Qiudong Wang

STRANGE ATTRACTORS IN PERIODICALLY KICKED CHUA'S CIRCUIT

In this paper, we discuss a new mechanism for chaos in light of some new developments in the theory of dynamical systems. It was shown in [Wang & Young, 2002b] that strange attractors occur when an...

RANK ONE CHAOS: THEORY AND APPLICATIONS

The main purpose of this tutorial is to introduce to a more application-oriented audience a new chaos theory that is applicable to certain systems of differential equations. This new chaos theory, namely the theory of rank one maps, claims a comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain systems of differential equations, the existence of the indicated phenomenon of chaos can be verified through a well-defined computational process. Applications to the well-known Chuas and MLC circuits employing controlled switches are also presented to demonstrate the usefulness of the theory. We try to introduce this new chaos theory by using a balanced combination of examples, numerical simulations and theoretical discussions. We also try to create a standard reference for this theory that will hopefully be accessible to a more application-oriented audience.

A NEW CLASS OF CHAOTIC ATTRACTORS IN MURALI–LAKSHMANAN–CHUA CIRCUIT

In this paper, we study the existence of a new class of chaotic attractors, namely the rank-one attractors, in the MLC (Murali–Lakshmanan–Chua) circuit [Murali et al., 1994] by numerical simulations based on a theory of rank-one maps developed in [Wang & Young, 2005]. With the guidance of the theory in [Wang & Young, 2005], weakly stable limit cycles, found through Hopf bifurcations and other numerical means, are subjected to periodic pulses with long relaxation periods to produce rank-one attractors. The periodic pulses are applied directly as an input. Periodic pulses have been used before in various schemes of chaos. However, for this scheme of creating rank-one attractors to work, the applied periodic pulses must have short pulse widths and long relaxation periods. This is one of the key components in creating this new class of chaotic attractors.

Dynamical profile of a class of rank-one attractors

This paper is about dynamical properties of a class of rank one attractors. Roughly speaking, a rank one attractor is an attractor on which there is exactly one neutral or unstable direction and all other directions are contracted strongly. In [WY2], we identified a class of well behaved rank one maps (called G in that paper). These maps are characterized by hyperbolic behavior away from “critical structures”, which are localized sources of nonhyperbolicity. The present paper contains an in-depth study of the geometric and ergodic theories of T ∈ G. We examine these maps from several different angles: Lyapunov exponents, SRB measures, basins of attraction, statistics of time series, global geometric and combinatorial structures, symbolic coding and periodic points. In short, we seek to build a dynamical profile for the class of maps T ∈ G, to connect them to the existing literature on hyperbolic dynamics. Since the statement of our results in Section 1 are quite straightforward, we would like to use the rest of this introduction to better acquaint the reader with the class of maps G: Where did it come from? Why is it interesting? Do maps of the type in G appear naturally? Are our results here relevant in applications? The class G has its origin in the Hénon family with 0 < b << 1, studied in the groundbreaking work [BC2] and subsequent papers, e.g. [BY1, BY2, BV]. These ideas were taken to a larger context for the first time in [WY1], where the formulas of the Hénon maps were replaced by generic geometric conditions. The aim of [WY1] was to make contact with more general rank one phenomena in dynamical systems (see below). The systems treated in [WY1] are 2D; their generalization to arbitrary dimensions is the class G studied in [WY2]. We believe this is the context to which the body of ideas begun in [BC2] truly belongs. A justification for studying G is that it provides a unique window into the workings of nonuniform hyperbolicity. While abstract nonuniform hyperbolic theory (as in e.g. [P, R, LY]) is fairly well developed, there are few concrete examples, the majority of which (e.g. billiards and the Lorenz attractor) have a priori invariant cones or separation of stable and unstable directions. The same is true for some works on partially hyperbolic systems. The maps in G admit no continuous families of invariant cones, and derivative growth occurs in genuinely nonuniform ways. But the mechanism that leads to the loss of hyperbolicity is known, and as we will show, conditions imposed on critical orbits (required for T to be in G) translate into a fair amount of control on the hyperbolic properties of other orbits in the system. It is a model of controlled nonuniform hyperbolicity. Equally important to us is that the maps in G arise naturally in applications, in differential equations modeling commonly occurring phenomena. For example, strange attractors with one direction of instability have been shown to appear shortly after the breakdown

RANK ONE CHAOS IN SWITCH-CONTROLLED MURALI–LAKSHMANAN–CHUA CIRCUIT

In this paper, we investigate the creation of strange attractors in a switch-controlled MLC (Murali–Lakshmanan–Chua) circuit. The design and use of this circuit is motivated by a recent mathematical theory of rank one attractors developed by Wang and Young. Strange attractors are created by periodically kicking a weakly stable limit cycle emerging from the center of a supercritical Hopf bifurcation, and are found in numerical simulations by following a recipe-like algorithm. Rigorous conditions for chaos are derived and various switch control schemes, such as synchronous, asynchronous, single-, and multi-pulse, are investigated in numerical simulations.

Rank one chaos in switch-controlled piecewise linear Chua's circuit

In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chuas original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations.

Experimental verification of rank 1 chaos in switch-controlled Chua circuit

In this paper, we provide the first experimental proof for the existence of rank 1 chaos in the switch-controlled Chua circuit by following a step-by-step procedure given by the theory of rank 1 maps. At the center of this procedure is a periodically kicked limit cycle obtained from the unforced system. Then, this limit cycle is subjected to periodic kicks by adding externally controlled switches to the original circuit. Both the smooth nonlinearity and the piecewise linear cases are considered in this experimental investigation. Experimental results are found to be in concordance with the conclusions of the theory.

Strange attractors and their periodic repetition.

In this paper, we present some important findings regarding a comprehensive characterization of dynamical behavior in the vicinity of two periodically perturbed homoclinic solutions. Using the Duffing system, we illustrate that the overall dynamical behavior of the system, including strange attractors, is organized in the form of an asymptotic invariant pattern as the magnitude of the applied periodic forcing approaches zero. Moreover, this invariant pattern repeats itself with a multiplicative period with respect to the magnitude of the forcing. This multiplicative period is an explicitly known function of the system parameters. The findings from the numerical experiments are shown to be in great agreement with the theoretical expectations.