Chaos In High-dimensional Systems Research Articles

HETEROCLINICAL REPELLERS IMPLY CHAOS

In this paper, we prove that chaos in the sense of Li–Yorke and of Devaney is prevalent in discrete systems admitting the so-called heteroclinical repellers, which are similar to the transversely heteroclinical orbits in both continuous and discrete systems and are corresponding to the snap-back repeller proposed by Marotto for proving the existence of chaos in higher-dimensional systems. In addition, the concept of heteroclinical repellers is generalized to be applicable to the case with degenerate transformations. In the end, some illustrative examples are provided to illustrate the theoretical results.

Mechanisms of extensive spatiotemporal chaos in Rayleigh-Benard convection

Spatially extended dynamical systems exhibit complex behaviour in both space and time--spatiotemporal chaos. Analysis of dynamical quantities (such as fractal dimensions and Lyapunov exponents) has provided insights into low-dimensional systems; but it has proven more difficult to understand spatiotemporal chaos in high-dimensional systems, despite abundant data describing its statistical properties. Initial attempts have been made to extend the dynamical approach to higher-dimensional systems, demonstrating numerically that the spatiotemporal chaos in several simple models is extensive (the number of dynamical degrees of freedom scales with the system volume). Here we report a computational investigation of a phenomenon found in nature, 'spiral defect' chaos in Rayleigh-Benard convection, in which we find that the spatiotemporal chaos in this state is extensive and characterized by about a hundred dynamical degrees of freedom. By studying the detailed space-time evolution of the dynamical degrees of freedom, we find that the mechanism for the generation of chaotic disorder is spatially and temporally localized to events associated with the creation and annihilation of defects.

Synchronization of chaos in high dimensional systems

Recent work pioneered by Pecora and Carroll has considered the possibility of exploiting the phenomenon of chaos synchronization to achieve secure communication. But, theoretical and experimental models studied thus far limit to low dimensional systems with one positive Lyapunov exponent. Consequently, messages masked by such simple chaotic processes are shown to be easily extracted in some cases. In this work we investigate high dimensional implementation of the Pecoia-Carroll synchronization paradigm. In particular, in regard to potential applications to communication, we address the question of whether by transmitting just one scalar signal one can achieve synchronism in chaotic systems with two or more positive Lyapunov exponents (hyperchaos). Using both numerical and analytical examples we argue that under very general conditions the answer to the above question is affirmative.

Control and synchronization of chaos in high dimensional systems: Review of some recent results.

Controlling chaos and synchronization of chaos have evolved for a number of years as essentially two separate areas of research. Only recently it has been realized that both subjects share a common root in control theory. In addition, as limitations of low dimensional chaotic systems in modeling real world phenomena become increasingly apparent, investigations into the control and synchronization of high dimensional chaotic systems are beginning to attract more interest. We review some recent advances in control and synchronization of chaos in high dimensional systems. Efforts will be made to stress the common origins of the two subjects. (c) 1997 American Institute of Physics.

Folded Bifurcation in Coupled Asymmetric Logistic Maps

A system of two coupled logistic maps, one periodic and the other chaotic, is studied. It is found that with the variation of the coupling strength, the system displays several curious features such as the appearance of quadrupling of period, occurrence of isolated period three attractor and the coexistence of the Hopf and pitchfork bifurcations. Possible applications and extensions are discus· sed. Chaos in higher dimensional systems is one of the focal subjects of physics today. Along with the approach starting from modeling physical systems with many degrees of freedom, there emerged a new approach, developed by Kaneko, to couple many one-dimensional maps 1 H> to study the behavior of the system as a whole. However, this model can only be applied to study the dynamics of a single medium such as pattern formation in a fluid. What happens if two media border each other? One may naturally be lead to a model of coupled logistic maps with different strength parameters. Thus it is appropriate to inquire whether loosening the condition of strict identity might bring any new feature while keeping both maps logistic to hold the redundant controlling parameters minimal. Even two logistic maps coupled to each other may serve as a dynamical model of driven coupled oscillators. It has been found that two coupled identical maps possess several characteristic features which are typical of higher dimensional chaos. 1 >. 4 > In this paper, we report the results of a numerical investigation of a system of two logistic maps with different strength parameters such that one map lies in a period one stable attractor or a bifurcation point and the other in chaotic region when decoupled. Several new features previously unobserved are found. Most notable among them is the appearance of a period four cycle straight from the stable period one cycle. The other peculiar feature is the almost simultaneous occurrence of periods four, eight and sixteen right after the Hopf bifurcation. This results in a very intriguing metamor­ phose of the attractor when one changes the coupling parameter. The system we study is two linearly coupled maps

Controlling chaos in high dimensional systems.

Recently formulated techniques for controlling chaotic dynamics face a fundamental problem when the system is high dimensional, and this problem is present even when the chaotic attractor is low dimensional. Here we introduce a procedure for controlling a chaotic time signal of an arbitrarily high dimensional system, without assuming any knowledge of the underlying dynamical equations. Specifically, we formulate a feedback control that requires modeling the local dynamics of only a single or a few of the possibly infinite number of phase-space variables.