Dissipative Chaotic Systems Research Articles

THE PENDULUM WEAVES ALL KNOTS AND LINKS

The behavior of a dissipative chaotic dynamical system is determined by the skeleton of its strange attractor, which consists of an uncountably infinite set of unstable periodic orbits. Each orbit is a topological knot, and the set is an infinite link. The types of knots and links supported by the system may be determined by collapsing the attractor along its local stable manifolds to form a template, a branched two manifold with boundary that supports the same set of knots and links as the original attractor. We show that the strange attractor of a chaotic, vertically forced physical pendulum can be collapsed to a template that supports all knots and links. Thus, one of the simplest and most well known dynamical systems is capable of the most complex behavior possible.

Dissipative Quantum Chaos: Transition from Wave Packet Collapse to Explosion

Using the quantum trajectories approach, we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable nonlinear dynamics the explosion practically disappears leaving place to collapse.

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lü–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rössler system at least a fourth-order polynomial is required to properly represent its energy.

Quantum Ratchets in Dissipative Chaotic Systems

Using the method of quantum trajectories, we study a quantum chaotic dissipative ratchet appearing for particles in a pulsed asymmetric potential in the presence of a dissipative environment. The system is characterized by directed transport emerging from a quantum strange attractor. This model exhibits, in the limit of small effective Planck constant, a transition from quantum to classical behavior, in agreement with the correspondence principle. We also discuss parameter values suitable for the implementation of the quantum ratchet effect with cold atoms in optical lattices.

Crossover from classical to quantum behavior of the Duffing oscillator through a pseudo-Lyapunov-exponent

We discuss the quantum-classical correspondence (QCC) in a specific dissipative chaotic system, the Duffing oscillator. The quantum version of the Duffing oscillator is treated as an open quantum system and analyzed numerically by the use of quantum state diffusion (QSD). We consider a pseudo-Lyapunov exponent and investigate it in detail, varying the Planck constant effectively. We show that there exists a critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we find that a dissipation effect suppresses the occurrence of chaos in the quantum region, while it, combined with the periodic external force, plays a crucial role in the chaotic behavior of the classical system.

Control of chaos by capture and release.

We propose and demonstrate a different method for the control of flip saddles in dissipative chaotic systems. Due to the dynamics of a flip saddle, the stable manifolds of a target orbit and its perturbation can be modeled as a pair of concentric Möbius bands. Over the period of the target orbit, these bands rotate relative to one another. The method of capture and release (CR) takes advantage of this rotation, and captures a nearby system state in the perturbed stable manifold, releasing it when the rotation of the Möbius bands brings them into alignment. Unlike the method of Ott, Grebogi, and Yorke and most of its variants, CR does not rely on the unstable component of the flow to push the system state onto the stable manifold; the system state is evolving in a stable subspace for the duration of the control perturbation.

Crossover from Classical to Quantum Behavior in Duffing Oscillator through “Pseudo-Lyapunov Exponent”

We discuss the quantum–classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) that is a certain formulation for quantum open systems and a useful tool for analyzing nonlinear problems and their classical limit. We define a “pseudo-Lyapunov exponent” and investigate the behavior of this quantity varying Planck constant effectively. We show that in a dissipative system there exists some critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that the role of dissipation changes for quantized Duffing oscillator since the effect of dissipation is important for the occurrence of chaos in classical mechanics but suppresses it in quantum region.

Targeting in dissipative chaotic systems: A survey.

The large number of unstable equilibrium modes embedded in the strange attractor of dissipative chaotic systems usually presents a sufficiently rich repertoire for the choice of the desirable motion as a target. Once the system is close enough to the chosen target local stabilization techniques can be employed to capture the system within the desired motion. The ergodic behavior of chaotic systems on their strange attractors guarantees that the system will eventually visit a close neighborhood of the target. However, for arbitrary initial conditions within the basin of attraction of the strange attractor the waiting time for such a visit may be intolerably long. In order to reduce the long waiting time it usually becomes indispensable to employ an appropriate method of targeting, which refers to the task of steering the system toward the close neighborhood of the target. This paper provides a survey of targeting methods proposed in the literature for dissipative chaotic systems. (c) 2002 American Institute of Physics.

The crossover from classical to quantum behavior in Duffing oscillator based on quantum state diffusion

The classical Duffing oscillator is a dissipative chaotic system, and so there is not a definite Hamiltonian. We quantize the Duffing oscillator on the basis of quantum state diffusion (QSD) which is a formulation for open quantum systems and a useful tool for analyzing nonlinear problems and classical limits. We can define a ‘Lyapunov exponent’, which corresponds to the classical one in the proper limit, and investigate the behavior of the system by varying the Planck constant effectively. We show that there exists a critical stage, where the behavior of the system crosses over from classical to quantum one.

Stability оf numerical estimations of time series characteristics

The numerical methods of a data analysis (finite ordered sequences of natural binary codes), corresponding to fragments оf trajectories, obtained by the numerical solving of а finite number of nonlinear ordinary differential equations are discussed. These equations represent the determined dissipative chaotic dynamic systems. Measuring properties of such sequences are characterized by the estimation which assumes the code, obtained as a result of processing of source sequence of data by given algorithm, realized on the computer and not supposing the participation of the expert. The concept оf stability оf the estimation is formalized. The stability of obtained estimations in relation to numerically simulated small variations оf registration scheme parameters and to parameters of the algorithm of processing is investigated. Examples of data sequences with typical parameters (digit capacity, time step, length of sequence) for many real experiments are considered. It is shown that estimation, obtained by the algorithm based on the analysis оf properties essentially depended оn behavior оf trajectory in every point, is unstable. At the same time, the estimation, obtained by the algorithm based on the analysis of properties, essentially depended оn some «average» for different points behavior оf trajectory, is stable.

Multidimensional reactive rate calculations in dissipative chaotic systems

Using concepts from transient chaos and stochastic dynamics, we develop a perturbative solution for multidimensional activated rate processes. The solution is applicable to the underdamped regime where system dynamics prevails over bath fluctuations. The baseline of the method is the partition of the multidimensional reactive flux in a chaotic system to a sum of independent fluxes in one-dimensional systems. The partition is based on the underlying dynamics of the multidimensional system. The method is fast and explains the high and low temperature dependence of multidimensional reaction rates.

Anticipating chaotic synchronization

Dissipative chaotic systems with a time-delayed feedback can drive near-identical systems in such a way that the driven systems anticipate the drivers by synchronizing with their (arbitrarily distant) future states. This counterintuitive behavior is globally stable, robust, and a pure result of the interplay between delayed feedback and dissipation. Thus it constitutes a rather universal phenomenon of nonlinear dynamics. For small anticipation times, anticipating synchronization also occurs in chaotic systems without a memory term in the driver.

Controlling Chaos Using Input–Output Linearization Approach

In this article the input–output linearization approach is used for controlling chaos. It is shown that by using only partial states, the entire chaotic system is stabilizable, provided the zero dynamics is stable. Generally speaking, trajectories of chaotic systems do not grow exponentially and are usually bounded. In particular, for dissipative chaotic systems the stable zero dynamics can always be found. Hence the stabilization as well as tracking periodic signals are possible. The Lorenz system is used to inform the discussion. Simulation results are presented to show the effectiveness of the approach.

Complexity in Hamiltonian-driven dissipative chaotic dynamical systems

The existence of symmetry in chaotic dynamical systems often leads to one or several low-dimensional invariant subspaces in the phase space. We demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is Hamiltonian but the full system is dissipative. In particular, an infinite number of distinct attractors can coexist. These attractors can be quasiperiodic, strange nonchaotic, and chaotic with different positive Lyapunov exponents. Finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors. However, arbitrarily small perturbations can lead to dynamically distinct chaotic attractors. This work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom. \textcopyright{} 1996 The American Physical Society.

Chaos at structural phase transitions

Abstract It is well known that the structural phase transitions in ferroelectric materials are connected with strong nonlinear properties. So we should expect all features of nonlinear dynamical systems such as period-doubling-cascades and chaotic behaviour in a dynamical system that contains ferroelectric materials. The experimentally investigated nonlinear dynamical system is a series resonance circuit, consisting of a linear inductance and a nonlinear capacitance. Different ferroelectric materials have been used as a nonlinear capacitance. There are two directions for our investigations: (i) The response-function of the nonlinear dynamical system and analysis methods of the nonlinear dynamics for dissipative chaotic systems (e. g. phase portrait, Fourier spectra, correlation dimension and Lyapunov exponents) are used for the study of structural phase-transitions and for the characterization of the large signal behaviour of ferroelectric materials. (ii) Controlling the chaotic behaviour of the system by applying a small perturbation of an external control parameter to it. This way the chaotic motion of the system can be switched over to a regular one by stabilizing one of the many regular instable states of the system.

SYNTHESIZING SELF-SYNCHRONIZING CHAOTIC SYSTEMS

A systematic approach is developed for synthesizing N-dimensional dissipative chaotic systems which possess the self-synchronization property. The ability to synthesize new chaotic systems further enhances the usefulness of synchronized chaotic systems for communications.

Chaotic systems: counting the number of periods

Abstract Characterization of chaotic motion may proceed both at an averaged “macroscopic” level, using such notions as Lyapunov exponents, dimensions and entropies, and at a “microscopic” level. In the latter case, the number of periodic orbits, being a topological invariant, plays an important role. For various one-dimensional mappings, the counting problem itself has many interesting facets and may be solved more or less completely in different ways. Recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher-dimensional dissipative chaotic systems.