterized four different Hamiltonian reservoir types. There is no tendency for any of these two-temperature Hamiltonian simulations to transfer heat from the hot to the cold degrees of freedom. Evidently steady heat flow simulations require energy sources and sinks, and are therefore incompatible with Hamiltonian mechanics.

  17. Double nonlinear resonance in ferromagnets and other dynamic systems

    NASA Astrophysics Data System (ADS)

    Bakai, A. S.

    2010-08-01

    The phenomenon of double nonlinear resonances in nonlinear oscillators of general type is described. The results are used to describe a double nonlinear ferromagnetic resonance in a uniaxial ferromagnet. The possibility of a similar resonance in the system of brain biocurrents is considered.

  18. Nonlinear resonance. [Univ. of Washington

    SciTech Connect

    Not Available

    1993-01-01

    A brief summary of progress is given in the following areas: sustained resonance in non-Hamiltonian systems, two simultaneous sustained resonances, bursting oscillators, and the interaction of a strong shock with weak disturbances. Work will encompass linearly unstable, weakly nonlinear waves.

  19. Consensus Tracking for Multiagent Systems with Nonlinear Dynamics

    PubMed Central

    2014-01-01

    This paper concerns the problem of consensus tracking for multiagent systems with a dynamical leader. In particular, it proposes the corresponding explicit control laws for multiple first-order nonlinear systems, second-order nonlinear systems, and quite general nonlinear systems based on the leader-follower and the tree shaped network topologies. Several numerical simulations are given to verify the theoretical results. PMID:25197689

  20. Nonlinear Behavior in Optical and Other Systems

    DTIC Science & Technology

    1987-09-01

    numerical analysis). Others will be devoted to ’state of the art ’ discussions of specific problems (e.g. nonlinear waveguides, Anderson localization). It is...Nonlinearity and Statistical Physics. Approximate Cost of Workshop: $5,312. STATE OF THE ART DEVELOPMfENTS IN NONLINEAR OPTICS Organizers: J. Moloney, A... Art Developments in Nonlinear Optics V. List of Preprints and Reprints with Abstracts ANTICIPATED WORKSHOPS 1987 - 1988 I. Workshop on Singularities

  1. Passive dynamic controllers for nonlinear mechanical systems

    NASA Technical Reports Server (NTRS)

    Juang, Jer-Nan; Wu, Shih-Chin; Phan, Minh; Longman, Richard W.

    1991-01-01

    A methodology for model-independant controller design for controlling large angular motion of multi-body dynamic systems is outlined. The controlled system may consist of rigid and flexible components that undergo large rigid body motion and small elastic deformations. Control forces/torques are applied to drive the system and at the same time suppress the vibration due to flexibility of the components. The proposed controller consists of passive second-order systems which may be designed with little knowledge of the system parameter, even if the controlled system is nonlinear. Under rather general assumptions, the passive design assures that the closed loop system has guaranteed stability properties. Unlike positive real controller design, stabilization can be accomplished without direct velocity feedback. In addition, the second-order passive design allows dynamic feedback controllers with considerable freedom to tune for desired system response, and to avoid actuator saturation. After developing the basic mathematical formulation of the design methodology, simulation results are presented to illustrate the proposed approach to a flexible six-degree-of-freedom manipulator.

  2. Lowest eigenvalues of random Hamiltonians

    SciTech Connect

    Shen, J. J.; Zhao, Y. M.; Arima, A.; Yoshinaga, N.

    2008-05-15

    In this article we study the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues is applicable to many different systems. We improve the accuracy of the formula by considering the third central moment. We show that these formulas are applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.

  3. Adiabatic elimination in nonlinear dynamical systems

    NASA Astrophysics Data System (ADS)

    Lugiato, L. A.; Mandel, P.; Narducci, L. M.

    1984-03-01

    The problem of the adiabatic elimination of selected dynamical variables in the description of nonlinear systems is reconsidered, with emphasis on the identification of suitable criteria for the global validity of this procedure. The problem is analyzed in detail using as a guideline the one-mode homogeneously broadened laser model, with an injected signal and an arbitrary population difference for added flexibility. Five conditions for the global validity of the adiabatic limit are proposed, after consideration not only of the relative size of the time scales involved, but also of the magnitude of all parameters, of the physical variables, and of their fluctuations. From the analysis, it is considered evident that the main conclusions are model independent and not at all restricted to the specific features of the dynamical system selected as a test case.

  4. Nonlinear dynamic analysis of flexible multibody systems

    NASA Technical Reports Server (NTRS)

    Bauchau, Olivier A.; Kang, Nam Kook

    1991-01-01

    Two approaches are developed to analyze the dynamic behavior of flexible multibody systems. In the first approach each body is modeled with a modal methodology in a local non-inertial frame of reference, whereas in the second approach, each body is modeled with a finite element methodology in the inertial frame. In both cases, the interaction among the various elastic bodies is represented by constraint equations. The two approaches were compared for accuracy and efficiency: the first approach is preferable when the nonlinearities are not too strong but it becomes cumbersome and expensive to use when many modes must be used. The second approach is more general and easier to implement but could result in high computation costs for a large system. The constraints should be enforced in a time derivative fashion for better accuracy and stability.

  5. Constrained tracking control for nonlinear systems.

    PubMed

    Khani, Fatemeh; Haeri, Mohammad

    2017-09-01

    This paper proposes a tracking control strategy for nonlinear systems without needing a prior knowledge of the reference trajectory. The proposed method consists of a set of local controllers with appropriate overlaps in their stability regions and an on-line switching strategy which implements these controllers and uses some augmented intermediate controllers to ensure steering the system states to the desired set points without needing to redesign the controller for each value of set point changes. The proposed approach provides smooth transient responses despite switching among the local controllers. It should be mentioned that the stability regions of the proposed controllers could be estimated off-line for a range of set-point changes. The efficiencies of the proposed algorithm are illustrated via two example simulations. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

  6. A constructive approach for nonlinear system identification using multilayer perceptrons.

    PubMed

    Choi, J Y; Van Landingham, H F; Bingulac, S

    1996-01-01

    This paper combines a conventional method of multivariable system identification with a dynamic multi-layer perceptron (MLP) to achieve a constructive method of nonlinear system identification. The class of nonlinear systems is assumed to operate nominally around an equilibrium point in the neighborhood of which a linearized model exists to represent the system, although normal operation is not limited to the linear region. The result is an accurate discrete-time nonlinear model, extended from a MIMO linear model, which captures the nonlinear behavior of the system.

  7. Application of nonlinear time series models to driven systems

    SciTech Connect

    Hunter, N.F. Jr.

    1990-01-01

    In our laboratory we have been engaged in an effort to model nonlinear systems using time series methods. Our objectives have been, first, to understand how the time series response of a nonlinear system unfolds as a function of the underlying state variables, second, to model the evolution of the state variables, and finally, to predict nonlinear system responses. We hope to address the relationship between model parameters and system parameters in the near future. Control of nonlinear systems based on experimentally derived parameters is also a planned topic of future research. 28 refs., 15 figs., 2 tabs.

  8. On state representations of nonlinear implicit systems

    NASA Astrophysics Data System (ADS)

    Pereira da Silva, Paulo Sergio; Batista, Simone

    2010-03-01

    This work considers a semi-implicit system Δ, that is, a pair (S, y), where S is an explicit system described by a state representation ? , where x(t) ∈ ℝ n and u(t) ∈ ℝ m , which is subject to a set of algebraic constraints y(t) = h(t, x(t), u(t)) = 0, where y(t) ∈ ℝ l . An input candidate is a set of functions v = (v 1, …, v s ), which may depend on time t, on x, and on u and its derivatives up to a finite order. The problem of finding a (local) proper state representation ż = g(t, z, v) with input v for the implicit system Δ is studied in this article. The main result shows necessary and sufficient conditions for the solution of this problem, under mild assumptions on the class of admissible state representations of Δ. These solvability conditions rely on an integrability test that is computed from the explicit system S. The approach of this article is the infinite-dimensional differential geometric setting of Fliess, Lévine, Martin, and Rouchon (1999) ('A Lie-Bäcklund Approach to Equivalence and Flatness of Nonlinear Systems', IEEE Transactions on Automatic Control, 44(5), (922-937)).

  9. Asymmetric Heat Conduction in Nonlinear Systems

    NASA Astrophysics Data System (ADS)

    Hu, Bambi

    2008-12-01

    Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name two hundred years ago, a first-principle derivation of this law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are still far from clear. In this talk I'll give a review of recent works done on this subject. I'll also report our latest work on asymmetric heat conduction in nonlinear systems. The study of heat condution is not only of theoretical interest but also of practical interest. The study of electric conduction has led to the invention of such important electric devices such as electric diodes and transistors. The study of heat conduction may also lead to the invention of thermal diodes and transistors in the future. Note from Publisher: This article contains the abstract only.

  10. Bifurcations and Patterns in Nonlinear Dissipative Systems

    SciTech Connect

    Guenter Ahlers

    2005-05-27

    This project consists of experimental investigations of heat transport, pattern formation, and bifurcation phenomena in non-linear non-equilibrium fluid-mechanical systems. These issues are studies in Rayleigh-B\\'enard convection, using both pure and multicomponent fluids. They are of fundamental scientific interest, but also play an important role in engineering, materials science, ecology, meteorology, geophysics, and astrophysics. For instance, various forms of convection are important in such diverse phenomena as crystal growth from a melt with or without impurities, energy production in solar ponds, flow in the earth's mantle and outer core, geo-thermal stratifications, and various oceanographic and atmospheric phenomena. Our work utilizes computer-enhanced shadowgraph imaging of flow patterns, sophisticated digital image analysis, and high-resolution heat transport measurements.

  11. Hamiltonian of a many-electron system with single-electron and electron-pair states in a two-dimensional periodic potential

    NASA Astrophysics Data System (ADS)

    Hai, Guo-Qiang; Peeters, François M.

    2015-01-01

    Based on the metastable electron-pair energy band in a two-dimensional (2D) periodic potential obtained previously by Hai and Castelano [J. Phys.: Condens. Matter 26, 115502 (2014)], we present in this work a Hamiltonian of many electrons consisting of single electrons and electron pairs in the 2D system. The electron-pair states are metastable of energies higher than those of the single-electron states at low electron density. We assume two different scenarios for the single-electron band. When it is considered as the lowest conduction band of a crystal, we compare the obtained Hamiltonian with the phenomenological model Hamiltonian of a boson-fermion mixture proposed by Friedberg and Lee [Phys. Rev. B 40, 6745 (1989)]. Single-electron-electron-pair and electron-pair-electron-pair interaction terms appear in our Hamiltonian and the interaction potentials can be determined from the electron-electron Coulomb interactions. When we consider the single-electron band as the highest valence band of a crystal, we show that holes in this valence band are important for stabilization of the electron-pair states in the system.

  12. Nondivergent and negative susceptibilities around critical points of a long-range Hamiltonian system with two order parameters

    NASA Astrophysics Data System (ADS)

    Yamaguchi, Yoshiyuki Y.; Sawai, Daiki

    2017-05-01

    The linear response is investigated in a long-range Hamiltonian system from the viewpoint of dynamics, which is described by the Vlasov equation in the large-population limit. Because of the existence of the Casimir invariants of the Vlasov dynamics, an external field does not drive the system to the forced thermal equilibrium in general, and the linear response is suppressed. With the aid of a linear response theory based on the Vlasov dynamics, we compute the suppressed linear response in a system having two order parameters, which introduce the conjugate two external fields and the susceptibility matrix of size 2 accordingly. Moreover, the two order parameters bring three phases and there are three types of second-order phase transitions between them. For each type of phase transition, all the critical exponents for elements of the susceptibility matrix are computed. The critical exponents reveal that some elements of the matrices do not diverge even at critical points, while the mean-field theory predicts divergences. The linear response theory also suggests the appearance of negative off-diagonal elements; in other words, an applied external field decreases the value of an order parameter. These theoretical predictions are confirmed by direct numerical simulations of the Vlasov equation.

  13. Chaos and Order in Weakly Coupled Systems of Nonlinear Oscillators

    NASA Astrophysics Data System (ADS)

    Bruhn, B.

    1987-01-01

    We consider in this paper perturbations of two degree of freedom Hamiltonian systems which contain periodic and heteroclinic orbits. The Melnikov-Keener condition is used to proof the existence of horseshoes in the dynamics. The same condition is applied to prove a high degree of order in the motion of the swinging Atwood's machine. For some selected parameter values the theoretical predictions are checked by numerical calculations.

  14. Nonlinear identification of MDOF systems using Volterra series approximation

    NASA Astrophysics Data System (ADS)

    Prawin, J.; Rao, A. Rama Mohan

    2017-02-01

    Most of the practical engineering structures exhibit nonlinearity due to nonlinear dynamic characteristics of structural joints, nonlinear boundary conditions and nonlinear material properties. Meanwhile, the presence of non-linearity in the system can lead to a wide range of structural behavior, for example, jumps, limit cycles, internal resonances, modal coupling, super and sub-harmonic resonances, etc. In this paper, we present a Volterra series approximation approach based on the adaptive filter concept for nonlinear identification of multi-degree of freedom systems, without sacrificing the benefits associated with the traditional Volterra series approach. The effectiveness of the proposed approach is demonstrated using two classical single degrees of freedom systems (breathing crack problem and Duffing Holmes oscillator) and later we extend to multi-degree of freedom systems.

  15. Adaptive control for a class of second-order nonlinear systems with unknown input nonlinearities.

    PubMed

    Zhang, T; Guay, M

    2003-01-01

    An adaptive controller is developed for a class of second-order nonlinear dynamic systems with input nonlinearities using artificial neural networks (ANN). The unknown input nonlinearities are continuous and monotone and satisfy a sector constraint. In contrast to conventional Lyapunov-based design techniques, an alternative Lyapunov function, which depends on both system states and control input variable, is used for the development of a control law and a learning algorithm. The proposed adaptive controller guarantees the stability of the closed-loop system and convergence of the output tracking error to an adjustable neighbour of the origin.

  16. Nonlinear Programming for Large, Sparse Systems

    DTIC Science & Technology

    1976-08-01

    REFERENCES [1] J. Abadie, "Application of the GRG algorithm to optimal control problems," in: J. Abadie, ed., Integer and nonlinear programming (North... Optimization Laboratory Department of Operations DDG OC 13 1976 St an f or d 143qt] : 1University - 0 Stanford California 94305 NONLINEAR PROGRAMVING FOR...characterized by having n-m "nonbasic" variables equal to their upper or lower bound. With nonlinear problems we cannot expect an optimal solution to

  17. Nonlinear state estimation and feedback control of nonlinear and bilinear distributed parameter systems

    NASA Technical Reports Server (NTRS)

    Balas, M. J.

    1980-01-01

    This paper presents a theory of nonlinear state observers for nonlinear and bilinear distributed parameter systems. Convergence results are proved for these observers. Linear feedback control derived from such state observers is applied to the distributed parameter system and conditions are presented for closed-loop stability. The emphasis is on finite dimensional state observers and controllers (which can be implemented with on-line computers) and conditions for their successful operation with infinite dimensional distributed parameter systems.

  18. Friction in a Model of Hamiltonian Dynamics

    NASA Astrophysics Data System (ADS)

    Fröhlich, Jürg; Gang, Zhou; Soffer, Avy

    2012-10-01

    We study the motion of a heavy tracer particle weakly coupled to a dense ideal Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations describing a process of emission of Cerenkov radiation of sound waves into the Bose-Einstein condensate along the particle's trajectory. The emission of Cerenkov radiation results in a friction force with memory acting on the tracer particle and causing it to decelerate until it comes to rest. "A moving body will come to rest as soon as the force pushing it no longer acts on it in the manner necessary for its propulsion."—— Aristotle

  19. Nonlinear behavior in small neural systems

    NASA Astrophysics Data System (ADS)

    Wheeler, Diek Winters

    This work addresses the nonlinear behavior of one or two model neurons under the influence of different stimuli, whether they be forms of chaos control or varieties of added noise. This is a step towards the ultimate objective of exploring the notion that a neural system might utilize a mechanism such as a memory-searching chaotic attractor to locate and retrieve stable-memory limit cycles. The biological realism of the Hopfield neuron models is discussed, and the concept of an ``effective'' neuron is introduced. The dynamical effects of adding inertial/inductance terms to an effective-neuron system are presented along with arguments for the biological relevance of such terms. A two neuron system with one or two inertial terms added is shown to exhibit chaos. The chaos is confirmed by Lyapunov exponents, power spectra, and phase-space plots. The effects of multiplicative and additive noise on the dynamics of a two effective-neuron system are investigated. One of the neurons possesses an added inertial term so the system is able to generate chaotic dynamics. The multiplicative noise is added to the connection parameter J 11, and the additive noise is added to the equation for U 2 like an external driving force. Using J11 as a bifurcation parameter, the system is examined as it passes from limit cycle dynamics to chaotic dynamics. Both types of noise are found to lower the bifurcation point with respect to its deterministic value, and both cause the dynamics to expand in phase space. For equivalent levels of noise, additive noise is found to have a stronger effect on the dynamics than multiplicative noise. The bifurcation points are explored by means of ensembles of the largest Lyapunov exponents derived from the stochastic dynamics. A brief overview is presented of the current state of control theory in chaotic systems. One control method, Hübler's [74] technique of using aperiodic forces to drive nonlinear oscillators to resonance, is analyzed. The technique is

  20. Multiple Time-Step Dual-Hamiltonian Hybrid Molecular Dynamics — Monte Carlo Canonical Propagation Algorithm

    PubMed Central

    Weare, Jonathan; Dinner, Aaron R.; Roux, Benoît

    2016-01-01

    A multiple time-step integrator based on a dual Hamiltonian and a hybrid method combining molecular dynamics (MD) and Monte Carlo (MC) is proposed to sample systems in the canonical ensemble. The Dual Hamiltonian Multiple Time-Step (DHMTS) algorithm is based on two similar Hamiltonians: a computationally expensive one that serves as a reference and a computationally inexpensive one to which the workload is shifted. The central assumption is that the difference between the two Hamiltonians is slowly varying. Earlier work has shown that such dual Hamiltonian multiple time-step schemes effectively precondition nonlinear differential equations for dynamics by reformulating them into a recursive root finding problem that can be solved by propagating a correction term through an internal loop, analogous to RESPA. Of special interest in the present context, a hybrid MD-MC version of the DHMTS algorithm is introduced to enforce detailed balance via a Metropolis acceptance criterion and ensure consistency with the Boltzmann distribution. The Metropolis criterion suppresses the discretization errors normally associated with the propagation according to the computationally inexpensive Hamiltonian, treating the discretization error as an external work. Illustrative tests are carried out to demonstrate the effectiveness of the method. PMID:26918826

  1. Hamiltonians defined by biorthogonal sets

    NASA Astrophysics Data System (ADS)

    Bagarello, Fabio; Bellomonte, Giorgia

    2017-04-01

    In some recent papers, studies on biorthogonal Riesz bases have found renewed motivation because of their connection with pseudo-Hermitian quantum mechanics, which deals with physical systems described by Hamiltonians that are not self-adjoint but may still have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed in some previous papers. However, in many physical models, one has to deal not with orthonormal bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of G -quasi basis, and we show a series of conditions under which a definition of non-self-adjoint Hamiltonian with purely point real spectra is still possible.

  2. Generalized James' effective Hamiltonian method

    NASA Astrophysics Data System (ADS)

    Shao, Wenjun; Wu, Chunfeng; Feng, Xun-Li

    2017-03-01

    James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method only corresponds to the second-order perturbation theory and cannot be exploited to treat problems which should be solved by using the third- or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two recently published examples [L. Garziano et al., Phys. Rev. Lett. 117, 043601 (2016), 10.1103/PhysRevLett.117.043601; Ken K. W. Ma and C. K. Law, Phys. Rev. A 92, 023842 (2015), 10.1103/PhysRevA.92.023842]; our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method, respectively. For some specific problems, this method can simplify the calculating procedure and the resultant effective Hamiltonian is more general.

  3. Collective Hamiltonian for chiral modes

    NASA Astrophysics Data System (ADS)

    Chen, Q. B.; Zhang, S. Q.; Zhao, P. W.; Jolos, R. V.; Meng, J.

    2013-02-01

    A collective model is proposed to describe the chiral rotation and vibration and applied to a system with one h11/2 proton particle and one h11/2 neutron hole coupled to a triaxial rigid rotor. The collective Hamiltonian is constructed from the potential energy and mass parameter obtained in the tilted axis cranking approach. By diagonalizing the collective Hamiltonian with a box boundary condition, it is found that for the chiral rotation, the partner states become more degenerate with the increase of the cranking frequency, and for the chiral vibrations, their important roles for the collective excitation are revealed at the beginning of the chiral rotation region.

  4. Bohr Hamiltonian with time-dependent potential

    NASA Astrophysics Data System (ADS)

    Naderi, L.; Hassanabadi, H.; Sobhani, H.

    2016-04-01

    In this paper, Bohr Hamiltonian has been studied with the time-dependent potential. Using the Lewis-Riesenfeld dynamical invariant method appropriate dynamical invariant for this Hamiltonian has been constructed and the exact time-dependent wave functions of such a system have been derived due to this dynamical invariant.

  5. Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay-Turbiner-Winternitz (TTW) and the Post-Winternitz (PW) systems

    NASA Astrophysics Data System (ADS)

    Rañada, Manuel F.

    2017-08-01

    The existence of quasi-bi-Hamiltonian structures for the Tremblay-Turbiner-Winternitz (TTW) and the Post-Winternitz (PW) systems is studied. We first recall that the superintegrability of these two systems is related with the existence of certain complex functions endowed with interesting Poisson bracket properties, and then we prove the existence of several quasi-bi-Hamiltonian structures making use of these complex functions. This is done in two steps: first with complex 2-forms (wedge product of the differentials of the complex functions) and then with several real 2-forms. The properties of these geometric structures and the associated recursion operators are also analyzed. The paper can be considered as divided in two parts. In the first part a we study the TTW system (related to the harmonic oscillator) and in the second part we study the PW system (related to the Kepler problem).

  6. Spline approximations for nonlinear hereditary control systems

    NASA Technical Reports Server (NTRS)

    Daniel, P. L.

    1982-01-01

    A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.

  7. Nonlinear dynamic analysis for coupled vehicle-bridge vibration system on nonlinear foundation

    NASA Astrophysics Data System (ADS)

    Zhou, Shihua; Song, Guiqiu; Wang, Rongpeng; Ren, Zhaohui; Wen, Bangchun

    2017-03-01

    In this paper, the nonlinear dynamics of a parametrically excited coupled vehicle-bridge vibration system (CVBVS) is investigated, and the coupled system is subjected to a time-dependent transverse load including a constant value together with a harmonic time-variant component. The dynamic equations of the CVBVS are established by using the generalized Lagrange's equation. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the continuous governing equation. The influences of parametric excitation with nonlinear support stiffness, mass ratio, excitation amplitude and position relation on the dynamic behaviors are studied for the interaction between vehicle and the bridge. The analysis results indicate that the nonlinear dynamic characteristics are strongly attributed to the interaction of the coupled system. Nonlinear support stiffness of foundation and mass ratio can lead to complex dynamic behaviors such as jump discontinuous phenomenon, periodic, quasi-periodic and chaotic motions. Vibration amplitude increases depending on the position, where the maximum vibration displacement does not occur at the center of the bridge. The excitation amplitude has an obvious influence on the nonlinear dynamic behaviors and the increase of the excitation amplitude makes the vibration strengthen. The bifurcation diagram and 3-D frequency spectrum are used to analyze the complex nonlinear dynamic behaviors of the CVBVS. The presented results can provide an insight to the understanding of the vibration characteristics of the coupled vehicle-bridge vibration system in engineering.

  8. Analysing periodicity, nonlinearity and transitional characteristics of nonlinear dynamic systems with Periodicity Ratio (PR)

    NASA Astrophysics Data System (ADS)

    Dai, L.; Han, L.

    2011-12-01

    The multiple-periodicity, nonlinearity and transitional characteristics of nonlinear dynamic systems subjected to external excitations are studied in this research. Diagnoses of the number and changing multiple-periodicities of Duffing's systems are performed with implementation of the Periodicity Ratio (PR). The multiple-periodicity diagram is generated such that the periodicities and nonlinearity of the systems with respect to the system parameters can be graphically studied. The stability and convergence of the systems are investigated. The results of the research show that the number of period of periodicity of the systems increases continuously when certain system parameters increase. Transitional characteristics of the systems are also investigated. Both Lyapunov Exponents and Periodicity Ratio are implemented to diagnose the transitional routes of the systems. New symmetrical transition characters from periodicity to quasi-periodicity and chaos are displayed in terms of PR values. Comparing to Lyapunov Exponents, the Periodicity Ratio discloses more detailed and accurate transition information.

  9. Nonlinear phase noise in coherent optical OFDM transmission systems.

    PubMed

    Zhu, Xianming; Kumar, Shiva

    2010-03-29

    We derive an analytical formula to estimate the variance of nonlinear phase noise caused by the interaction of amplified spontaneous emission (ASE) noise with fiber nonlinearity such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) in coherent orthogonal frequency division multiplexing (OFDM) systems. The analytical results agree very well with numerical simulations, enabling the study of the nonlinear penalties in long-haul coherent OFDM systems without extensive numerical simulation. Our results show that the nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM, which is in contrast to traditional WDM systems where ASE-FWM interaction is negligible in quasi-linear systems. We also found that fiber chromatic dispersion can reduce the nonlinear phase noise. The variance of the total phase noise increases linearly with the bit rate, and does not depend significantly on the number of subcarriers for systems with moderate fiber chromatic dispersion.

  10. Periodic equatorial water flows from a Hamiltonian perspective

    NASA Astrophysics Data System (ADS)

    Ionescu-Kruse, Delia; Martin, Calin Iulian

    2017-04-01

    The main result of this paper is a Hamiltonian formulation of the nonlinear governing equations for geophysical periodic stratified water flows in the equatorial f-plane approximation allowing for piecewise constant vorticity.

  11. Non-Noether symmetries of Hamiltonian systems with conformable fractional derivatives

    NASA Astrophysics Data System (ADS)

    Lin-Li, Wang; Jing-Li, Fu

    2016-01-01

    In this paper, we present the fractional Hamilton’s canonical equations and the fractional non-Noether symmetry of Hamilton systems by the conformable fractional derivative. Firstly, the exchanging relationship between isochronous variation and fractional derivatives, and the fractional Hamilton principle of the system under this fractional derivative are proposed. Secondly, the fractional Hamilton’s canonical equations of Hamilton systems based on the Hamilton principle are established. Thirdly, the fractional non-Noether symmetries, non-Noether theorem and non-Noether conserved quantities for the Hamilton systems with the conformable fractional derivatives are obtained. Finally, an example is given to illustrate the results. Project supported by the National Natural Science Foundation of China (Grant Nos. 11272287 and 11472247), the Program for Changjiang Scholars and Innovative Research Team in University, China (Grant No. IRT13097), and the Key Science and Technology Innovation Team Project of Zhejiang Province, China (Grant No. 2013TD18).

  12. Investigation of the Dynamical Structure and Diffusion in a System of Hamiltonian Type: 4-Dimensional Symplectic Map

    NASA Astrophysics Data System (ADS)

    Todorovic, N.

    2009-09-01

    The Nekhoroshev theorem (Nekhoroshev 1977) is one of the most important theorems in modern Hamiltonian dynamics. This theorem applies to quasi integrable Hamiltonian systems of type H(I,\\varphi)=h(I)+\\varepsilon f (I, \\varphi), where h(I) is the integrable approximation, f(I, \\varphi) the perturbing function, \\varepsilon is a small perturbing parameter, Iin R^n are the actions and \\varphi in T^n the angles of the system. With some additional geometrical and analytical properties, the theorem provides the stability of actions in exponentially long times. In recent years it has been shown that with some modifications the Nekhoroshev theorem can be applied to the problems in Solar system dynamics (Morbidelli and Guzzo 1997, Guzzo et al 2002, Efthymiopoulos and Sándor 2105, Pavlović and Guzzo 2008). In this work, we are interested to observe numerically a Nekhoroshev like behavior on a model given with a 4-dimensional symplectic map. The model is not in the quasi-integrable form, i.e. independently from the perturbation it contains some additional hyperbolic structures (they appear in the model as primary resonances). Since the hyperbolic structures exist even for zero perturbation, the system will belong to the class of the so called a priori unstable systems. The main numerical tool used here was the Fast Lyapunov Indicator- FLI, introduced in (Froschlé et al. 1997, 2000). As an indicator of chaotic motion, FLI gives very precise and fast information about the chaoticity of an orbit. Also, among regular orbits, FLI is able to differentiate resonant from nonresonant motions. This property of FLI allows us to visualize the studied system and to obtain the Arnold web of the model (Froschlé et al. 2000). In such a way it was possible to observe the transition from a stable Nekhoroshev like structure (regular orbits dominate) to a globally chaotic system where resonances overlap, also known as Chirikov regime. Numerically, this transition happens when between 50

  13. Observers for Systems with Nonlinearities Satisfying an Incremental Quadratic Inequality

    NASA Technical Reports Server (NTRS)

    Acikmese, Ahmet Behcet; Corless, Martin

    2004-01-01

    We consider the problem of state estimation for nonlinear time-varying systems whose nonlinearities satisfy an incremental quadratic inequality. These observer results unifies earlier results in the literature; and extend it to some additional classes of nonlinearities. Observers are presented which guarantee that the state estimation error exponentially converges to zero. Observer design involves solving linear matrix inequalities for the observer gain matrices. Results are illustrated by application to a simple model of an underwater.

  14. Observers for Systems with Nonlinearities Satisfying an Incremental Quadratic Inequality

    NASA Technical Reports Server (NTRS)

    Acikmese, Ahmet Behcet; Corless, Martin

    2004-01-01

    We consider the problem of state estimation for nonlinear time-varying systems whose nonlinearities satisfy an incremental quadratic inequality. These observer results unifies earlier results in the literature; and extend it to some additional classes of nonlinearities. Observers are presented which guarantee that the state estimation error exponentially converges to zero. Observer design involves solving linear matrix inequalities for the observer gain matrices. Results are illustrated by application to a simple model of an underwater.

  15. Dynamics of nonlinear dissipative systems in the vicinity of resonance

    NASA Astrophysics Data System (ADS)

    Plaksiy, K. Y.; Mikhlin, Y. V.

    2015-01-01

    The behavior of nonlinear dissipative 2-DOF mechanical systems in the vicinity of resonance is studied in this paper. Namely, the free resonance vibrations of a spring-mass-pendulum system and the forced resonance vibrations of a 2-DOF dissipative system containing a nonlinear absorber are considered. A reduced system stated with respect to the system energy, the arctangent of the vibration amplitudes ratio, and the phase difference, is obtained and analyzed. The nonlinear normal mode approach is used in this analysis. Conditions for vibration energy localization are discussed.

  16. Identification of the nonlinear vibration system of power transformers

    NASA Astrophysics Data System (ADS)

    Jing, Zheng; Hai, Huang; Pan, Jie; Yanni, Zhang

    2017-01-01

    This paper focuses on the identification of the nonlinear vibration system of power transformers. A Hammerstein model is used to identify the system with electrical inputs and the vibration of the transformer tank as the output. The nonlinear property of the system is modelled using a Fourier neural network consisting of a nonlinear element and a linear dynamic block. The order and weights of the network are determined based on the Lipschitz criterion and the back-propagation algorithm. This system identification method is tested on several power transformers. Promising results for predicting the transformer vibration and extracting system parameters are presented and discussed.

  17. Input-to-state stable nonlinear filtering for a class of continuous-time delayed nonlinear systems

    NASA Astrophysics Data System (ADS)

    Ahn, Choon Ki

    2013-06-01

    This paper investigates the input-to-state stable (ISS) nonlinear filtering problem for a class of continuous-time delayed nonlinear systems with external disturbance. A new delay-dependent nonlinear ISS filter is established through available measurements to estimate the states of delayed nonlinear systems, such that the filtering error system is both exponentially and input-to-state stable for any bounded external disturbance. The design of the nonlinear ISS filter for these nonlinear systems is achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using standard numerical packages. An illustrative example is given to demonstrate the effectiveness of the proposed filter.

  18. Non equilibrium dynamics of isolated disordered systems: the classical Hamiltonian p-spin model

    NASA Astrophysics Data System (ADS)

    Cugliandolo, Leticia F.; Lozano, Gustavo S.; Nessi, Emilio N.

    2017-08-01

    We study the dynamics of a classical disordered macroscopic model completely isolated from the environment reproducing, in a classical setting, the ‘quantum quench’ protocol. We show that, depending on the pre and post quench parameters, the system approaches equilibrium, succeeding to act as a bath on itself, or remains out of equilibrium, in two different ways. In the latter one, the system stays confined in a metastable state in which it undergoes stationary dynamics characterised by a single temperature. In the other, the system ages and its dynamics are characterised by two temperatures associated with observations made at short and long time differences (high and low frequencies). The parameter dependence of the asymptotic states is rationalised in terms of a dynamic phase diagram with one equilibrium and two out of equilibrium phases. Aspects of pre-thermalisation are observed and discussed. Similarities and differences with the dynamics of the dissipative model are also explained.

  19. Robust nonlinear variable selective control for networked systems

    NASA Astrophysics Data System (ADS)

    Rahmani, Behrooz

    2016-10-01

    This paper is concerned with the networked control of a class of uncertain nonlinear systems. In this way, Takagi-Sugeno (T-S) fuzzy modelling is used to extend the previously proposed variable selective control (VSC) methodology to nonlinear systems. This extension is based upon the decomposition of the nonlinear system to a set of fuzzy-blended locally linearised subsystems and further application of the VSC methodology to each subsystem. To increase the applicability of the T-S approach for uncertain nonlinear networked control systems, this study considers the asynchronous premise variables in the plant and the controller, and then introduces a robust stability analysis and control synthesis. The resulting optimal switching-fuzzy controller provides a minimum guaranteed cost on an H2 performance index. Simulation studies on three nonlinear benchmark problems demonstrate the effectiveness of the proposed method.

  20. Tools for Nonlinear Control Systems Design

    NASA Technical Reports Server (NTRS)

    Sastry, S. S.

    1997-01-01

    This is a brief statement of the research progress made on Grant NAG2-243 titled "Tools for Nonlinear Control Systems Design", which ran from 1983 till December 1996. The initial set of PIs on the grant were C. A. Desoer, E. L. Polak and myself (for 1983). From 1984 till 1991 Desoer and I were the Pls and finally I was the sole PI from 1991 till the end of 1996. The project has been an unusually longstanding and extremely fruitful partnership, with many technical exchanges, visits, workshops and new avenues of investigation begun on this grant. There were student visits, long term.visitors on the grant and many interesting joint projects. In this final report I will only give a cursory description of the technical work done on the grant, since there was a tradition of annual progress reports and a proposal for the succeeding year. These progress reports cum proposals are attached as Appendix A to this report. Appendix B consists of papers by me and my students as co-authors sorted chronologically. When there are multiple related versions of a paper, such as a conference version and journal version they are listed together. Appendix C consists of papers by Desoer and his students as well as 'solo' publications by other researchers supported on this grant similarly chronologically sorted.

  1. Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems

    NASA Astrophysics Data System (ADS)

    Calleja, Renato C.; Celletti, Alessandra; de la Llave, Rafael

    2017-08-01

    Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We consider a family of conformally symplectic maps \

  2. Horseshoes in Perturbations of Hamiltonian Systems with two Degress of Freedom.

    DTIC Science & Technology

    1981-05-01

    Edition, Addison Wesley. V.I. Arnold [1964]. Instability of dynamical systems with several degrees of freedom. Dokl . Akad . Nauk . SSSR 156:9-12. V.I...bodies, 4th ed. Cambridge Univ. Press, Cambridge. S.L. Ziglin [1980]. Nonintegrability of a problem on the motion of fourpoint vortices, Soy. Math. Dokl . 21, 296-299. 1. dJ -- V Wit- -z""fL

  3. Robust online Hamiltonian learning

    NASA Astrophysics Data System (ADS)

    Granade, Christopher E.; Ferrie, Christopher; Wiebe, Nathan; Cory, D. G.

    2012-10-01

    In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.

  4. Identification of systems containing nonlinear stiffnesses using backbone curves

    NASA Astrophysics Data System (ADS)

    Londoño, Julián M.; Cooper, Jonathan E.; Neild, Simon A.

    2017-02-01

    This paper presents a method for the dynamic identification of structures containing discrete nonlinear stiffnesses. The approach requires the structure to be excited at a single resonant frequency, enabling measurements to be made in regimes of large displacements where nonlinearities are more likely to be significant. Measured resonant decay data is used to estimate the system backbone curves. Linear natural frequencies and nonlinear parameters are identified using these backbone curves assuming a form for the nonlinear behaviour. Numerical and experimental examples, inspired by an aerospace industry test case study, are considered to illustrate how the method can be applied. Results from these models demonstrate that the method can successfully deliver nonlinear models able to predict the response of the test structure nonlinear dynamics.

  5. A Survey of Repetitive Control for Nonlinear Systems

    NASA Astrophysics Data System (ADS)

    Quan, Quan; Cai, Kai-Yuan

    2010-10-01

    In aerospace engineering and industry, control tasks are often of a periodic nature, while repetitive control is especially suitable for tracking and rejection of periodic exogenous signals. Because of limited research effort on nonlinear systems, we give a survey of repetitive control for nonlinear systems in this paper. First, a brief introduction of repetitive control is presented. Then, after giving a brief overview of repetitive control for linear systems, this paper summarizes design methods and existing problems of repetitive control for nonlinear systems in detail. Lastly, relationships between repetitive control and other control schemes are analyzed to recognize repetitive control from different aspects more insightfully.

  6. System Identification for Nonlinear Control Using Neural Networks

    NASA Technical Reports Server (NTRS)

    Stengel, Robert F.; Linse, Dennis J.

    1990-01-01

    An approach to incorporating artificial neural networks in nonlinear, adaptive control systems is described. The controller contains three principal elements: a nonlinear inverse dynamic control law whose coefficients depend on a comprehensive model of the plant, a neural network that models system dynamics, and a state estimator whose outputs drive the control law and train the neural network. Attention is focused on the system identification task, which combines an extended Kalman filter with generalized spline function approximation. Continual learning is possible during normal operation, without taking the system off line for specialized training. Nonlinear inverse dynamic control requires smooth derivatives as well as function estimates, imposing stringent goals on the approximating technique.

  7. Derivation of Hamiltonians for accelerators

    SciTech Connect

    Symon, K.R.

    1997-09-12

    In this report various forms of the Hamiltonian for particle motion in an accelerator will be derived. Except where noted, the treatment will apply generally to linear and circular accelerators, storage rings, and beamlines. The generic term accelerator will be used to refer to any of these devices. The author will use the usual accelerator coordinate system, which will be introduced first, along with a list of handy formulas. He then starts from the general Hamiltonian for a particle in an electromagnetic field, using the accelerator coordinate system, with time t as independent variable. He switches to a form more convenient for most purposes using the distance s along the reference orbit as independent variable. In section 2, formulas will be derived for the vector potentials that describe the various lattice components. In sections 3, 4, and 5, special forms of the Hamiltonian will be derived for transverse horizontal and vertical motion, for longitudinal motion, and for synchrobetatron coupling of horizontal and longitudinal motions. Hamiltonians will be expanded to fourth order in the variables.

  8. Asymptotic Stability of Interconnected Passive Non-Linear Systems

    NASA Technical Reports Server (NTRS)

    Isidori, A.; Joshi, S. M.; Kelkar, A. G.

    1999-01-01

    This paper addresses the problem of stabilization of a class of internally passive non-linear time-invariant dynamic systems. A class of non-linear marginally strictly passive (MSP) systems is defined, which is less restrictive than input-strictly passive systems. It is shown that the interconnection of a non-linear passive system and a non-linear MSP system is globally asymptotically stable. The result generalizes and weakens the conditions of the passivity theorem, which requires one of the systems to be input-strictly passive. In the case of linear time-invariant systems, it is shown that the MSP property is equivalent to the marginally strictly positive real (MSPR) property, which is much simpler to check.

  9. A new, challenging benchmark for nonlinear system identification

    NASA Astrophysics Data System (ADS)

    Tiso, Paolo; Noël, Jean-Philippe

    2017-02-01

    The progress accomplished during the past decade in nonlinear system identification in structural dynamics is considerable. The objective of the present paper is to consolidate this progress by challenging the community through a new benchmark structure exhibiting complex nonlinear dynamics. The proposed structure consists of two offset cantilevered beams connected by a highly flexible element. For increasing forcing amplitudes, the system sequentially features linear behaviour, localised nonlinearity associated with the buckling of the connecting element, and distributed nonlinearity resulting from large elastic deformations across the structure. A finite element-based code with time integration capabilities is made available at https://sem.org/nonlinear-systems-imac-focus-group/. This code permits the numerical simulation of the benchmark dynamics in response to arbitrary excitation signals.

  10. Nonlinear normal modes in electrodynamic systems: A nonperturbative approach

    SciTech Connect

    Kudrin, A. V. Kudrina, O. A.; Petrov, E. Yu.

    2016-06-15

    We consider electromagnetic nonlinear normal modes in cylindrical cavity resonators filled with a nonlinear nondispersive medium. The key feature of the analysis is that exact analytic solutions of the nonlinear field equations are employed to study the mode properties in detail. Based on such a nonperturbative approach, we rigorously prove that the total energy of free nonlinear oscillations in a distributed conservative system, such as that considered in our work, can exactly coincide with the sum of energies of the normal modes of the system. This fact implies that the energy orthogonality property, which has so far been known to hold only for linear oscillations and fields, can also be observed in a nonlinear oscillatory system.

  11. Universal two-body-Hamiltonian quantum computing

    NASA Astrophysics Data System (ADS)

    Nagaj, Daniel

    2012-03-01

    We present a Hamiltonian quantum-computation scheme universal for quantum computation. Our Hamiltonian is a sum of a polynomial number (in the number of gates L in the quantum circuit) of constant-norm, time-independent, two-body interaction terms. Furthermore, each qubit in the system interacts only with a constant number of other qubits in a three-layer, geometrically local layout. The computer runs in three steps—it starts in a simple initial product state, evolves according to a time-independent Hamiltonian for time of order L2 (up to logarithmic factors), and finishes with a two-qubit measurement. Our model improves previous universal two-local-Hamiltonian constructions, as it avoids using perturbation gadgets and large energy-penalty terms in the Hamiltonian, which would result in a large required run time.

  12. On the benefit of DMT modulation in nonlinear VLC systems.

    PubMed

    Qian, Hua; Cai, Sunzeng; Yao, Saijie; Zhou, Ting; Yang, Yang; Wang, Xudong

    2015-02-09

    In a visible light communication (VLC) system, the nonlinear characteristic of the light emitting diode (LED) in transmitter is a limiting factor of system performance. Modern modulation signals with large peak-to-power-ratio (PAPR) suffers uneven distortion. The nonlinear response directly impacts the intensity modulation and direct detection VLC system with pulse-amplitude modulation (PAM). The amplitude of the PAM signal is distorted unevenly and large signal is vulnerable to noise. Orthogonal linear transformations, such as discrete multi-tone (DMT) modulation, can spread the nonlinear effects evenly to each data symbol, thus perform better than PAM signals. In this paper, we provide theoretical analysis on the benefit of DMT modulation in nonlinear VLC system. We show that the DMT modulation is a better choice than the PAM modulation for the VLC system as the DMT modulation is more robust against nonlinearity. We also show that the post-distortion nonlinear elimination method, which is applied at the receiver, can be a reliable solution to the nonlinear VLC system. Simulation results show that the post-distortion greatly improves the system performance for the DMT modulation.

  13. A study of nonlinear flight control system designs

    NASA Astrophysics Data System (ADS)

    Tian, Lijun

    This thesis discusses both normal aircraft flight control where the control surfaces are the primary effectors, and unconventional emergency flight control by engines only. It has long been realized that nonlinearity in aircraft dynamics is a prominent consideration in design of high-performance conventional flight control systems. The engine-only flight control problem also faces strong nonlinearity, although due to different reasons. A nonlinear predictive control method and an approximate receding-horizon control method are used for normal and engine-only flight control system designs for an F-18 aircraft. The comparison of the performance with that of linear flight controllers provides some insight into when nonlinear controllers may render a much improved performance. The concept of nonlinear flight control system design is extended to output tracking control problem. The capability of the nonlinear controller to stabilize the aircraft and accomplish output tracking control for non-minimum phase system is successfully demonstrated. Numerical simulation results of longitudinal motion based on two typical flight conditions for an F-18 aircraft is presented to illustrate some of these aspects. It is suggested in this thesis that nonlinear flight control system design, particularly the engine-only controller design and output tracking control design for non-minimum phase system by using a nonlinear method is more effective for the highly nonlinear environment. The recently developed continuous-time predictive control approach and an approximate receding-horizon control method are shown to be effective methods in the situation while the conventional linear or popular nonlinear control designs are either ineffective or inapplicable.

  14. Adaptive control under arbitrary switching for a class of switched nonlinear systems with nonlinear parameterisation

    NASA Astrophysics Data System (ADS)

    Wang, C. Y.; Jiao, X. H.

    2015-10-01

    This paper is devoted to discuss arbitrarily switching control problem for a class of nonlinearly parameterised nonlinear switched systems. Compared with the existing results, improvements are that a systematic procedure is given for an explicit construction of a common smooth adaptive controller independent of the switching signals. Meanwhile, the developed design method can be extended to the adaptive arbitrarily switching stabilisation problem for a class of cascade switched nonlinear systems. The theoretical analysis is presented for the Lyapunov stability of the resulting closed-loop switched system and the convergence of the original switched system states at the equilibrium under arbitrary switching. Moreover, the effectiveness and feasibility of the developed method are demonstrated by both a numerical example and a chemical system.

  15. Analysis and design of robust decentralized controllers for nonlinear systems

    SciTech Connect

    Schoenwald, D.A.

    1993-07-01

    Decentralized control strategies for nonlinear systems are achieved via feedback linearization techniques. New results on optimization and parameter robustness of non-linear systems are also developed. In addition, parametric uncertainty in large-scale systems is handled by sensitivity analysis and optimal control methods in a completely decentralized framework. This idea is applied to alleviate uncertainty in friction parameters for the gimbal joints on Space Station Freedom. As an example of decentralized nonlinear control, singular perturbation methods and distributed vibration damping are merged into a control strategy for a two-link flexible manipulator.

  16. Control design for a class of nonlinear parameter varying systems

    NASA Astrophysics Data System (ADS)

    Cai, Xiushan; Liu, Yang; Zhang, Wei

    2015-07-01

    Stabilisation for a class of one-sided Lipschitz nonlinear parameter varying systems is dealt with in this paper. First, the nonlinear parameter varying system is represented as a subsystem of a differential inclusion. Sufficient conditions for exponential stabilisation for the differential inclusion are given by solving linear matrix inequalities. Then a continuous control law is designed to stabilise the differential inclusion. It leads to stabilising the nonlinear parameter varying system. Finally, a simulation example is presented to show the validity and advantages of the proposed method.

  17. Nonlinear system identification and control based on modular neural networks.

    PubMed

    Puscasu, Gheorghe; Codres, Bogdan

    2011-08-01

    A new approach for nonlinear system identification and control based on modular neural networks (MNN) is proposed in this paper. The computational complexity of neural identification can be greatly reduced if the whole system is decomposed into several subsystems. This is obtained using a partitioning algorithm. Each local nonlinear model is associated with a nonlinear controller. These are also implemented by neural networks. The switching between the neural controllers is done by a dynamical switcher, also implemented by neural networks, that tracks the different operating points. The proposed multiple modelling and control strategy has been successfully tested on simulated laboratory scale liquid-level system.

  18. Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems

    NASA Astrophysics Data System (ADS)

    Kay, Alastair

    2007-09-01

    Here we present a problem related to the local Hamiltonian problem (identifying whether the ground-state energy falls within one of two ranges) which is restricted to being translationally invariant. We prove that for Hamiltonians with a fixed local dimension and O(log(N)) -body local terms, or local dimension N and two-body terms, there are instances where finding the ground-state energy is quantum-Merlin-Arthur-complete and simulating the dynamics is BQP-complete (BQP denotes “bounded error, quantum polynomial time”). We discuss the implications for the computational complexity of finding ground states of these systems and hence for any classical approximation techniques that one could apply including density-matrix renormalization group, matrix product states, and multiscale entanglement renormalization ansatz. One important example is a one-dimensional lattice of bosons with nearest-neighbor hopping at constant filling fraction—i.e., a generalization of the Bose-Hubbard model.

  19. An experimental study of nonlinear dynamic system identification

    NASA Technical Reports Server (NTRS)

    Stry, Greselda I.; Mook, D. Joseph

    1990-01-01

    A technique for robust identification of nonlinear dynamic systems is developed and illustrated using both simulations and analog experiments. The technique is based on the Minimum Model Error optimal estimation approach. A detailed literature review is included in which fundamental differences between the current approach and previous work is described. The most significant feature of the current work is the ability to identify nonlinear dynamic systems without prior assumptions regarding the form of the nonlinearities, in constrast to existing nonlinear identification approaches which usually require detailed assumptions of the nonlinearities. The example illustrations indicate that the method is robust with respect to prior ignorance of the model, and with respect to measurement noise, measurement frequency, and measurement record length.

  20. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression.

    PubMed

    Ding, Bo; Fang, Huajing

    2017-03-31

    This paper is concerned with the fault prediction for the nonlinear stochastic system with incipient faults. Based on the particle filter and the reasonable assumption about the incipient faults, the modified fault estimation algorithm is proposed, and the system state is estimated simultaneously. According to the modified fault estimation, an intuitive fault detection strategy is introduced. Once each of the incipient fault is detected, the parameters of which are identified by a nonlinear regression method. Then, based on the estimated parameters, the future fault signal can be predicted. Finally, the effectiveness of the proposed method is verified by the simulations of the Three-tank system.

  1. Quantum mechanical hamiltonian models of turing machines

    NASA Astrophysics Data System (ADS)

    Benioff, Paul

    1982-11-01

    Quantum mechanical Hamiltonian models, which represent an aribtrary but finite number of steps of any Turing machine computation, are constructed here on a finite lattice of spin-1/2 systems. Different regions of the lattice correspond to different components of the Turing machine (plus recording system). Successive states of any machine computation are represented in the model by spin configuration states. Both time-independent and time-dependent Hamiltonian models are constructed here. The time-independent models do not dissipate energy or degrade the system state as they evolve. They operate close to the quantum limit in that the total system energy uncertainty/computation speed is close to the limit given by the time-energy uncertainty relation. However, the model evolution is time global and the Hamiltonian is more complex. The time-dependent models do not degrade the system state. Also they are time local and the Hamiltonian is less complex.

  2. Computational studies of nonlinear dispersive plasma systems

    NASA Astrophysics Data System (ADS)

    Qian, Xin

    Plasma systems with dispersive waves are ubiquitous. Dispersive waves have the property that their wave velocity depends on the wave number of the wave. These waves show up in weakly as well as strongly coupled plasmas, and play a significant role in the underlying plasma dynamics. Dispersive waves bring new challenges to the computer simulation of nonlinear phenomena. The goal of this thesis is to discuss two computational studies of plasma phenomena, one drawn from strongly coupled complex or dusty plasmas, and the other from weakly coupled hydrogen plasmas. In the realm of dusty plasmas, we focus on the problem of three-dimensional (3D) Mach cones which we study by means of Molecular Dynamics (MD) simulations, assuming that the dust particles interact via a Yukawa potential. While laboratory and MD simulations have explored thoroughly the properties of Mach cones in 2D, elucidating the important role of dispersive waves in the formation of multiple cones, the simulations presented in this thesis represent the first 3D MD studies of Mach cones in strongly coupled dusty plasmas. These results have qualitative similarities with experimental observations on 3D Mach cones from the PK-3 plus project, which studies complex plasmas under microgravity conditions aboard the International Space station. In the realm of weakly coupled plasmas, we present results on the application of non-oscillatory central schemes to Hall MHD reconnection problems, in which the presence of dispersive whistler waves presents a formidable challenge for numerical algorithms that rely on explicit time-stepping schemes. In particular, we focus on the semi-discrete central formulation of Kurganov and Tadmor (2000), which has the advantage that it allow for larger time steps, and with significantly smaller numerical viscosity, than fully discrete schemes. We implement the Hall MHD equations through the CentPACK software package that implements the Kurganov-Tadmor formulation for a wide range of

  3. Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems.

    PubMed

    Dong, Jiuxiang; Wang, Youyi; Yang, Guang-Hong

    2010-12-01

    This paper considers the output feedback control problem for nonlinear discrete-time systems, which are represented by a type of fuzzy systems with local nonlinear models. By using the estimations of the states and nonlinear functions in local models, sufficient conditions for designing observer-based controllers are given for discrete-time nonlinear systems. First, a separation property, i.e., the controller and the observer can be independently designed, is proved for the class of fuzzy systems. Second, a two-step procedure with cone complementarity linearization algorithms is also developed for solving the H( ∞) dynamic output feedback (DOF) control problem. Moreover, for the case where the nonlinear functions in local submodels are measurable, a convex condition for designing H(∞) controllers is given by a new DOF control scheme. In contrast to the existing methods, the new methods can design output feedback controllers with fewer fuzzy rules as well as less computational burden, which is helpful for controller designs and implementations. Lastly, numerical examples are given to illustrate the effectiveness of the proposed methods.

  4. Distributed Adaptive Neural Control for Stochastic Nonlinear Multiagent Systems.

    PubMed

    Wang, Fang; Chen, Bing; Lin, Chong; Li, Xuehua

    2016-11-14

    In this paper, a consensus tracking problem of nonlinear multiagent systems is investigated under a directed communication topology. All the followers are modeled by stochastic nonlinear systems in nonstrict feedback form, where nonlinearities and stochastic disturbance terms are totally unknown. Based on the structural characteristic of neural networks (in Lemma 4), a novel distributed adaptive neural control scheme is put forward. The raised control method not only effectively handles unknown nonlinearities in nonstrict feedback systems, but also copes with the interactions among agents and coupling terms. Based on the stochastic Lyapunov functional method, it is indicated that all the signals of the closed-loop system are bounded in probability and all followers' outputs are convergent to a neighborhood of the output of leader. At last, the efficiency of the control method is testified by a numerical example.

  5. Supersymmetry of tridiagonal Hamiltonians

    NASA Astrophysics Data System (ADS)

    Yamani, Hashim A.; Mouayn, Zouhair

    2014-07-01

    A positive semi-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set, allows a definition of forward- and backward-shift operators that can be used to define the matrix representation of its supersymmetric partner Hamiltonian H( + ) with respect to the same basis. We find explicit relationships connecting the matrix elements of both Hamiltonians. We present a method to obtain the orthogonal polynomials in the eigenstate expansion problem attached to H( + ) starting from those polynomials arising in the same problem for H. This connection is established by using the notion of kernel polynomials. We apply the obtained results to two known solvable models with different kinds of spectrum.

  6. Optimal second order sliding mode control for nonlinear uncertain systems.

    PubMed

    Das, Madhulika; Mahanta, Chitralekha

    2014-07-01

    In this paper, a chattering free optimal second order sliding mode control (OSOSMC) method is proposed to stabilize nonlinear systems affected by uncertainties. The nonlinear optimal control strategy is based on the control Lyapunov function (CLF). For ensuring robustness of the optimal controller in the presence of parametric uncertainty and external disturbances, a sliding mode control scheme is realized by combining an integral and a terminal sliding surface. The resulting second order sliding mode can effectively reduce chattering in the control input. Simulation results confirm the supremacy of the proposed optimal second order sliding mode control over some existing sliding mode controllers in controlling nonlinear systems affected by uncertainty.

  7. An experimental study of nonlinear dynamic system identification

    NASA Technical Reports Server (NTRS)

    Stry, Greselda I.; Mook, D, Joseph

    1991-01-01

    A technique based on the Minimum Model Error optimal estimation approach is employed for robust identification of a nonlinear dynamic system. A simple harmonic oscillator with quadratic position feedback was simulated on an analog computer. With the aid of analog measurements and an assumed linear model, the Minimum Model Error Algorithm accurately identifies the quadratic nonlinearity. The tests demonstrate that the method is robust with respect to prior ignorance of the nonlinear system model and with respect to measurement record length, regardless of initial conditions.

  8. Computational power of symmetric Hamiltonians

    NASA Astrophysics Data System (ADS)

    Kay, Alastair

    2008-07-01

    The presence of symmetries, be they discrete or continuous, in a physical system typically leads to a reduction in the problem to be solved. Here we report that neither translational invariance nor rotational invariance reduce the computational complexity of simulating Hamiltonian dynamics; the problem is still bounded error, quantum polynomial time complete, and is believed to be hard on a classical computer. This is achieved by designing a system to implement a universal quantum interface, a device which enables control of an arbitrary computation through the control of a fixed number of spins, and using it as a building block to entirely remove the need for control, except in the system initialization. Finally, it is shown that cooling such Hamiltonians to their ground states in the presence of random magnetic fields solves a Quantum-Merlin-Arthur-complete problem.

  9. A Student's Guide to Lagrangians and Hamiltonians

    NASA Astrophysics Data System (ADS)

    Hamill, Patrick

    2013-11-01

    Part I. Lagrangian Mechanics: 1. Fundamental concepts; 2. The calculus of variations; 3. Lagrangian dynamics; Part II. Hamiltonian Mechanics: 4. Hamilton's equations; 5. Canonical transformations: Poisson brackets; 6. Hamilton-Jacobi theory; 7. Continuous systems; Further reading; Index.

  10. New stability conditions for nonlinear time varying delay systems

    NASA Astrophysics Data System (ADS)

    Elmadssia, S.; Saadaoui, K.; Benrejeb, M.

    2016-07-01

    In this paper, new practical stability conditions for a class of nonlinear time varying delay systems are proposed. The study is based on the use of a specific state space description, known as the Benrejeb characteristic arrow form matrix, and aggregation techniques to obtain delay-dependent stability conditions. Application of this method to delayed Lurie-Postnikov nonlinear systems is given. Illustrative examples are presented to show the effectiveness of the proposed approach.

  11. Nonlinear filter based decision feedback equalizer for optical communication systems.

    PubMed

    Han, Xiaoqi; Cheng, Chi-Hao

    2014-04-07

    Nonlinear impairments in optical communication system have become a major concern of optical engineers. In this paper, we demonstrate that utilizing a nonlinear filter based Decision Feedback Equalizer (DFE) with error detection capability can deliver a better performance compared with the conventional linear filter based DFE. The proposed algorithms are tested in simulation using a coherent 100 Gb/sec 16-QAM optical communication system in a legacy optical network setting.

  12. Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs

    NASA Astrophysics Data System (ADS)

    Bourgain, J.

    2004-04-01

    This is a survey of recent investigations of quasi-periodic localization on lattices (of both methods based on perturbation theory and non-perturbative methods) and of applications of KAM theories in connection with infinite-dimensional Hamiltonian systems. The focus is on applications of these investigations to the Schrödinger equation and the wave equation with periodic boundary conditions, and to non-linear random Schrödinger equations with short-range potentials.

  13. Features and states of microscopic particles in nonlinear quantum-mechanics systems

    NASA Astrophysics Data System (ADS)

    Pang, Xiao-Feng

    2008-06-01

    In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schrödinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy, momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles

  14. Applications of nonlinear systems theory to control design

    NASA Technical Reports Server (NTRS)

    Hunt, L. R.; Villarreal, Ramiro

    1988-01-01

    For most applications in the control area, the standard practice is to approximate a nonlinear mathematical model by a linear system. Since the feedback linearizable systems contain linear systems as a subclass, the procedure of approximating a nonlinear system by a feedback linearizable one is examined. Because many physical plants (e.g., aircraft at the NASA Ames Research Center) have mathematical models which are close to feedback linearizable systems, such approximations are certainly justified. Results and techniques are introduced for measuring the gap between the model and its truncated linearizable part. The topic of pure feedback systems is important to the study.

  15. Self-characterization of linear and nonlinear adaptive optics systems

    NASA Astrophysics Data System (ADS)

    Hampton, Peter J.; Conan, Rodolphe; Keskin, Onur; Bradley, Colin; Agathoklis, Pan

    2008-01-01

    We present methods used to determine the linear or nonlinear static response and the linear dynamic response of an adaptive optics (AO) system. This AO system consists of a nonlinear microelectromechanical systems deformable mirror (DM), a linear tip-tilt mirror (TTM), a control computer, and a Shack-Hartmann wavefront sensor. The system is modeled using a single-input-single-output structure to determine the one-dimensional transfer function of the dynamic response of the chain of system hardware. An AO system has been shown to be able to characterize its own response without additional instrumentation. Experimentally determined models are given for a TTM and a DM.

  16. Robust adaptive dynamic programming and feedback stabilization of nonlinear systems.

    PubMed

    Jiang, Yu; Jiang, Zhong-Ping

    2014-05-01

    This paper studies the robust optimal control design for a class of uncertain nonlinear systems from a perspective of robust adaptive dynamic programming (RADP). The objective is to fill up a gap in the past literature of adaptive dynamic programming (ADP) where dynamic uncertainties or unmodeled dynamics are not addressed. A key strategy is to integrate tools from modern nonlinear control theory, such as the robust redesign and the backstepping techniques as well as the nonlinear small-gain theorem, with the theory of ADP. The proposed RADP methodology can be viewed as an extension of ADP to uncertain nonlinear systems. Practical learning algorithms are developed in this paper, and have been applied to the controller design problems for a jet engine and a one-machine power system.

  17. Future directions of nonlinear dynamics in physical and biological systems

    SciTech Connect

    Christiansen, P.L.; Eilbeck, J.C.; Parmentier, R.D.

    1992-01-01

    Early in 1990 a scientific committee was formed for the purpose of organizing a high-level scientific meeting on Future Directions of Nonlinear Dynamics in Physical and Biological Systems, in honor of Alwyn Scott's 60th birthday (December 25, 1991). As preparations for the meeting proceeded, they were met with an unusually broad-scale and high level of enthusiasm on the part of the international nonlinear science community, resulting in a participation by 168 scientists from 23 different countries in the conference, which was held July 23 to August 1 1992. The contributions to this present volume have been grouped into the following chapters: (1) Integrability, solitons and coherent structures; (2) Nonlinear evolution equations and diffusive systems; (3) Chaotic and stochastic dynamics; (4) Classical and quantum lattices and fields; (5) Superconductivity and superconducting devices; (6) Nonlinear optics; (7) Davydov solitons and biomolecular dynamics; and (8) Biological systems and Neurophysics.

  18. 3-D Mesh Generation Nonlinear Systems

    SciTech Connect

    Christon, M. A.; Dovey, D.; Stillman, D. W.; Hallquist, J. O.; Rainsberger, R. B

    1994-04-07

    INGRID is a general-purpose, three-dimensional mesh generator developed for use with finite element, nonlinear, structural dynamics codes. INGRID generates the large and complex input data files for DYNA3D, NIKE3D, FACET, and TOPAZ3D. One of the greatest advantages of INGRID is that virtually any shape can be described without resorting to wedge elements, tetrahedrons, triangular elements or highly distorted quadrilateral or hexahedral elements. Other capabilities available are in the areas of geometry and graphics. Exact surface equations and surface intersections considerably improve the ability to deal with accurate models, and a hidden line graphics algorithm is included which is efficient on the most complicated meshes. The primary new capability is associated with the boundary conditions, loads, and material properties required by nonlinear mechanics programs. Commands have been designed for each case to minimize user effort. This is particularly important since special processing is almost always required for each load or boundary condition.

  19. Dynamical supersymmetric Dirac Hamiltonians

    SciTech Connect

    Ginocchio, J.N.

    1986-01-01

    Using the language of quantum electrodynamics, the Dirac Hamiltonian of a neutral fermion interacting with a tensor field is examined. A supersymmetry found for a general Dirac Hamiltonian of this type is discussed, followed by consideration of the special case of a harmonic electric potential. The square of the Dirac Hamiltonian of a neutral fermion interacting via an anomalous magnetic moment in an electric potential is shown to be equivalent to a three-dimensional supersymmetric Schroedinger equation. It is found that for a potential that grows as a power of r, the lowest energy of the Hamiltonian equals the rest mass of the fermion, and the Dirac eigenfunction has only an upper component which is normalizable. It is also found that the higher energy states have upper and lower components which form a supersymmetric doublet. 15 refs. (LEW)

  20. Revisiting the (E + A) ⊗ (e + a) problems of polyatomic systems with trigonal symmetry: general expansions of their vibronic Hamiltonians.

    PubMed

    Zeng, Tao; Seidu, Issaka

    2017-05-10

    In this work, we derive general expansions in vibrational coordinates for the (E + A) ⊗ (e + a) vibronic Hamiltonians of molecules with one and only one C3 axis. We first derive the expansion for the lowest C3 symmetry. Additional symmetry elements systematically eliminate terms in the expansion. We compare our expansions with the previous results for two cases, the and the C3 (E + A) ⊗ e. The first comparison demonstrates the robustness, completeness, conciseness, and convenience of our formalism. There is a systematic discrepancy in the second comparison. We discuss the origin of the discrepancy and use a numerical example to corroborate our expansion. Our formalism covers 153 vibronic problems in 6 point groups. It also gives general expansions for the spin-orbit vibronic Hamiltonians of the p-type (E + A) ⊗ (e + a) problems.

  1. Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities.

    PubMed

    Chen, Mou; Ge, Shuzhi Sam; How, Bernard Voon Ee

    2010-05-01

    In this paper, robust adaptive neural network (NN) control is investigated for a general class of uncertain multiple-input-multiple-output (MIMO) nonlinear systems with unknown control coefficient matrices and input nonlinearities. For nonsymmetric input nonlinearities of saturation and deadzone, variable structure control (VSC) in combination with backstepping and Lyapunov synthesis is proposed for adaptive NN control design with guaranteed stability. In the proposed adaptive NN control, the usual assumption on nonsingularity of NN approximation for unknown control coefficient matrices and boundary assumption between NN approximation error and control input have been eliminated. Command filters are presented to implement physical constraints on the virtual control laws, then the tedious analytic computations of time derivatives of virtual control laws are canceled. It is proved that the proposed robust backstepping control is able to guarantee semiglobal uniform ultimate boundedness of all signals in the closed-loop system. Finally, simulation results are presented to illustrate the effectiveness of the proposed adaptive NN control.

  2. Quasi-Hamiltonian structure and Hojman construction

    NASA Astrophysics Data System (ADS)

    Carinena, Jose F.; Guha, Partha; Ranada, Manuel F.

    2007-08-01

    Given a smooth vector field [Gamma] and assuming the knowledge of an infinitesimal symmetry X, Hojman [S. Hojman, The construction of a Poisson structure out of a symmetry and a conservation law of a dynamical system, J. Phys. A Math. Gen. 29 (1996) 667-674] proposed a method for finding both a Poisson tensor and a function H such that [Gamma] is the corresponding Hamiltonian system. In this paper, we approach the problem from geometrical point of view. The geometrization leads to the clarification of several concepts and methods used in Hojman's paper. In particular, the relationship between the nonstandard Hamiltonian structure proposed by Hojman and the degenerate quasi-Hamiltonian structures introduced by Crampin and Sarlet [M. Crampin, W. Sarlet, Bi-quasi-Hamiltonian systems, J. Math. Phys. 43 (2002) 2505-2517] is unveiled in this paper. We also provide some applications of our construction.

  3. Duality and integrability: Electromagnetism, linearized gravity, and massless higher spin gauge fields as bi-Hamiltonian systems

    SciTech Connect

    Barnich, Glenn; Troessaert, Cedric

    2009-04-15

    In the reduced phase space of electromagnetism, the generator of duality rotations in the usual Poisson bracket is shown to generate Maxwell's equations in a second, much simpler Poisson bracket. This gives rise to a hierarchy of bi-Hamiltonian evolution equations in the standard way. The result can be extended to linearized Yang-Mills theory, linearized gravity, and massless higher spin gauge fields.

  4. A Hamiltonian approach to Thermodynamics

    SciTech Connect

    Baldiotti, M.C.; Fresneda, R.; Molina, C.

    2016-10-15

    In the present work we develop a strictly Hamiltonian approach to Thermodynamics. A thermodynamic description based on symplectic geometry is introduced, where all thermodynamic processes can be described within the framework of Analytic Mechanics. Our proposal is constructed on top of a usual symplectic manifold, where phase space is even dimensional and one has well-defined Poisson brackets. The main idea is the introduction of an extended phase space where thermodynamic equations of state are realized as constraints. We are then able to apply the canonical transformation toolkit to thermodynamic problems. Throughout this development, Dirac’s theory of constrained systems is extensively used. To illustrate the formalism, we consider paradigmatic examples, namely, the ideal, van der Waals and Clausius gases. - Highlights: • A strictly Hamiltonian approach to Thermodynamics is proposed. • Dirac’s theory of constrained systems is extensively used. • Thermodynamic equations of state are realized as constraints. • Thermodynamic potentials are related by canonical transformations.

  5. Parameter and Structure Inference for Nonlinear Dynamical Systems

    NASA Technical Reports Server (NTRS)

    Morris, Robin D.; Smelyanskiy, Vadim N.; Millonas, Mark

    2006-01-01

    A great many systems can be modeled in the non-linear dynamical systems framework, as x = f(x) + xi(t), where f() is the potential function for the system, and xi is the excitation noise. Modeling the potential using a set of basis functions, we derive the posterior for the basis coefficients. A more challenging problem is to determine the set of basis functions that are required to model a particular system. We show that using the Bayesian Information Criteria (BIC) to rank models, and the beam search technique, that we can accurately determine the structure of simple non-linear dynamical system models, and the structure of the coupling between non-linear dynamical systems where the individual systems are known. This last case has important ecological applications.

  6. Fault diagnosis for a class of nonlinear systems via ESO.

    PubMed

    Yan, Bingyong; Tian, Zuohua; Shi, Songjiao; Weng, Zhengxin

    2008-10-01

    In this paper, a novel fault detection and identification (FDI) scheme for a class of nonlinear systems is presented. First of all, an augment system is constructed by making the unknown system faults as an extended system state. Then based on the ESO theory, a novel fault diagnosis filter is constructed to diagnose the nonlinear system faults. An extension to a class of nonlinear uncertain systems is then made. An outstanding feature of this scheme is that it can simultaneously detect and identify the shape and magnitude of the system faults in real time without training the network compared with the neural network-based FDI schemes. Finally, simulation examples are given to illustrate the feasibility and effectiveness of the proposed approach.

  7. Robust H∞ filtering for discrete nonlinear delayed stochastic systems with missing measurements and randomly occurring nonlinearities

    NASA Astrophysics Data System (ADS)

    Liu, Yurong; Alsaadi, Fuad E.; Yin, Xiaozhou; Wang, Yamin

    2015-02-01

    In this paper, we are concerned with the robust H∞ filtering problem for a class of nonlinear discrete time-delay stochastic systems. The system under consideration involves parameter uncertainties, stochastic disturbances, time-varying delays and sector nonlinearities. Both missing measurements and randomly occurring nonlinearities are described via the binary switching sequences satisfying a conditional probability distribution, and the nonlinearities are assumed to be sector bounded. The problem addressed is the design of a full-order filter such that, for all admissible uncertainties, nonlinearities and time-delays, the dynamics of the filtering error is constrained to be robustly exponentially stable in the mean square, and a prescribed ? disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and some new techniques, sufficient conditions are first established to ensure the existence of the desired filtering parameters. Then, the explicit expression of the desired filter gains is described in terms of the solution to a linear matrix inequality. Finally, a numerical example is exploited to show the usefulness of the results derived.

  8. Effective Hamiltonians for Rapidly Driven Many-Body Lattice Systems: Induced Exchange Interactions and Density-Dependent Hoppings

    NASA Astrophysics Data System (ADS)

    Itin, A. P.; Katsnelson, M. I.

    2015-08-01

    We consider 1D lattices described by Hubbard or Bose-Hubbard models, in the presence of periodic high-frequency perturbations, such as uniform ac force or modulation of hopping coefficients. Effective Hamiltonians for interacting particles are derived using an averaging method resembling classical canonical perturbation theory. As is known, a high-frequency force may renormalize hopping coefficients, causing interesting phenomena such as coherent destruction of tunneling and creation of artificial gauge fields. We find explicitly additional corrections to the effective Hamiltonians due to interactions, corresponding to nontrivial processes such as single-particle density-dependent tunneling, correlated pair hoppings, nearest neighbor interactions, etc. Some of these processes arise also in multiband lattice models, and are capable of giving rise to a rich variety of quantum phases. The apparent contradiction with other methods, e.g., Floquet-Magnus expansion, is explained. The results may be useful for designing effective Hamiltonian models in experiments with ultracold atoms, as well as in the field of ultrafast nonequilibrium magnetism. An example of manipulating exchange interaction in a Mott-Hubbard insulator is considered, where our corrections play an essential role.

  9. Effective Hamiltonians for Rapidly Driven Many-Body Lattice Systems: Induced Exchange Interactions and Density-Dependent Hoppings.

    PubMed

    Itin, A P; Katsnelson, M I

    2015-08-14

    We consider 1D lattices described by Hubbard or Bose-Hubbard models, in the presence of periodic high-frequency perturbations, such as uniform ac force or modulation of hopping coefficients. Effective Hamiltonians for interacting particles are derived using an averaging method resembling classical canonical perturbation theory. As is known, a high-frequency force may renormalize hopping coefficients, causing interesting phenomena such as coherent destruction of tunneling and creation of artificial gauge fields. We find explicitly additional corrections to the effective Hamiltonians due to interactions, corresponding to nontrivial processes such as single-particle density-dependent tunneling, correlated pair hoppings, nearest neighbor interactions, etc. Some of these processes arise also in multiband lattice models, and are capable of giving rise to a rich variety of quantum phases. The apparent contradiction with other methods, e.g., Floquet-Magnus expansion, is explained. The results may be useful for designing effective Hamiltonian models in experiments with ultracold atoms, as well as in the field of ultrafast nonequilibrium magnetism. An example of manipulating exchange interaction in a Mott-Hubbard insulator is considered, where our corrections play an essential role.

  10. Gauge-invariant hydrogen-atom Hamiltonian

    SciTech Connect

    Sun Weimin; Wang Fan; Chen Xiangsong; Lue Xiaofu

    2010-07-15

    For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this puzzle was recently provided by us [X.-S. Chen et al., Phys. Rev. Lett. 100, 232002 (2008)]. Based on the separation of the electromagnetic potential into pure-gauge and gauge-invariant parts, we have proposed a new set of momentum and Hamiltonian operators which satisfy both the requirement of gauge invariance and the relevant commutation relations. In this paper we report a check for the case of the hydrogen-atom problem: Starting from the Hamiltonian of the coupled electron, proton, and electromagnetic field, under the infinite proton mass approximation, we derive the gauge-invariant hydrogen-atom Hamiltonian and verify explicitly that this Hamiltonian is different from the Dirac Hamiltonian, which is the time translation generator of the system. The gauge-invariant Hamiltonian is the energy operator, whose eigenvalue is the energy of the hydrogen atom. It is generally time dependent. In this case, one can solve the energy eigenvalue equation at any specific instant of time. It is shown that the energy eigenvalues are gauge independent, and by suitably choosing the phase factor of the time-dependent eigenfunction, one can ensure that the time-dependent eigenfunction satisfies the Dirac equation.

  11. Hamiltonian time integrators for Vlasov-Maxwell equations

    NASA Astrophysics Data System (ADS)

    He, Yang; Qin, Hong; Sun, Yajuan; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian

    2015-12-01

    Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.

  12. Hamiltonian time integrators for Vlasov-Maxwell equations

    SciTech Connect

    He, Yang; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian; Qin, Hong; Sun, Yajuan

    2015-12-15

    Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.

  13. Effective Floquet Hamiltonians for dipolar and quadrupolar coupled N-spin systems in solid state nuclear magnetic resonance under magic angle spinning.

    PubMed

    Pandey, Manoj Kumar; Krishnan, Mangala Sunder

    2010-11-07

    Spin dynamics under magic angle spinning has been studied using different theoretical approaches and also by extensive numerical simulation programs. In this article we present a general theoretical approach that leads to analytic forms for effective Hamiltonians for an N-spin dipolar and quadrupolar coupled system under magic angle spinning (MAS) conditions, using a combination of Floquet theory and van Vleck (contact) transformation. The analytic forms presented are shown to be useful for the study of MAS spin dynamics in solids with the help of a number of simulations in two, three, and four coupled, spin-1/2 systems as well as spins in which quadrupolar interactions are also present.

  14. A permutation characterization of Sturm global attractors of Hamiltonian type

    NASA Astrophysics Data System (ADS)

    Fiedler, Bernold; Rocha, Carlos; Wolfrum, Matthias

    We consider Neumann boundary value problems of the form u=u+f on the interval 0⩽x⩽π for dissipative nonlinearities f=f(u). A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the much more general case f=f(x,u,u). We present a permutation characterization for the global attractors in the restrictive class of nonlinearities f=f(u). In this class the stationary solutions of the parabolic equation satisfy the second order ODE v+f(v)=0 and we obtain the permutation characterization from a characterization of the set of 2 π-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation.

  15. The analysis on nonlinear control of the aircraft arresting system

    NASA Astrophysics Data System (ADS)

    Song, Jinchun; Du, Tianrong

    2005-12-01

    The aircraft arresting system is a complicated nonlinear system. This paper analyzes the mechanical-hydraulic structure of aircraft arresting system composed of electro hydraulic valve and establishes the dynamic equation of the aircraft arresting system. Based on the state-feedback linearization of nonlinear system, a PD-based controller is synthesized. Simulation studies indicate, while arresting the different type aircraft, the proposed controller has fast response, good tracking performance and strong robustness. By tuning the parameters of the PD controller, a satisfactory control performance can be guaranteed.

  16. Coherent nonlinear structures in ITG-Zonal flow system

    NASA Astrophysics Data System (ADS)

    Singh, Rameswar; Singh, Raghvendra; Kaw, Predhiman; Diamond, Patrick H.

    2013-10-01

    Nonlinear stationary structure formation in the coupled ion temperature gradient (ITG) - Zonal flow system is investigated. The ITG turbulence is described by a wave-kinetic equation for the action density of ITG mode and the longer scale zonal mode is described by a dynamical equation for the m = n = 0 component of the potential. In a moving frame, two populations of trappped and untrapped drift wave trajectories are shown to exist. This novel effect leads to formation of nonlinear stationary structures. It is shown that the ITG turbulence can self-consistently sustain coherent, radialy propagating modulation envelope structures such as solitons, shocks, nonlinear wave trains, etc.

  17. Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems

    PubMed Central

    Lü, Xin-You; Zhang, Wei-Min; Ashhab, Sahel; Wu, Ying; Nori, Franco

    2013-01-01

    We investigate a hybrid electro-optomechanical system that allows us to realize controllable strong Kerr nonlinearities even in the weak-coupling regime. We show that when the controllable electromechanical subsystem is close to its quantum critical point, strong photon-photon interactions can be generated by adjusting the intensity (or frequency) of the microwave driving field. Nonlinear optical phenomena, such as the appearance of the photon blockade and the generation of nonclassical states (e.g., Schrödinger cat states), are demonstrated in the weak-coupling regime, making the observation of strong Kerr nonlinearities feasible with currently available optomechanical technology. PMID:24126279

  18. Applications of nonlinear system identification to structural health monitoring.

    SciTech Connect

    Farrar, C. R.; Sohn, H.; Robertson, A. N.

    2004-01-01

    The process of implementing a damage detection strategy for aerospace, civil and mechanical engineering infrastructure is referred to as structural health monitoring (SHM). In many cases damage causes a structure that initially behaves in a predominantly linear manner to exhibit nonlinear response when subject to its operating environment. The formation of cracks that subsequently open and close under operating loads is an example of such damage. The damage detection process can be significantly enhanced if one takes advantage of these nonlinear effects when extracting damage-sensitive features from measured data. This paper will provide an overview of nonlinear system identification techniques that are used for the feature extraction process. Specifically, three general approaches that apply nonlinear system identification techniques to the damage detection process are discussed. The first two approaches attempt to quantify the deviation of the system from its initial linear characteristics that is a direct result of damage. The third approach is to extract features from the data that are directly related to the specific nonlinearity associated with the damaged condition. To conclude this discussion, a summary of outstanding issues associated with the application of nonlinear system identification techniques to the SHM problem is presented.

  19. Adaptive control of nonlinear systems with actuator failures and uncertainties

    NASA Astrophysics Data System (ADS)

    Tang, Xidong

    2005-11-01

    Actuator failures have damaging effect on the performance of control systems, leading to undesired system behavior or even instability. Actuator failures are unknown in terms of failure time instants, failure patterns, and failure parameters. For system safety and reliability, the compensation of actuator failures is of both theoretical and practical significance. This dissertation is to further the study of adaptive designs for actuator failure compensation to nonlinear systems. In this dissertation a theoretical framework for adaptive control of nonlinear systems with actuator failures and system uncertainties is established. The contributions are the development of new adaptive nonlinear control schemes to handle unknown actuator failures for convergent tracking performance, the specification of conditions as a guideline for applications and system designs, and the extension of the adaptive nonlinear control theory. In the dissertation, adaptive actuator failure compensation is studied for several classes of nonlinear systems. In particular, adaptive state feedback schemes are developed for feedback linearizable systems and parametric strict-feedback systems. Adaptive output feedback schemes are deigned for output-feedback systems and a class of systems with unknown state-dependent nonlinearities. Furthermore, adaptive designs are addressed for MIMO systems with actuator failures, based on two grouping techniques: fixed grouping and virtual grouping. Theoretical issues such as controller structures, actuation schemes, zero dynamics, observation, grouping conditions, closed-loop stability, and tracking performance are extensively investigated. For each scheme, design conditions are clarified, and detailed stability and performance analysis is presented. A variety of applications including a wing-rock model, twin otter aircraft, hypersonic aircraft, and cooperative multiple manipulators are addressed with simulation results showing the effectiveness of the

  20. Non-linear system identification in flow-induced vibration

    SciTech Connect

    Spanos, P.D.; Zeldin, B.A.; Lu, R.

    1996-12-31

    The paper introduces a method of identification of non-linear systems encountered in marine engineering applications. The non-linearity is accounted for by a combination of linear subsystems and known zero-memory non-linear transformations; an equivalent linear multi-input-single-output (MISO) system is developed for the identification problem. The unknown transfer functions of the MISO system are identified by assembling a system of linear equations in the frequency domain. This system is solved by performing the Cholesky decomposition of a related matrix. It is shown that the proposed identification method can be interpreted as a {open_quotes}Gram-Schmidt{close_quotes} type of orthogonal decomposition of the input-output quantities of the equivalent MISO system. A numerical example involving the identification of unknown parameters of flow (ocean wave) induced forces on offshore structures elucidates the applicability of the proposed method.

  1. Aeroelasticity of Nonlinear Tail / Rudder Systems with Freeplay

    NASA Astrophysics Data System (ADS)

    Rishel, Evan

    This thesis details the development of a linear/nonlinear three degree of freedom aeroelastic system designed and manufactured at the University of Washington (UW). Describing function analysis was carried out in the frequency domain. Time domain simulations were carried out to account for all types of motion. Nonlinear aeroelastic behavior may lead to limit cycles which can be captured in the frequency domain using describing function approximation and numerically using Runga-Kutta integration. Linear and nonlinear aeroelastic tests were conducted in the UW 3x3 low-speed wind tunnel to determine the linear flutter speed and frequency of the system as well as its nonlinear behavior when freeplay is introduced. The test data is presented along with the results of the MATLAB-based simulations. The correlation between test and numerical results is very high.

  2. Simulation program of nonlinearities applied to telecommunication systems

    NASA Technical Reports Server (NTRS)

    Thomas, C.

    1979-01-01

    In any satellite communication system, the problems of distorsion created by nonlinear devices or systems must be considered. The subject of this paper is the use of the Fast Fourier Transform (F.F.T.) in the prediction of the intermodulation performance of amplifiers, mixers, filters. A nonlinear memory-less model is chosen to simulate amplitude and phase nonlinearities of the device in the simulation program written in FORTRAN 4. The experimentally observed nonlinearity parameters of a low noise 3.7-4.2 GHz amplifier are related to the gain and phase coefficients of Fourier Service Series. The measured results are compared with those calculated from the simulation in the cases where the input signal is composed of two, three carriers and noise power density.

  3. Nonlinear Dynamics and Chaotic Motions in Feedback Controlled Elastic Systems.

    DTIC Science & Technology

    1985-08-01

    mechanical oscillator ", "On slowly varying oscillations ", "Knotted Orbits and bifurcation sequences in periodically forced oscillations ", "Dynamics of a...each P.I. 2.1 Analytical Studies of Feedback Controlled Oscillators (P.J. Holmes, S. Wiggins (Grad. Student)) 2.1.1 Bifurcation studies. Local and...global bifurcation studies of nonlinear oscillators subject to linear and nonlinear feedback have been completed. The systems treated have the form x

  4. Observer Based Compensators for Nonlinear Systems

    DTIC Science & Technology

    1989-03-31

    Automation, vol. 4, no. 1, 1988. [42] Poincare, H., Oeuvres, Tome 1, Gauthier- Villars , Paris, 1928. [43] Su, R., "On the linear equivalents of nonlinear...Control Theory, M. Fliess and M. Hazewinkel (eds.). D. Reidel, Dordrehct, to appear. [161 H. Poincare, Oeuvres, Tome 1 (Gauthier- Villars , Paris 1928). 117...one can choose a metric G on N .M G [ Gil 0 (49 def ffL (2)_ i(2) +() 2 G 0 (49) QP(x,u)dxdu (42) 2 and find a solution to 7(2) min I 1 (50) We want to

  5. Adaptive Control of Nonlinear Flexible Systems

    DTIC Science & Technology

    1994-05-26

    nonlinear plants which admit a finite- dimensional state-space description of the form S= f(Z) + g(z)u for which the State-Space Exact Linearization Problem...robust state-feedback law and the sensi- i tivity of the exact - linearization based control law. 2.6.3 Example 2 I Consider the following one state...is also available for exact linearization , Now apply the certainty equivalence based control one can bring an input-output approach to a particu- law

  6. Hierarchical robust nonlinear switching control design for propulsion systems

    NASA Astrophysics Data System (ADS)

    Leonessa, Alexander

    1999-09-01

    The desire for developing an integrated control system- design methodology for advanced propulsion systems has led to significant activity in modeling and control of flow compression systems in recent years. In this dissertation we develop a novel hierarchical switching control framework for addressing the compressor aerodynamic instabilities of rotating stall and surge. The proposed control framework accounts for the coupling between higher-order modes while explicitly addressing actuator rate saturation constraints and system modeling uncertainty. To develop a hierarchical nonlinear switching control framework, first we develop generalized Lyapunov and invariant set theorems for nonlinear dynamical systems wherein all regularity assumptions on the Lyapunov function and the system dynamics are removed. In particular, local and global stability theorems are given using lower semicontinuous Lyapunov functions. Furthermore, generalized invariant set theorems are derived wherein system trajectories converge to a union of largest invariant sets contained in intersections over finite intervals of the closure of generalized Lyapunov level surfaces. The proposed results provide transparent generalizations to standard Lyapunov and invariant set theorems. Using the generalized Lyapunov and invariant set theorems, a nonlinear control-system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed. Specifically, using equilibria- dependent Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. The switching nonlinear controller architecture is designed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria. The proposed framework provides a

  7. Operator-based robust nonlinear control system design for MIMO nonlinear plants with unknown coupling effects

    NASA Astrophysics Data System (ADS)

    Deng, Mingcong; Bi, Shuhui

    2010-09-01

    In this article, operator-based robust nonlinear control system design for multi-input multi-output (MIMO) plants with unknown coupling effects is considered. That is, by using operator-based robust nonlinear control design, coupling effects existing in the MIMO nonlinear plants can be decoupled based on a feedback design and robust right coprime factorisation approach, the coupling effects caused by controllers and plant outputs can be stabilised by using definition of Lipschitz operator and contraction mapping theorem, and output tracking performance can be realised by a tracking design scheme. Finally, a simulation example about temperature control process of 3-input/3-output aluminum plate is given to support the theoretical analysis.

  8. Nonperturbative analytical approximate solutions in intrinsically nonlinear systems

    NASA Astrophysics Data System (ADS)

    Kindall, Kevin Gaylynn

    The basis for obtaining analytical approximations in this dissertation is a new nonperturbative iterative approach that preserves the intrinsic nonlinearity of the system. The traditional method for approaching nonlinear equations has been the small amplitude approximation of classical perturbation theory. However, it is becoming increasingly evident that intrinsic nonlinearity or persistence of the interaction is a primary feature of the solutions for the nonlinear equations that have been solved. Although perturbation theory may be useful in certain physical domains, it is a domain which excludes the effects of the persistent interaction, since perturbation theory nullifies any intrinsically nonlinear property. The method of solution used here proceeds by analogy to the well-known result that second order, linear ordinary differential equations can be transformed to a Riccati equation by a change in dependent variable. An analogous transformation for nonlinear partial differential equations leads to a set of integro- differential equations for which the basic structure is Riccati. Approximations are introduced in the integral part of the integro-differential equation which allow for systematic iteration while making no expansion in powers of the coupling constant. Two sets of differential equations are examined: the Maxwell-Bloch set and the Rossler set. The importance of the former lies in its importance to the phenomenon of optical bistability. The latter represents the minimal set necessary to display chaos. In each case, their intrinsic nonlinearity is demonstrated, and nonperturbative approximate solutions are constructed.

  9. Experimental nonlinear laser systems: Bigger data for better science?

    SciTech Connect

    Kane, D. M.; Toomey, J. P.; McMahon, C.; Noblet, Y.; Argyris, A.; Syvridis, D.

    2014-10-06

    Bigger data is supporting knowledge discovery in nonlinear laser systems as will be demonstrated with examples from three semiconductor laser based systems – one with optical feedback, a photonic integrated circuit (PIC) chaotic laser and a frequency shifted feedback laser system.

  10. Adaptive Neural Network Based Control of Noncanonical Nonlinear Systems.

    PubMed

    Zhang, Yanjun; Tao, Gang; Chen, Mou

    2016-09-01

    This paper presents a new study on the adaptive neural network-based control of a class of noncanonical nonlinear systems with large parametric uncertainties. Unlike commonly studied canonical form nonlinear systems whose neural network approximation system models have explicit relative degree structures, which can directly be used to derive parameterized controllers for adaptation, noncanonical form nonlinear systems usually do not have explicit relative degrees, and thus their approximation system models are also in noncanonical forms. It is well-known that the adaptive control of noncanonical form nonlinear systems involves the parameterization of system dynamics. As demonstrated in this paper, it is also the case for noncanonical neural network approximation system models. Effective control of such systems is an open research problem, especially in the presence of uncertain parameters. This paper shows that it is necessary to reparameterize such neural network system models for adaptive control design, and that such reparameterization can be realized using a relative degree formulation, a concept yet to be studied for general neural network system models. This paper then derives the parameterized controllers that guarantee closed-loop stability and asymptotic output tracking for noncanonical form neural network system models. An illustrative example is presented with the simulation results to demonstrate the control design procedure, and to verify the effectiveness of such a new design method.

  11. Nonlinear system identification based on internal recurrent neural networks.

    PubMed

    Puscasu, Gheorghe; Codres, Bogdan; Stancu, Alexandru; Murariu, Gabriel

    2009-04-01

    A novel approach for nonlinear complex system identification based on internal recurrent neural networks (IRNN) is proposed in this paper. The computational complexity of neural identification can be greatly reduced if the whole system is decomposed into several subsystems. This approach employs internal state estimation when no measurements coming from the sensors are available for the system states. A modified backpropagation algorithm is introduced in order to train the IRNN for nonlinear system identification. The performance of the proposed design approach is proven on a car simulator case study.

  12. Geometric framework for phase synchronization in coupled noisy nonlinear systems

    NASA Astrophysics Data System (ADS)

    Balakrishnan, J.

    2006-03-01

    A geometric approach is introduced for understanding the phenomenon of phase synchronization in coupled nonlinear systems in the presence of additive noise. We show that the emergence of cooperative behavior through a change of stability via a Hopf bifurcation entails the spontaneous appearance of a gauge structure in the system, arising from the evolution of the slow dynamics, but induced by the fast variables. The conditions for the oscillators to be synchronised in phase are obtained. The role of weak noise appears to be to drive the system towards a more synchronized behavior. Our analysis provides a framework to explain recent experimental observations on noise-induced phase synchronization in coupled nonlinear systems.

  13. Integrable Nonlinear Schrödinger System on a Triangular-Lattice Ribbon

    NASA Astrophysics Data System (ADS)

    Vakhnenko, Oleksiy O.

    2015-01-01

    An integrable nonlinear Schrödinger system on a triangular-lattice ribbon, whose geometric configuration is similar to that of (1,1) armchair boron nanotube, is studied in detail. The system Hamiltonian formulation is shown to underline an essentially nontrivial Poisson structure associated with four basic field variables appearing as nearly amplitudes of the probability to find the lattice sites being excited and with two concomitant field variables maintaining the finite background. The coupling parameters of the system are allowed to be complex-valued ones thus permitting to model external magnetic fluxes threading the elementary plackets of a lattice in terms of Peierls phases. An alternative version of zero-curvature representation given in terms of 2 × 2 auxiliary spectral and evolution matrices is proved to support the constructive integrability of the system by means of Darboux-Bäcklund dressing method. In the framework of Darboux approach the one-soliton solution is found explicitly and analyzed with special attention to the principal differences between the bare and physical soliton parameters.

  14. Robust Online Hamiltonian Learning

    NASA Astrophysics Data System (ADS)

    Granade, Christopher; Ferrie, Christopher; Wiebe, Nathan; Cory, David

    2013-05-01

    In this talk, we introduce a machine-learning algorithm for the problem of inferring the dynamical parameters of a quantum system, and discuss this algorithm in the example of estimating the precession frequency of a single qubit in a static field. Our algorithm is designed with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online, during experimental data collection, or can be used as a tool for post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. Finally, we discuss the performance of the our algorithm by appeal to the Cramer-Rao bound. This work was financially supported by the Canadian government through NSERC and CERC and by the United States government through DARPA. NW would like to acknowledge funding from USARO-DTO.

  15. A nonlinear complementarity approach for the national energy modeling system

    SciTech Connect

    Gabriel, S.A.; Kydes, A.S.

    1995-03-08

    The National Energy Modeling System (NEMS) is a large-scale mathematical model that computes equilibrium fuel prices and quantities in the U.S. energy sector. At present, to generate these equilibrium values, NEMS sequentially solves a collection of linear programs and nonlinear equations. The NEMS solution procedure then incorporates the solutions of these linear programs and nonlinear equations in a nonlinear Gauss-Seidel approach. The authors describe how the current version of NEMS can be formulated as a particular nonlinear complementarity problem (NCP), thereby possibly avoiding current convergence problems. In addition, they show that the NCP format is equally valid for a more general form of NEMS. They also describe several promising approaches for solving the NCP form of NEMS based on recent Newton type methods for general NCPs. These approaches share the feature of needing to solve their direction-finding subproblems only approximately. Hence, they can effectively exploit the sparsity inherent in the NEMS NCP.

  16. Towards Port-Hamiltonian Approach for Modeling and Control of Two-wheeled Wheelchair

    NASA Astrophysics Data System (ADS)

    Aula, A.; Akmeliawati, R.; Ahmad, S.; Altalmas, T. M.; Sidek, S. N.

    2013-12-01

    This paper introduces the modeling and control design of a two-wheeled wheelchair (TWW) based on structure-preserving port-Hamiltonian concept. In this paper, a model of TWW with features, including space-saving, four to two-wheel transformation, and adjustable seat height is proposed to increased mobility and independence of the user. Then, the mathematical model of a TWW in its balanced mode is derived. The model is based on the total energy in the system. The system is divided into subsystems whereby the interconnections which exist are utilized. The nonlinearity of the model is preserved using port-controlled Hamiltonian (PCH) system and made to advantage. The proposed controlled is designed based on the idea of PCH such that the energy balance in the system can be achieved while stabilizing the system.

  17. Variable structure control of nonlinear systems through simplified uncertain models

    NASA Technical Reports Server (NTRS)

    Sira-Ramirez, Hebertt

    1986-01-01

    A variable structure control approach is presented for the robust stabilization of feedback equivalent nonlinear systems whose proposed model lies in the same structural orbit of a linear system in Brunovsky's canonical form. An attempt to linearize exactly the nonlinear plant on the basis of the feedback control law derived for the available model results in a nonlinearly perturbed canonical system for the expanded class of possible equivalent control functions. Conservatism tends to grow as modeling errors become larger. In order to preserve the internal controllability structure of the plant, it is proposed that model simplification be carried out on the open-loop-transformed system. As an example, a controller is developed for a single link manipulator with an elastic joint.

  18. Nonlinear dynamical system identification using unscented Kalman filter

    NASA Astrophysics Data System (ADS)

    Rehman, M. Javvad ur; Dass, Sarat Chandra; Asirvadam, Vijanth Sagayan

    2016-11-01

    Kalman Filter is the most suitable choice for linear state space and Gaussian error distribution from decades. In general practical systems are not linear and Gaussian so these assumptions give inconsistent results. System Identification for nonlinear dynamical systems is a difficult task to perform. Usually, Extended Kalman Filter (EKF) is used to deal with non-linearity in which Jacobian method is used for linearizing the system dynamics, But it has been observed that in highly non-linear environment performance of EKF is poor. Unscented Kalman Filter (UKF) is proposed here as a better option because instead of analytical linearization of state space, UKF performs statistical linearization by using sigma point calculated from deterministic samples. Formation of the posterior distribution is based on the propagation of mean and covariance through sigma points.

  19. A nonlinear filtering process diagnostic system for the Space Station

    NASA Technical Reports Server (NTRS)

    Yoel, Raymond R.; Buchner, M.; Loparo, K.; Cubukcu, Arif

    1988-01-01

    A nonlinear filtering process diagnostic system, terrestrial simulation and real time implementation studies is presented. Possible applications to Space Station subsystem elements are discussed. A process diagnostic system using model based nonlinear filtering for systems with random structure was shown to provide improvements in stability, robustness, and overall performance in comparison to linear filter based systems. A suboptimal version of the nonlinear filter (zero order approximation filter, or ZOA filter) was used in simulation studies, initially, with a pressurized water reactor model and then with water/steam heat exchanger models. Finally, a real time implementation for leak detection in a water/steam heat exchanger was conducted using the ZOA filter and heat exchanger models.

  20. Performance evaluation of nonlinear weighted T-system

    NASA Astrophysics Data System (ADS)

    Benfekir, A.; Hamaci, S.; Boimond, J.-L.; Labadi, K.

    2013-10-01

    This article deals with the analysis of discrete event systems which can be modelled by timed event graphs with multipliers (TEGMs). These graphs are an extension of weighted T-systems studied in the Petri net literature. These models do not admit a linear representation in (min, +) algebra. This nonlinearity is due to the presence of weights on arcs. To mitigate this problem of nonlinearity and to apply some basic results used to analyse the performances of linear systems in dioid algebra, we propose a linearisation method of mathematical model reflecting the behaviour of a TEGM in order to obtain a (min, +) linear model.

  1. Recent results of nonlinear estimators applied to hereditary systems.

    NASA Technical Reports Server (NTRS)

    Schiess, J. R.; Roland, V. R.; Wells, W. R.

    1972-01-01

    An application of the extended Kalman filter to delayed systems to estimate the state and time delay is presented. Two nonlinear estimators are discussed and the results compared with those of the Kalman filter. For all the filters considered, the hereditary system was treated with the delay in the pure form and by using Pade approximations of the delay. A summary of the convergence properties of the filters studied is given. The results indicate that the linear filter applied to the delayed system performs inadequately while the nonlinear filters provide reasonable estimates of both the state and the parameters.

  2. Recent results of nonlinear estimators applied to hereditary systems.

    NASA Technical Reports Server (NTRS)

    Schiess, J. R.; Roland, V. R.; Wells, W. R.

    1972-01-01

    An application of the extended Kalman filter to delayed systems to estimate the state and time delay is presented. Two nonlinear estimators are discussed and the results compared with those of the Kalman filter. For all the filters considered, the hereditary system was treated with the delay in the pure form and by using Pade approximations of the delay. A summary of the convergence properties of the filters studied is given. The results indicate that the linear filter applied to the delayed system performs inadequately while the nonlinear filters provide reasonable estimates of both the state and the parameters.

  3. Convex aggregative modelling of infinite memory nonlinear systems

    NASA Astrophysics Data System (ADS)

    Wachel, Paweł

    2016-08-01

    The convex aggregation technique is applied for modelling general class of nonlinear systems with unknown structure and infinite memory. The finite sample size properties of the algorithm are formally established and compared to the standard least-squares counterpart of the method. The proposed algorithm demonstrates its advantages when the a-priori knowledge and the measurement data are both scarce, that is, when the information about the actual system structure is unknown or uncertain and the measurement set is small and disturbed by a noise. Numerical experiments illustrate application and practical benefits of the method for various nonlinear systems.

  4. A model of a nonlinear DNA-protein interaction system with Killingbeck potential and its stability

    NASA Astrophysics Data System (ADS)

    Syahroni, E.; Suparmi, A.; Cari, C.; Fuad, A.

    2016-11-01

    In this paper, we presented a model of a nonlinear DNA-protein interaction system. The interaction system consisted of a molecule of protein bound with a single chain of DNA. The interaction between DNA chain, especially adenine and thymine, and DNA-protein bound to glutamine and adenine. The forms of these bonds are adapted from the hydrogen bonds. The Killingbeack potential were used to describe both of the interactions. We proposed the Hamiltonian equation to describe the general model of interaction. The interaction model is satisfied when a protein molecule triggers pulses on a DNA chain. An initial shift in position of protein xm should trigger the shift in position of DNA ym , or alter the state. However, an initial shift in DNA, yn , should not alter the state of a rest protein (i.e. xm = 0), otherwise, the protein would not steadily bind. We also investigated the stability of the model from the DNA-protein interaction with Lyapunov function. The stability of system can be determined when we obtained the equilibrium point.

  5. Nonlinear analysis for image stabilization in IR imaging system

    NASA Astrophysics Data System (ADS)

    Xie, Zhan-lei; Lu, Jin; Luo, Yong-hong; Zhang, Mei-sheng

    2009-07-01

    In order to acquire stabilization image for IR imaging system, an image stabilization system is required. Linear method is often used in current research on the system and a simple PID controller can meet the demands of common users. In fact, image stabilization system is a structure with nonlinear characters such as structural errors, friction and disturbances. In up-grade IR imaging system, although conventional PID controller is optimally designed, it cannot meet the demands of higher accuracy and fast responding speed when disturbances are present. To get high-quality stabilization image, nonlinear characters should be rejected. The friction and gear clearance are key factors and play an important role in the image stabilization system. The friction induces static error of system. When the system runs at low speed, stick-slip and creeping induced by friction not only decrease resolution and repeating accuracy, but also increase the tracking error and the steady state error. The accuracy of the system is also limited by gear clearance, and selfexcited vibration is brought on by serious clearance. In this paper, effects of different nonlinear on image stabilization precision are analyzed, including friction and gear clearance. After analyzing the characters and influence principle of the friction and gear clearance, a friction model is established with MATLAB Simulink toolbox, which is composed of static friction, Coulomb friction and viscous friction, and the gear clearance non-linearity model is built, providing theoretical basis for the future engineering practice.

  6. Nonperturbative embedding for highly nonlocal Hamiltonians

    NASA Astrophysics Data System (ADS)

    Subaşı, Yiǧit; Jarzynski, Christopher

    2016-07-01

    The need for Hamiltonians with many-body interactions arises in various applications of quantum computing. However, interactions beyond two-body are difficult to realize experimentally. Perturbative gadgets were introduced to obtain arbitrary many-body effective interactions using Hamiltonians with at most two-body interactions. Although valid for arbitrary k -body interactions, their use is limited to small k because the strength of interaction is k th order in perturbation theory. In this paper we develop a nonperturbative technique for obtaining effective k -body interactions using Hamiltonians consisting of at most l -body interactions with l system plus ancilla qubits (possibly using a gate-based device), then evolve with a new Hamiltonian which is more local than the original one (using an analog device), and finally reverse the unitary transformation. The net effect of this procedure is shown to be equivalent to evolving the system with the original nonlocal Hamiltonian. This technique does not suffer from the aforementioned shortcoming of perturbative methods and requires only one ancilla qubit for each k -body interaction irrespective of the value of k . It works best for Hamiltonians with a few many-body interactions involving a large number of qubits and can be used together with perturbative gadgets to embed Hamiltonians of considerable complexity in proper subspaces of two-local Hamiltonians. We describe how our technique can be implemented in a hybrid (gate-based and adiabatic) as well as solely adiabatic quantum computing scheme.

  7. Algebraic function operator expectation value based quantum eigenstate determination: A case of twisted or bent Hamiltonian, or, a spatially univariate quantum system on a curved space

    SciTech Connect

    Baykara, N. A.

    2015-12-31

    Recent studies on quantum evolutionary problems in Demiralp’s group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one.

  8. Stability properties of nonlinear dynamical systems and evolutionary stable states

    NASA Astrophysics Data System (ADS)

    Gleria, Iram; Brenig, Leon; Rocha Filho, Tarcísio M.; Figueiredo, Annibal

    2017-03-01

    In this paper we address the problem of stability in a general class of non-linear systems. We establish a link between the concepts of asymptotic stable interior fixed points of square Quasi-Polynomial systems and evolutionary stable states, a property of some payoff matrices arising from evolutionary games.

  9. Observer Design for a Class of MIMO Nonlinear Systems (Preprint)

    DTIC Science & Technology

    2006-06-01

    without control), because it covers an important class of dynamic systems such as the Van der Pol equation and Duffing oscillator [5], [13] — both of...1992. [5] J. Guckenheimer and P. Holmes, Nonlinear oscillations , dynamical systems, and bifurcations of vector fields, Springer, NY, 1983. [6] A

  10. A bias identification and state estimation methodology for nonlinear systems

    NASA Technical Reports Server (NTRS)

    Caglayan, A. K.; Lancraft, R. E.

    1983-01-01

    A computational algorithm for the identification of input and output biases in discrete-time nonlinear stochastic systems is derived by extending the separate bias estimation results for linear systems to the extended Kalman filter formulation. The merits of the approach are illustrated by identifying instrument biases using a terminal configured vehicle simulation.

  11. Transient stability and control of renewable generators based on Hamiltonian surface shaping and power flow control. Part II, analysis.

    SciTech Connect

    Robinett, Rush D., III; Wilson, David Gerald

    2010-11-01

    The swing equations for renewable generators connected to the grid are developed and a wind turbine is used as an example. The swing equations for the renewable generators are formulated as a natural Hamiltonian system with externally applied non-conservative forces. A two-step process referred to as Hamiltonian Surface Shaping and Power Flow Control (HSSPFC) is used to analyze and design feedback controllers for the renewable generators system. This formulation extends previous results on the analytical verification of the Potential Energy Boundary Surface (PEBS) method to nonlinear control analysis and design and justifies the decomposition of the system into conservative and non-conservative systems to enable a two-step, serial analysis and design procedure. The first step is to analyze the system as a conservative natural Hamiltonian system with no externally applied non-conservative forces. The Hamiltonian surface of the swing equations is related to the Equal-Area Criterion and the PEBS method to formulate the nonlinear transient stability problem. This formulation demonstrates the effectiveness of proportional feedback control to expand the stability region. The second step is to analyze the system as natural Hamiltonian system with externally applied non-conservative forces. The time derivative of the Hamiltonian produces the work/rate (power flow) equation which is used to ensure balanced power flows from the renewable generators to the loads. The Second Law of Thermodynamics is applied to the power flow equations to determine the stability boundaries (limit cycles) of the renewable generators system and enable design of feedback controllers that meet stability requirements while maximizing the power generation and flow to the load. Necessary and sufficient conditions for stability of renewable generators systems are determined based on the concepts of Hamiltonian systems, power flow, exergy (the maximum work that can be extracted from an energy flow) rate

  12. Numerical studies of identification in nonlinear distributed parameter systems

    NASA Technical Reports Server (NTRS)

    Banks, H. T.; Lo, C. K.; Reich, Simeon; Rosen, I. G.

    1989-01-01

    An abstract approximation framework and convergence theory for the identification of first and second order nonlinear distributed parameter systems developed previously by the authors and reported on in detail elsewhere are summarized and discussed. The theory is based upon results for systems whose dynamics can be described by monotone operators in Hilbert space and an abstract approximation theorem for the resulting nonlinear evolution system. The application of the theory together with numerical evidence demonstrating the feasibility of the general approach are discussed in the context of the identification of a first order quasi-linear parabolic model for one dimensional heat conduction/mass transport and the identification of a nonlinear dissipation mechanism (i.e., damping) in a second order one dimensional wave equation. Computational and implementational considerations, in particular, with regard to supercomputing, are addressed.

  13. Applications of equivalent linearization approaches to nonlinear piping systems

    SciTech Connect

    Park, Y.; Hofmayer, C.; Chokshi, N.

    1997-04-01

    The piping systems in nuclear power plants, even with conventional snubber supports, are highly complex nonlinear structures under severe earthquake loadings mainly due to various mechanical gaps in support structures. Some type of nonlinear analysis is necessary to accurately predict the piping responses under earthquake loadings. The application of equivalent linearization approaches (ELA) to seismic analyses of nonlinear piping systems is presented. Two types of ELA`s are studied; i.e., one based on the response spectrum method and the other based on the linear random vibration theory. The test results of main steam and feedwater piping systems supported by snubbers and energy absorbers are used to evaluate the numerical accuracy and limitations.

  14. Identification of continuous-time nonlinear systems: The nonlinear difference equation with moving average noise (NDEMA) framework

    NASA Astrophysics Data System (ADS)

    Zhang, B.; Billings, S. A.

    2015-08-01

    Although a vast number of techniques for the identification of nonlinear discrete-time systems have been introduced, the identification of continuous-time nonlinear systems is still extremely difficult. In this paper, the Nonlinear Difference Equation with Moving Average noise (NDEMA) model which is a general representation of nonlinear systems and contains, as special cases, both continuous-time and discrete-time models, is first proposed. Then based on this new representation, a systematic framework for the identification of nonlinear continuous-time models is developed. The new approach can not only detect the model structure and estimate the model parameters, but also work for noisy nonlinear systems. Both simulation and experimental examples are provided to illustrate how the new approach can be applied in practice.

  15. Nonlinear control structures based on embedded neural system models.

    PubMed

    Lightbody, G; Irwin, G W

    1997-01-01

    This paper investigates in detail the possible application of neural networks to the modeling and adaptive control of nonlinear systems. Nonlinear neural-network-based plant modeling is first discussed, based on the approximation capabilities of the multilayer perceptron. A structure is then proposed to utilize feedforward networks within a direct model reference adaptive control strategy. The difficulties involved in training this network, embedded within the closed-loop are discussed and a novel neural-network-based sensitivity modeling approach proposed to allow for the backpropagation of errors through the plant to the neural controller. Finally, a novel nonlinear internal model control (IMC) strategy is suggested, that utilizes a nonlinear neural model of the plant to generate parameter estimates over the nonlinear operating region for an adaptive linear internal model, without the problems associated with recursive parameter identification algorithms. Unlike other neural IMC approaches the linear control law can then be readily designed. A continuous stirred tank reactor was chosen as a realistic nonlinear case study for the techniques discussed in the paper.

  16. Dynamic analysis of nonlinear rotor-housing systems

    NASA Technical Reports Server (NTRS)

    Noah, Sherif T.

    1988-01-01

    Nonlinear analysis methods are developed which will enable the reliable prediction of the dynamic behavior of the space shuttle main engine (SSME) turbopumps in the presence of bearing clearances and other local nonlinearities. A computationally efficient convolution method, based on discretized Duhamel and transition matrix integral formulations, is developed for the transient analysis. In the formulation, the coupling forces due to the nonlinearities are treated as external forces acting on the coupled subsystems. Iteration is utilized to determine their magnitudes at each time increment. The method is applied to a nonlinear generic model of the high pressure oxygen turbopump (HPOTP). As compared to the fourth order Runge-Kutta numerical integration methods, the convolution approach proved to be more accurate and more highly efficient. For determining the nonlinear, steady-state periodic responses, an incremental harmonic balance method was also developed. The method was successfully used to determine dominantly harmonic and subharmonic responses fo the HPOTP generic model with bearing clearances. A reduction method similar to the impedance formulation utilized with linear systems is used to reduce the housing-rotor models to their coordinates at the bearing clearances. Recommendations are included for further development of the method, for extending the analysis to aperiodic and chaotic regimes and for conducting critical parameteric studies of the nonlinear response of the current SSME turbopumps.

  17. Error estimates for approximate dynamic systems. [linear and nonlinear control systems of different dimensions

    NASA Technical Reports Server (NTRS)

    Gunderson, R. W.; George, J. H.

    1974-01-01

    Two approaches are investigated for obtaining estimates on the error between approximate and exact solutions of dynamic systems. The first method is primarily useful if the system is nonlinear and of low dimension. The second requires construction of a system of v-functions but is useful for higher dimensional systems, either linear or nonlinear.

  18. Error estimates for approximate dynamic systems. [linear and nonlinear control systems of different dimensions

    NASA Technical Reports Server (NTRS)

    Gunderson, R. W.; George, J. H.

    1974-01-01

    Two approaches are investigated for obtaining estimates on the error between approximate and exact solutions of dynamic systems. The first method is primarily useful if the system is nonlinear and of low dimension. The second requires construction of a system of v-functions but is useful for higher dimensional systems, either linear or nonlinear.

  19. Nonlinear Modes in Finite-Dimensional PT-Symmetric Systems

    NASA Astrophysics Data System (ADS)

    Zezyulin, D. A.; Konotop, V. V.

    2012-05-01

    By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schrödinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra does not imply similarity of the structure or stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. In addition, the stability is not directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use a graph representation of PT-symmetric networks allowing for a simple illustration of linearly equivalent networks and indicating their possible experimental design.

  20. A tensor approach to modeling of nonhomogeneous nonlinear systems

    NASA Technical Reports Server (NTRS)

    Yurkovich, S.; Sain, M.

    1980-01-01

    Model following control methodology plays a key role in numerous application areas. Cases in point include flight control systems and gas turbine engine control systems. Typical uses of such a design strategy involve the determination of nonlinear models which generate requested control and response trajectories for various commands. Linear multivariable techniques provide trim about these motions; and protection logic is added to secure the hardware from excursions beyond the specification range. This paper reports upon experience in developing a general class of such nonlinear models based upon the idea of the algebraic tensor product.

  1. Nonlinear system guidance in the presence of transmission zero dynamics

    NASA Technical Reports Server (NTRS)

    Meyer, G.; Hunt, L. R.; Su, R.

    1995-01-01

    An iterative procedure is proposed for computing the commanded state trajectories and controls that guide a possibly multiaxis, time-varying, nonlinear system with transmission zero dynamics through a given arbitrary sequence of control points. The procedure is initialized by the system inverse with the transmission zero effects nulled out. Then the 'steady state' solution of the perturbation model with the transmission zero dynamics intact is computed and used to correct the initial zero-free solution. Both time domain and frequency domain methods are presented for computing the steady state solutions of the possibly nonminimum phase transmission zero dynamics. The procedure is illustrated by means of linear and nonlinear examples.

  2. Anti-synchronization of two hyperchaotic systems via nonlinear control

    NASA Astrophysics Data System (ADS)

    Al-Sawalha, M. Mossa; Noorani, M. S. M.

    2009-08-01

    Based on the nonlinear control theory, the anti-synchronization between two different hyperchaotic systems is investigated. Through rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in hyperchaotic systems. Numerical simulations are performed for the hyperchaotic Chen system and the hyperchaotic Lü system to demonstrate the effectiveness of the proposed control strategy.

  3. Geometrically Induced Nonlinearity in Materials and Structural Systems

    NASA Astrophysics Data System (ADS)

    Ebrahimi, Hamid

    For structural analysis there are three sources of nonlinear behavior. The corresponding nonlinear effects are identified by material, geometry and boundary condition nonlinearities. Here in the present work we focused on nonlinear behavior of structural systems that arises from geometry and specifically tackled three problems: nonlinearity in shell structures, nonlinearity in scale-substrate systems and nonlinearity is cellular solids. Firstly, we present a new instability that is observed in the indentation of a highly ellipsoidal shell by a horizontal plate. Above a critical indentation depth, the plate loses contact with the shell in a series of well-defined `blisters' along the long axis of the ellipsoid. We characterize the onset of this instability and explain it using scaling arguments, numerical simulations and experiments. We also characterize the properties of the blistering pattern by showing how the number of blisters and their size depend on both the geometrical properties of the shell and the indentation but not on the shell's elastic modulus. This blistering instability may be used to determine the thickness of highly ellipsoidal shells simply by squashing them between two plates. For the second problem, we investigate the nonlinear mechanical effects of biomimetic scale like attachments on the behavior of an elastic substrate brought about by the contact interaction of scales in pure bending using qualitative experiments, analytical models and detailed finite element analysis. Our results reveal the existence of three distinct kinematic phases of operation spanning linear, nonlinear and rigid behavior driven by kinematic interactions of scales. The response of the modified elastic beam strongly depends on the size and spatial overlap of rigid scales. The nonlinearity is perceptible even in relatively small strain regime and without invoking material level complexities of either the scales or the substrate. And lastly, we develop a new class of two

  4. Hamiltonian vector fields on almost symplectic manifolds

    NASA Astrophysics Data System (ADS)

    Vaisman, Izu

    2013-09-01

    Let (M, ω) be an almost symplectic manifold (ω is a nondegenerate, not closed, 2-form). We say that a vector field X of M is locally Hamiltonian if LXω = 0, d(i(X)ω) = 0, and it is Hamiltonian if, furthermore, the 1-form i(X)ω is exact. Such vector fields were considered in Fassò and Sansonetto ["Integrable almost-symplectic Hamiltonian systems," J. Math. Phys. 48, 092902 (2007)], 10.1063/1.2783937, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, nonzero, Hamiltonian vector fields, or even none. Therefore, it is important to have examples and it is our aim to provide such examples here. We also obtain some new general results. In particular, we show that the locally Hamiltonian vector fields generate a Dirac structure on M and we state a reduction theorem of the Marsden-Weinstein type. A final section is dedicated to almost symplectic structures on tangent bundles.

  5. On the nonlinear normal modes of free vibration of piecewise linear systems

    NASA Astrophysics Data System (ADS)

    Uspensky, B. V.; Avramov, K. V.

    2014-07-01

    A modification of the Shaw-Pierre nonlinear normal modes is suggested in order to analyze the vibrations of a piecewise linear mechanical systems with finite degrees of freedom. The use of this approach allows one to reduce to twice the dimension of the nonlinear algebraic equations system for nonlinear normal modes calculations in comparison with systems obtained by previous researchers. Two degrees of freedom and fifteen degrees of freedom nonlinear dynamical systems are investigated numerically by using nonlinear normal modes.

  6. A simple approach to nonlinear estimation of physical systems

    USGS Publications Warehouse

    Christakos, G.

    1988-01-01

    Recursive algorithms for estimating the states of nonlinear physical systems are developed. This requires some key hypotheses regarding the structure of the underlying processes. Members of this class of random processes have several desirable properties for the nonlinear estimation of random signals. An assumption is made about the form of the estimator, which may then take account of a wide range of applications. Under the above assumption, the estimation algorithm is mathematically suboptimal but effective and computationally attractive. It may be compared favorably to Taylor series-type filters, nonlinear filters which approximate the probability density by Edgeworth or Gram-Charlier series, as well as to conventional statistical linearization-type estimators. To link theory with practice, some numerical results for a simulated system are presented, in which the responses from the proposed and the extended Kalman algorithms are compared. ?? 1988.

  7. Direct adaptive control of partially known nonlinear systems.

    PubMed

    McLain, R B; Henson, M A; Pottmann, M

    1999-01-01

    A direct adaptive control strategy for a class of single-input/single-output nonlinear systems is presented. The major advantage of the proposed method is that a detailed dynamic nonlinear model is not required for controller design. The only information required about the plant is measurements of the state variables, the relative degree, and the sign of a Lie derivative which appears in the associated input-output linearizing control law. Unknown controller functions are approximated using locally supported radial basis functions that are introduced only in regions of the state space where the closed-loop system actually evolves. Lyapunov stability analysis is used to derive parameter update laws which ensure (under certain assumptions) the state vector remains bounded and the plant output asymptotically tracks the output of a linear reference model. The technique is successfully applied to a nonlinear biochemical reactor model.

  8. Federated nonlinear predictive filtering for the gyroless attitude determination system

    NASA Astrophysics Data System (ADS)

    Zhang, Lijun; Qian, Shan; Zhang, Shifeng; Cai, Hong

    2016-11-01

    This paper presents a federated nonlinear predictive filter (NPF) for the gyroless attitude determination system with star sensor and Global Positioning System (GPS) sensor. This approach combines the good qualities of both the NPF and federated filter. In order to combine them, the equivalence relationship between the NPF and classical Kalman filter (KF) is demonstrated from algorithm structure and estimation criterion. The main features of this approach include a nonlinear predictive filtering algorithm to estimate uncertain model errors and determine the spacecraft attitude by using attitude kinematics and dynamics, and a federated filtering algorithm to process measurement data from multiple attitude sensors. Moreover, a fault detection and isolation algorithm is applied to the proposed federated NPF to improve the estimation accuracy even when one sensor fails. Numerical examples are given to verify the navigation performance and fault-tolerant performance of the proposed federated nonlinear predictive attitude determination algorithm.

  9. Hybrid simulation theory for a classical nonlinear dynamical system

    NASA Astrophysics Data System (ADS)

    Drazin, Paul L.; Govindjee, Sanjay

    2017-03-01

    Hybrid simulation is an experimental and computational technique which allows one to study the time evolution of a system by physically testing a subset of it while the remainder is represented by a numerical model that is attached to the physical portion via sensors and actuators. The technique allows one to study large or complicated mechanical systems while only requiring a subset of the complete system to be present in the laboratory. This results in vast cost savings as well as the ability to study systems that simply can not be tested due to scale. However, the errors that arise from splitting the system in two requires careful attention, if a valid simulation is to be guaranteed. To date, efforts to understand the theoretical limitations of hybrid simulation have been restricted to linear dynamical systems. In this work we consider the behavior of hybrid simulation when applied to nonlinear dynamical systems. As a model problem, we focus on the damped, harmonically-driven nonlinear pendulum. This system offers complex nonlinear characteristics, in particular periodic and chaotic motions. We are able to show that the application of hybrid simulation to nonlinear systems requires a careful understanding of what one expects from such an experiment. In particular, when system response is chaotic we advocate the need for the use of multiple metrics to characterize the difference between two chaotic systems via Lyapunov exponents and Lyapunov dimensions, as well as correlation exponents. When system response is periodic we advocate the use of L2 norms. Further, we are able to show that hybrid simulation can falsely predict chaotic or periodic response when the true system has the opposite characteristic. In certain cases, we are able to show that control system parameters can mitigate this issue.

  10. Frequency bands of strongly nonlinear homogeneous granular systems.

    PubMed

    Lydon, Joseph; Jayaprakash, K R; Ngo, Duc; Starosvetsky, Yuli; Vakakis, Alexander F; Daraio, Chiara

    2013-07-01

    Recent numerical studies on an infinite number of identical spherical beads in Hertzian contact showed the presence of frequency bands [Jayaprakash, Starosvetsky, Vakakis, Peeters, and Kerschen, Nonlinear Dyn. 63, 359 (2011)]. These bands, denoted here as propagation and attenuation bands (PBs and ABs), are typically present in linear or weakly nonlinear periodic media; however, their counterparts are not intuitive in essentially nonlinear periodic media where there is a complete lack of classical linear acoustics, i.e., in "sonic vacua." Here, we study the effects of PBs and ABs on the forced dynamics of ordered, uncompressed granular systems. Through numerical and experimental techniques, we find that the dynamics of these systems depends critically on the frequency and amplitude of the applied harmonic excitation. For fixed forcing amplitude, at lower frequencies, the oscillations are large in amplitude and governed by strongly nonlinear and nonsmooth dynamics, indicating PB behavior. At higher frequencies the dynamics is weakly nonlinear and smooth, in the form of compressed low-amplitude oscillations, indicating AB behavior. At the boundary between the PB and the AB large-amplitude oscillations due to resonance occur, giving rise to collisions between beads and chaotic dynamics; this renders the forced dynamics sensitive to initial and forcing conditions, and hence unpredictable. Finally, we study asymptotically the near field standing wave dynamics occurring for high frequencies, well inside the AB.

  11. The coupled nonlinear dynamics of a lift system

    SciTech Connect

    Crespo, Rafael Sánchez E-mail: stefan.kaczmarczyk@northampton.ac.uk E-mail: huijuan.su@northampton.ac.uk; Kaczmarczyk, Stefan E-mail: stefan.kaczmarczyk@northampton.ac.uk E-mail: huijuan.su@northampton.ac.uk; Picton, Phil E-mail: stefan.kaczmarczyk@northampton.ac.uk E-mail: huijuan.su@northampton.ac.uk; Su, Huijuan E-mail: stefan.kaczmarczyk@northampton.ac.uk E-mail: huijuan.su@northampton.ac.uk

    2014-12-10

    Coupled lateral and longitudinal vibrations of suspension and compensating ropes in a high-rise lift system are often induced by the building motions due to wind or seismic excitations. When the frequencies of the building become near the natural frequencies of the ropes, large resonance motions of the system may result. This leads to adverse coupled dynamic phenomena involving nonplanar motions of the ropes, impact loads between the ropes and the shaft walls, as well as vertical vibrations of the car, counterweight and compensating sheave. Such an adverse dynamic behaviour of the system endangers the safety of the installation. This paper presents two mathematical models describing the nonlinear responses of a suspension/ compensating rope system coupled with the elevator car / compensating sheave motions. The models accommodate the nonlinear couplings between the lateral and longitudinal modes, with and without longitudinal inertia of the ropes. The partial differential nonlinear equations of motion are derived using Hamilton Principle. Then, the Galerkin method is used to discretise the equations of motion and to develop a nonlinear ordinary differential equation model. Approximate numerical solutions are determined and the behaviour of the system is analysed.

  12. Receptance method for active vibration control of a nonlinear system

    NASA Astrophysics Data System (ADS)

    Ghandchi Tehrani, Maryam; Wilmshurst, Laurence; Elliott, Stephen J.

    2013-09-01

    This paper presents the application of the receptance method to nonlinear systems for active vibration control. The method, previously established for linear systems, is extended to a class of single-degree-of-freedom nonlinear systems that can be characterised using describing functions. A significant advantage of the receptance method is that there is no requirement to know the system parameters such as mass, damping and stiffness terms, typically obtained using finite element methods. The method is particularly advantageous for nonlinear systems, since there is no requirement for nonlinear identification. A linear state feedback controller is applied to an example of a single-degree-of-freedom Duffing oscillator, to assign the peak resonance to a prescribed value using the established Sherman-Morrison receptance method. It is then demonstrated that an iterative form of the Sherman-Morrison receptance method is required for the accurate assignment of this peak resonance, in order to account for changes in the open-loop receptance. Both harmonic balance and Volterra series representations are investigated to approximate the receptance in the complex domain, and their advantages and disadvantages are discussed in a numerical example.

  13. Non-linear dynamic analysis of geared systems, part 2

    NASA Technical Reports Server (NTRS)

    Singh, Rajendra; Houser, Donald R.; Kahraman, Ahmet

    1990-01-01

    A good understanding of the steady state dynamic behavior of a geared system is required in order to design reliable and quiet transmissions. This study focuses on a system containing a spur gear pair with backlash and periodically time-varying mesh stiffness, and rolling element bearings with clearance type non-linearities. A dynamic finite element model of the linear time-invariant (LTI) system is developed. Effects of several system parameters, such as torsional and transverse flexibilities of the shafts and prime mover/load inertias, on free and force vibration characteristics are investigated. Several reduced order LTI models are developed and validated by comparing their eigen solution with the finite element model results. Several key system parameters such as mean load and damping ratio are identified and their effects on the non-linear frequency response are evaluated quantitatively. Other fundamental issues such as the dynamic coupling between non-linear modes, dynamic interactions between component non-linearities and time-varying mesh stiffness, and the existence of subharmonic and chaotic solutions including routes to chaos have also been examined in depth.

  14. Parameter identification for nonlinear aerodynamic systems

    NASA Technical Reports Server (NTRS)

    Pearson, Allan E.

    1991-01-01

    Work continues on frequency analysis for transfer function identification, both with respect to the continued development of the underlying algorithms and in the identification study of two physical systems. Some new results of a theoretical nature were recently obtained that lend further insight into the frequency domain interpretation of the research. Progress in each of those areas is summarized. Although not related to the system identification problem, some new results were obtained on the feedback stabilization of linear time lag systems.

  15. Seismic analysis of series isolation system based on geometry nonlinearity

    NASA Astrophysics Data System (ADS)

    Lin, Z. D.; Shi, H.; Xue, L.

    2017-08-01

    According to the system of rubber bearing serially connected with column, the mathematical model of serially isolated system based on geometric nonlinear is investigated by using Hamilton’s principle. The effects of axial pressure and difference column size to the series isolation system in seismic response is discussed. The series isolation system dynamics model based on geometric nonlinear is established considering the cross section rotated and the influence of the shear deformation and axial pressure. The differential quadrature element method is employed for discrete processing on governing equations and boundary conditions. Seismic response of series isolation system subjected to the far-field ground motions is solved numerically. Results show that: the slenderness ratio of cantilever column will significantly affect the seismic response of the isolation system under far-field ground motions, and it is particularly to response of the cantilever column.

  16. Model system-bath Hamiltonian and nonadiabatic rate constants for proton-coupled electron transfer at electrode-solution interfaces.

    PubMed

    Navrotskaya, Irina; Soudackov, Alexander V; Hammes-Schiffer, Sharon

    2008-06-28

    An extension of the Anderson-Newns-Schmickler model for electrochemical proton-coupled electron transfer (PCET) is presented. This model describes reactions in which electron transfer between a solute complex in solution and an electrode is coupled to proton transfer within the solute complex. The model Hamiltonian is derived in a basis of electron-proton vibronic states defined within a double adiabatic approximation for the electrons, transferring proton, and bath modes. The interaction term responsible for electronic transitions between the solute complex and the electrode depends on the proton donor-acceptor vibrational mode within the solute complex. This model Hamiltonian is used to derive the anodic and cathodic rate constants for nonadiabatic electrochemical PCET. The derivation is based on the master equations for the reduced density matrix of the electron-proton subsystem, which includes the electrons of the solute complex and the electrode, as well as the transferring proton. The rate constant expressions differ from analogous expressions for electrochemical electron transfer because of the summation over electron-proton vibronic states and the dependence of the couplings on the proton donor-acceptor vibrational motion. These differences lead to additional contributions to the total reorganization energy, an additional exponential temperature-dependent prefactor, and a temperature-dependent term in the effective activation energy that has different signs for the anodic and cathodic processes. This model can be generalized to describe both nonadiabatic and adiabatic electrochemical PCET reactions and provides the framework for the inclusion of additional effects, such as the breaking and forming of other chemical bonds.

  17. From nonlinear Schroedinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations

    SciTech Connect

    Yang Xiao; Du Dianlou

    2010-08-15

    The Poisson structure on C{sup N}xR{sup N} is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schroedinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.

  18. Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Liénard type nonlinear oscillators

    SciTech Connect

    Chithiika Ruby, V.; Senthilvelan, M.; Lakshmanan, M.; Chandrasekar, V. K.

    2015-01-15

    We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

  19. Simple nonlinear systems and navigating catastrophes

    NASA Astrophysics Data System (ADS)

    Harré, Michael S.; Atkinson, Simon R.; Hossain, Liaquat

    2013-06-01

    Tipping points are a common occurrence in complex adaptive systems. In such systems feedback dynamics strongly influence equilibrium points and they are one of the principal concerns of research in this area. Tipping points occur as small changes in system parameters result in disproportionately large changes in the global properties of the system. In order to show how tipping points might be managed we use the Maximum Entropy (MaxEnt) method developed by Jaynes to find the fixed points of an economic system in two different ways. In the first, economic agents optimise their choices based solely on their personal benefits. In the second they optimise the total benefits of the system, taking into account the effects of all agent's actions. The effect is to move the game from a recently introduced dual localised Lagrangian problem to that of a single global Lagrangian. This leads to two distinctly different but related solutions where localised optimisation provides more flexibility than global optimisation. This added flexibility allows an economic system to be managed by adjusting the relationship between macro parameters, in this sense such manipulations provide for the possibility of "steering" an economy around potential disasters.

  20. Solidification of ternary systems with a nonlinear phase diagram

    NASA Astrophysics Data System (ADS)

    Alexandrov, D. V.; Dubovoi, G. Yu.; Malygin, A. P.; Nizovtseva, I. G.; Toropova, L. V.

    2017-02-01

    The directional solidification of a ternary system with an extended phase transition region is theoretically studied. A mathematical model is developed to describe quasi-stationary solidification, and its analytical solution is constructed with allowance for a nonlinear liquidus line equation. A deviation of the liquidus equation from a linear function is shown to result in a substantial change in the solidification parameters.

  1. Finding sets of solutions to systems of nonlinear inequalities

    NASA Astrophysics Data System (ADS)

    Evtushenko, Yu. G.; Posypkin, M. A.; Rybak, L. A.; Turkin, A. V.

    2017-08-01

    The problem of approximating the set of all solutions to a system of nonlinear inequalities is studied. A method based on the concept of nonuniform coverings is proposed. It allows one to obtain an interior and exterior approximation of this set with a prescribed accuracy. The efficiency of the method is demonstrated by determining the workspace of a parallel robot.

  2. Photon nonlinear mixing in subcarrier multiplexed quantum key distribution systems.

    PubMed

    Capmany, José

    2009-04-13

    We provide, for the first time to our knowledge, an analysis of the influence of nonlinear photon mixing on the end to end quantum bit error rate (QBER) performance of subcarrier multiplexed quantum key distribution systems. The results show that negligible impact is to be expected for modulation indexes in the range of 2%.

  3. Adaptive Control of Flat MIMO Nonlinear Systems with Additive Disturbance

    DTIC Science & Technology

    2006-01-01

    Survey of Iterative Learning Control,” IEEE Control Systems Magazine, Vol. 26, No. 3, pp. 96–114, 2006. [2] Z. Cai, M.S. de Queiroz , and D.M. Dawson...427, 1996. [17] B. Xian, D.M. Dawson, M.S. de Queiroz , and J. Chen, “A Continuous Asymptotic Tracking Control Strategy for Uncertain Nonlinear Sys- tems

  4. Passive dynamic controllers for non-linear mechanical systems

    NASA Technical Reports Server (NTRS)

    Juang, Jer-Nan; Wu, Shih-Chin; Phan, Minh; Longman, Richard W.

    1992-01-01

    The objective is to develop active model-independent controllers for slewing and vibration control of nonlinear multibody flexible systems, including flexible robots. The topics are presented in viewgraph form and include: passive stabilization; work-energy rate principle; Liapunov theory; displacement feedback; dynamic controller; displacement and acceleration feedback; velocity feedback; displacement feedback; physical interaction; a 6-DOF robot; and simulation results.

  5. Application of dynamical systems theory to nonlinear aircraft dynamics

    NASA Technical Reports Server (NTRS)

    Culick, Fred E. C.; Jahnke, Craig C.

    1988-01-01

    Dynamical systems theory has been used to study nonlinear aircraft dynamics. A six degree of freedom model that neglects gravity has been analyzed. The aerodynamic model, supplied by NASA, is for a generic swept wing fighter and includes nonlinearities as functions of the angle of attack. A continuation method was used to calculate the steady states of the aircraft, and bifurcations of these steady states, as functions of the control deflections. Bifurcations were used to predict jump phenomena and the onset of periodic motion for roll coupling instabilities and high angle of attack maneuvers. The predictions were verified with numerical simulations.

  6. Extreme nonlinear optics of two-level systems

    SciTech Connect

    Tritschler, T.; Muecke, O. D.; Wegener, M.

    2003-09-01

    For Rabi frequencies comparable to, or even larger than, the transition frequency of a two-level system, the regime of extreme nonlinear optics is reached. Here, we give an overview of the radiated light intensity as a function of carrier frequency of light, transition frequency, Rabi frequency, spectrometer frequency, as well as of the shape and duration of the exciting optical pulses. The graphical representations reveal an amazing complexity and beauty of the nonlinear optical response. Analytical results within the ''square-wave approximation'' qualitatively reproduce many of the intricate features of the exact numerical calculations.

  7. Hamiltonian description of closed configurations of the vacuum magnetic field

    SciTech Connect

    Skovoroda, A. A.

    2015-05-15

    Methods of obtaining and using the Hamiltonians of closed vacuum magnetic configurations of fusion research systems are reviewed. Various approaches to calculate the flux functions determining the Hamiltonian are discussed. It is shown that the Hamiltonian description allows one not only to reproduce all traditional results, but also to study the behavior of magnetic field lines by using the theory of dynamic systems. The potentialities of the Hamiltonian formalism and its close relation to traditional methods are demonstrated using a large number of classical examples adopted from the fundamental works by A.I. Morozov, L.S. Solov’ev, and V.D. Shafranov.

  8. Action with Acceleration i: Euclidean Hamiltonian and Path Integral

    NASA Astrophysics Data System (ADS)

    Baaquie, Belal E.

    2013-10-01

    An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.

  9. Terminal Sliding Modes In Nonlinear Control Systems

    NASA Technical Reports Server (NTRS)

    Venkataraman, Subramanian T.; Gulati, Sandeep

    1993-01-01

    Control systems of proposed type called "terminal controllers" offers increased precision and stability of robotic operations in presence of unknown and/or changing parameters. Systems include special computer hardware and software implementing novel control laws involving terminal sliding modes of motion: closed-loop combination of robot and terminal controller converge, in finite time, to point of stable equilibrium in abstract space of velocity and/or position coordinates applicable to particular control problem.

  10. Terminal Sliding Modes In Nonlinear Control Systems

    NASA Technical Reports Server (NTRS)

    Venkataraman, Subramanian T.; Gulati, Sandeep

    1993-01-01

    Control systems of proposed type called "terminal controllers" offers increased precision and stability of robotic operations in presence of unknown and/or changing parameters. Systems include special computer hardware and software implementing novel control laws involving terminal sliding modes of motion: closed-loop combination of robot and terminal controller converge, in finite time, to point of stable equilibrium in abstract space of velocity and/or position coordinates applicable to particular control problem.

  11. On stability theory. [of nonlinear feedback control systems

    NASA Technical Reports Server (NTRS)

    Safonov, M. G.; Athans, M.

    1979-01-01

    It is found that under mild assumptions, feedback system stability can be concluded if one can 'topologically separate' the infinite-dimensional function space containing the system's dynamical input-output relations into two regions, one region containing the dynamical input-output relation of the 'feedforward' element of the system and the other region containing the dynamical output-input relation of the 'feedback' element. Nonlinear system stability criteria of both the input-output type and the state-space (Liapunov) type are interpreted in this context. The abstract generality and conceptual simplicity afforded by the topological separation perspective clarifies some of the basic issues underlying stability theory and serves to suggest improvements in existing stability criteria. A generalization of Zames' (1966) conic-relation stability criterion is proved, laying the foundation for improved multivariable generalizations of the frequency-domain circle stability criterion for nonlinear systems.

  12. Robust Decentralized Nonlinear Control for a Twin Rotor MIMO System

    PubMed Central

    Belmonte, Lidia María; Morales, Rafael; Fernández-Caballero, Antonio; Somolinos, José Andrés

    2016-01-01

    This article presents the design of a novel decentralized nonlinear multivariate control scheme for an underactuated, nonlinear and multivariate laboratory helicopter denominated the twin rotor MIMO system (TRMS). The TRMS is characterized by a coupling effect between rotor dynamics and the body of the model, which is due to the action-reaction principle originated in the acceleration and deceleration of the motor-propeller groups. The proposed controller is composed of two nested loops that are utilized to achieve stabilization and precise trajectory tracking tasks for the controlled position of the generalized coordinates of the TRMS. The nonlinear internal loop is used to control the electrical dynamics of the platform, and the nonlinear external loop allows the platform to be perfectly stabilized and positioned in space. Finally, we illustrate the theoretical control developments with a set of experiments in order to verify the effectiveness of the proposed nonlinear decentralized feedback controller, in which a comparative study with other controllers is performed, illustrating the excellent performance of the proposed robust decentralized control scheme in both stabilization and trajectory tracking tasks. PMID:27472338

  13. Robust Decentralized Nonlinear Control for a Twin Rotor MIMO System.

    PubMed

    Belmonte, Lidia María; Morales, Rafael; Fernández-Caballero, Antonio; Somolinos, José Andrés

    2016-07-27

    This article presents the design of a novel decentralized nonlinear multivariate control scheme for an underactuated, nonlinear and multivariate laboratory helicopter denominated the twin rotor MIMO system (TRMS). The TRMS is characterized by a coupling effect between rotor dynamics and the body of the model, which is due to the action-reaction principle originated in the acceleration and deceleration of the motor-propeller groups. The proposed controller is composed of two nested loops that are utilized to achieve stabilization and precise trajectory tracking tasks for the controlled position of the generalized coordinates of the TRMS. The nonlinear internal loop is used to control the electrical dynamics of the platform, and the nonlinear external loop allows the platform to be perfectly stabilized and positioned in space. Finally, we illustrate the theoretical control developments with a set of experiments in order to verify the effectiveness of the proposed nonlinear decentralized feedback controller, in which a comparative study with other controllers is performed, illustrating the excellent performance of the proposed robust decentralized control scheme in both stabilization and trajectory tracking tasks.

  14. Nonlinear dynamics and quantum entanglement in optomechanical systems.

    PubMed

    Wang, Guanglei; Huang, Liang; Lai, Ying-Cheng; Grebogi, Celso

    2014-03-21

    To search for and exploit quantum manifestations of classical nonlinear dynamics is one of the most fundamental problems in physics. Using optomechanical systems as a paradigm, we address this problem from the perspective of quantum entanglement. We uncover strong fingerprints in the quantum entanglement of two common types of classical nonlinear dynamical behaviors: periodic oscillations and quasiperiodic motion. There is a transition from the former to the latter as an experimentally adjustable parameter is changed through a critical value. Accompanying this process, except for a small region about the critical value, the degree of quantum entanglement shows a trend of continuous increase. The time evolution of the entanglement measure, e.g., logarithmic negativity, exhibits a strong dependence on the nature of classical nonlinear dynamics, constituting its signature.

  15. Nonlinearity as a resource for nonclassicality in anharmonic systems

    NASA Astrophysics Data System (ADS)

    Albarelli, Francesco; Ferraro, Alessandro; Paternostro, Mauro; Paris, Matteo G. A.

    2016-03-01

    Nonclassicality is a key ingredient for quantum enhanced technologies and experiments involving macroscopic quantum coherence. Considering various exactly solvable quantum-oscillator systems, we address the role played by the anharmonicity of their potential in the establishment of nonclassical features. Specifically, we show that a monotonic relation exists between the entropic nonlinearity of the considered potentials and their ground-state nonclassicality, as quantified by the negativity of the Wigner function. In addition, in order to clarify the role of squeezing, which is not captured by the negativity of the Wigner function, we focus on the Glauber-Sudarshan P function and address the nonclassicality-nonlinearity relation using the entanglement potential. Finally, we consider the case of a generic sixth-order potential confirming the idea that nonlinearity is a resource for the generation of nonclassicality and may serve as a guideline for the engineering of quantum oscillators.

  16. Nonlinear gyrokinetic equations for tokamak microturbulence

    SciTech Connect

    Hahm, T.S.

    1988-05-01

    A nonlinear electrostatic gyrokinetic Vlasov equation, as well as Poisson equation, has been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov-Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits. 13 refs.

  17. On discrete control of nonlinear systems with applications to robotics

    NASA Technical Reports Server (NTRS)

    Eslami, Mansour

    1989-01-01

    Much progress has been reported in the areas of modeling and control of nonlinear dynamic systems in a continuous-time framework. From implementation point of view, however, it is essential to study these nonlinear systems directly in a discrete setting that is amenable for interfacing with digital computers. But to develop discrete models and discrete controllers for a nonlinear system such as robot is a nontrivial task. Robot is also inherently a variable-inertia dynamic system involving additional complications. Not only the computer-oriented models of these systems must satisfy the usual requirements for such models, but these must also be compatible with the inherent capabilities of computers and must preserve the fundamental physical characteristics of continuous-time systems such as the conservation of energy and/or momentum. Preliminary issues regarding discrete systems in general and discrete models of a typical industrial robot that is developed with full consideration of the principle of conservation of energy are presented. Some research on the pertinent tactile information processing is reviewed. Finally, system control methods and how to integrate these issues in order to complete the task of discrete control of a robot manipulator are also reviewed.

  18. Staggered quantum walks with Hamiltonians

    NASA Astrophysics Data System (ADS)

    Portugal, R.; de Oliveira, M. C.; Moqadam, J. K.

    2017-01-01

    Quantum walks are recognizably useful for the development of new quantum algorithms, as well as for the investigation of several physical phenomena in quantum systems. Actual implementations of quantum walks face technological difficulties similar to the ones for quantum computers, though. Therefore, there is a strong motivation to develop new quantum-walk models which might be easier to implement. In this work we present an extension of the staggered quantum walk model that is fitted for physical implementations in terms of time-independent Hamiltonians. We demonstrate that this class of quantum walk includes the entire class of staggered quantum walk model, Szegedy's model, and an important subset of the coined model.

  19. 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

  20. Cumulant generating function formula of heat transfer in ballistic systems with lead-lead coupling and general nonlinear systems

    NASA Astrophysics Data System (ADS)

    Li, Huanan

    2013-03-01

    Based on a two-time observation protocol, we consider heat transfer in a given time interval tM in a lead-junction-lead system taking coupling between the leads into account. In view of the two-time observation, consistency conditions are carefully verified in our specific family of quantum histories. Furthermore, its implication is briefly explored. Then using the nonequilibrium Green's function method, we obtain an exact formula for the cumulant generating function for heat transfer between the two leads, valid in both transient and steady-state regimes. Also, a compact formula for the cumulant generating function in the long-time limit is derived, for which the Gallavotti-Cohen fluctuation symmetry is explicitly verified. In addition, we briefly discuss Di Ventra's repartitioning trick regarding whether the repartitioning procedure of the total Hamiltonian affects the nonequilibrium steady-state current fluctuation. All kinds of properties of nonequilibrium current fluctuations, such as the fluctuation theorem in different time regimes, could be readily given according to these exact formulas. Finally a practical formalism dealing with cumulants of heat transfer across general nonlinear quantum systems is established based on field theoretical/algebraic method.