Nonchaotic Systems Research Articles

Synchronization of non-chaotic dynamical systems

A synchronization mechanism driven by annealed noise is studied for two replicas of a coupled-map lattice which exhibits stable chaos (SC), i.e., irregular behavior despite a negative Lyapunov spectrum. We show that the observed synchronization transition, on changing the strength of the stochastic coupling between replicas, belongs to the directed percolation universality class. This result is consistent with the behavior of chaotic deterministic cellular automata (DCA), supporting the equivalence ansatz between SC models and DCA. The coupling threshold above which the two system replicas synchronize is strictly related to the propagation velocity of perturbations in the system.

Note on chaos and diffusion

Using standard definitions of chaos (as positive Kolmogorov–Sinai entropy) and diffusion (that multiple time distribution functions are Gaussian), we show numerically that both chaotic and nonchaotic systems exhibit diffusion, and hence that there is no direct logical connection between the two properties. This extends a previous result for two time distribution functions.

Adaptive estimation and control method for unstable periodic dynamics in spike trains.

Dynamical control of excitable biological systems is often complicated by the difficult and unreliable task of precontrol identification of unstable periodic orbits (UPO's). Here we show that, for both chaotic and nonchaotic systems, UPO's can be located, and their dynamics characterized, during control. Tracking of system nonstationarities emerges naturally from this approach. Such a method is potentially valuable for the control of spike trains of excitable biological systems, for which precontrol UPO identification is often impractical, and nonstationarities (natural or stimulation induced) are common.

The Synchronization of Strange Nonchaotic Systems Based on Contraction Maps

In this paper a new method of synchronization of strange nonchaotic systems is proposed. Strange nonchaotic attractors are obtained from a chaotic system.driven by another chao tic system. Because the Lyapunov exponent of the strange nonchaotic attractor is negative it can sup presses the sensitive dependence on initial conditions of the chaotic system. In order to synchronize the strange nonchaotic system-contraction map is used to synchronize the chaotic driving system. Simul ations indicate that this method can synchronize fast be robust to nose-easily be implemented and c an be used in the field of chaotic secure communication. 

Learning Control of Time-Delay Chaotic Systems and its Applications

The present paper proposes a learning control system that automatically stabilizes one-dimensional time-delayed chaotic systems. We give a systematic procedure to design the control system using a few pieces of uncertain information on the chaotic system. Furthermore, this control system can be applied to a technique that moves a stable orbit of nonchaotic systems to coexistent unstable points and stabilizes the moving orbit onto these unstable points. To check the theoretical results, we demonstrate some numerical experiments.

Nonchaotic attractors with highly fluctuating finite-time Lyapunov exponents

The implications of large fluctuations of finite-time Lyapunov exponents are discussed for nonchaotic systems. We show that for a nonchaotic system driven by quasiperiodic force, if its finite-time Lyapunov exponents periodically become positive, the resulting attractor can be strange but nonchaotic. With long enough expanding time intervals, a nearly periodic force can also lead to strange nonchaotic attractors. For the case of a periodic force, a special typical periodic attractor that is sensitive to micronoise is obtained. With the different finite computing precisions, different pseudoperiodic orbits can be obtained.

Spatial complex behavior in nonchaotic flow systems

The convective instability induces the phenomenon of spatial sensitivity to the boundary conditions. We introduce a quantity, the ``spatial'' Lyapunov exponent, to characterize this kind of complex behavior, in nonchaotic but convectively unstable flow systems. Then we establish a relation between this spatial-complexity index and the comoving Lyapunov exponent.

Mode Fluctuations as Fingerprints of Chaotic and Non-Chaotic Systems

The mode-fluctuation distribution P(W) is studied for chaotic as well as for non-chaotic quantum billiards. This statistic is discussed in the broader framework of the E(k,L) functions being the probability of finding k energy levels in a randomly chosen interval of length L, and the distribution of n(L), where n(L) is the number of levels in such an interval, and their cumulants ck(L). It is demonstrated that the cumulants provide a possible measure for the distinction between chaotic and non-chaotic systems. The vanishing of the normalized cumulants Ck, k ≥ 3, implies a Gaussian behaviour of P(W), which is realized in the case of chaotic systems, whereas non-chaotic systems display non-vanishing values for these cumulants leading to a non-Gaussian behaviour of P(W). For some integrable systems there exist rigorous proofs of the non-Gaussian behaviour which are also discussed. Our numerical results and the rigorous results for integrable systems suggest that a clear fingerprint of chaotic systems is provided by a Gaussian distribution of the mode-fluctuation distribution P(W).

Real-time, adaptive, model-independent control of low-dimensional chaotic and nonchaotic dynamical systems

Current model-independent control techniques are limited, from a practical standpoint, by their dependence on a precontrol learning stage. Here we develop a model-independent control technique, for chaotic and nonchaotic low-dimensional dynamical systems, that operates in real-time (i.e., it does not require a learning stage). We show that this technique is adaptive to system nonstationarities, robust to noise, and capable of stabilizing higher-order unstable periodic orbits. Because this technique is real-time, adaptive, and model-independent, it is practical for real-world systems.

Inducing stable periodic behaviour in a class of dynamical systems by parametric perturbations

We develop an analytic approach to the problem of the behaviour stabilization of dynamical systems by parametric perturbations. First, questions of realization of the stable dynamics in non-chaotic systems with continuous time are studied. It is analytically shown that by means of parametric perturbations it is possible to obtain the stable periodic behaviour in systems which do not possess stable oscillations in the autonomous case. Then, on the basis of these results we advance the following conjecture: assume that a system has a chaotic attractor. Then, if we successfully choose a parametric perturbation of such a system in those regions where its behaviour is chaotic, then one can expect that this perturbation leads to the appearance of the stable periodic orbits which are either unstable or non-existent in the initial unperturbed system. We present rigorous results which assert that for some discrete time dynamical systems such a conjecture is valid: for certain families of mappings it is possible to find perturbations that lead to stabilization of their chaotic dynamics. In the framework of such an approach we offer a goal-oriented non-feedback way for stabilization of the desired stable periodic behaviour.

Using noise and chaos control to control nonchaotic systems.

Here we show that chaos control techniques can be used to stabilize unstable periodic orbits in a nonchaotic system provided additive noise can be utilized (1) to determine the local dynamics of a chosen orbit, and (2) to move the system's trajectory into the neighborhood of the orbit so that control can be initiated. Using these techniques, we demonstrate that the qualitative dynamics of a nonchaotic system can be altered without using large controls or large parameter shifts. Unlike classical control methods, this approach requires no knowledge of the underlying system equations. (c) 1995 The American Physical Society

Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis

Chaos is usually regarded as a distinct alternative to random effects such as environmental fluctuations or external disturbances. We argue that strict separation between chaotic and stochastic dynamics in ecological systems is unnecessary and misleading, and we present a more comprehensive approach for systems subject to stochastic perturbations. The defining property of chaos is sensitive dependence on initial conditions. Chaotic systems are amplifiers that magnify perturbations; nonchaotic systems are mufflers that dampen perturbations. We also present statistical methods for detecting chaos in time-series data, based on using nonlinear time-series modeling to estimate the Lyapunov exponent λ, which gives the average rate at which perturbation effects grow (λ > 0) or decay (λ < 0). These methods allow for dynamic noise and can detect low-dimensional chaos with realistic amounts of data. Results for natural and laboratory populations span the entire range from noise-dominated and strongl...

Numerical Study of Sound Emission by 2D Regular and Chaotic Vortex Configurations

The far-field noise generated by a system of three Gaussian vortices lying over a flat boundary is numerically investigated using a two-dimensional vortex element method. The method is based on the discretization of the vorticity field into a finite number of smoothed vortex elements of spherical overlapping cores. The elements are convected in a Lagrangian reference along particle trajectories using the local velocity vector, given in terms of a desingularized Biot-Savart law. The initial structure of the vortex system is triangular; a one-dimensional family of initial configurations is constructed by keeping one side of the triangle fixed and vertical, and varying the abscissa of the centroid of the remaining vortex. The inviscid dynamics of this vortex configuration are first investigated using non-deformable vortices. Depending on the aspect ratio of the initial system, regular or chaotic motion occurs. Due to wall-related symmetries, the far-field sound always exhibits a time-independent quadrupolar directivity with maxima parallel end perpendicular to the wall. When regular motion prevails, the noise spectrum is dominated by discrete frequencies which correspond to the fundamental system frequency and its superharmonics. For chaotic motion, a broadband spectrum is obtained; computed soundlevels are substantially higher than in non-chaotic systems. A more sophisticated analysis is then performed which accounts for vortex core dynamics. Results show that the vortex cores are susceptible to inviscid instability which leads to violent vorticity reorganization within the core. This phenomenon has little effect on the large-scale features of the motion of the system or on low frequency sound emission. However, it leads to the generation of a high-frequency noise band in the acoustic pressure spectrum. The latter is observed in both regular and chaotic system simulations.

Chaotic family dynamics

This article illustrates the basic features of chaotic systems and demonstrates some of the differences between chaotic and nonchaotic systems. To facilitate the explanation of advanced mathematical ideas, this presentation is couched in terms of the interactions within two families: the Limit family and the Scroll family. A nonchaotic model of the Limit family leads to repetitive sequences of affection and hostility between husband and wife. The Scroll family model demonstrates the dynamics of a truly chaotic system; it leads to alternating episodes of mutual endearment and mutual aversion between mother and daughter. Problems of measurement that would accompany the use of these models are mentioned.

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.

Statistical properties of time-dependent wave functions for classically chaotic bound and scattering systems.

Statistical properties of time-dependent wave functions for classically chaotic bound and scattering systems with two degrees of freedom are studied numerically and analytically. For comparison, two-dimensional systems that are classically nonchaotic are also studied numerically. Our numerical results show that the wave functions for chaotic systems become spatially irregular after some characteristic time, whereas no evident irregularities are observed in the nonchaotic systems. Spatial fluctuations of irregular wave functions are shown to be reproduced quite accurately by the random plane-wave superposition approximation. This approximation predicts that the real and imaginary parts of an irregular wave function are uncorrelated Gaussian stochastic functions with the same spatial pair correlation function, which is expressed through a Bessel function. To assess this prediction, distribution functions and spatial pair distribution functions for each of the real and imaginary parts, which must be Gaussian for Gaussian stochastic functions, are computed numerically and found to be close to Gaussian. Analysis of the widths of the numerical pair distribution functions enables us to obtain a pair correlation function that is, in accordance with prediction, well approximated by the Bessel function.

Wave chaos in quantized classically nonchaotic systems.

Wave chaos in quantized classically nonchaotic systems.

Strong mode locking in systems far from chaos.

Based on a simple model of a driven oscillator without inertia, we demonstrate that entirely nonchaotic systems in general may develop a rich mode-locking structure in an appropriate range of control parameters. We show that for this structure the scaling properties are similar to those for a quasiperiodic system near the transition of chaos. Nevertheless, the structure originates from a different type of transition. In particular, the analysis serves to resolve the origin of the ``close-to-chaotic'' behavior in overdamped charge-density-wave systems. Also, this work provides a natural basis for studies of overdamped spatially extended systems.

Simulation of intense particle beams with regularly distributed Gaussian subbeams

Techniques for the simulation of intense particle beams are investigated with respect to the required number of simulation particles. It is shown that for nonchaotic systems it is advantageous if the particles initially are not distributed in a statistical manner but rather arranged in a regular pattern in phase space. This reduces the number of required simulation particles drastically. In the case of such an initially regular arrangement of particles the algorithm which assigns the charges of the particles to the computation mesh becomes of prime importance. The performances of different commonly used algorithms are investigated. The Gaussian assignment algorithm proved far superior to other more commonly used techniques, allowing simulations even at the theoretical limit of 1 particle per cell. Examples for very accurate simulations of beam dynamics with very few particles using an initially regular mesh of particles and Gaussian assignment are given.

Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition

We consider Markov processes arising from small random perturbations of non-chaotic dynamical systems. Under rather general conditions we prove that, with large probability, the distance between two arbitrary paths starting close to a same attractor of the unperturbed system decreases exponentially fast in time. The case of paths starting in different basins of attraction is also considered as well as some applications to the analysis of the invariant measure and to elliptic problems with small parameter in front to the second derivatives. The proof is based on a multiscale analysis of the typical trajectories of the Markov process; this analysis is done using techniques involved in the proof of Anderson localization for disordered quantum systems.