Control of Homoclinic Chaos by Weak Periodic Perturbations

Abstract

PREFACEAn exciting and extremely active area of multidisciplinary investigation during the past decade has been the problem of controlling chaotic systems. Indeed there have been a number of books written which have served to review a wide variety of chaos control theories, methods, and perspectives. The main reasons for such interest are the interdisciplinary character of the problem, the implicit promise of a better understanding of chaotic behavior, and the possibility of successful applications in such diverse areas of research as aerodynamics, biology, chemical engineering, epidemiology, electric power systems, electronics, fluid mechanics, laser physics, physiology, secure information processing, and so on. The subject of chaos control exhibits at present a huge spectrum of methods and techniques based on different perspectives. While useful, some of the aforementioned books have suffered from the too ambitious goal of attempting to discuss very concisely every method. On the other hand, many of the papers published on control (suppression/enhancement) of chaos by additional time-dependent excitations (forcing or parametric excitation) have been based on the results of computer simulations. That is why the author's goal was to write a monograph which would give a reasonably rigorous theory of a particular but highly relevant control technique: the suppression/enhancement of chaos by weak periodic excitations in low-dimensional, dissipative, and non-autonomous systems. Controlling chaos is therefore understood as a procedure which suppresses chaos when it is unwanted, and enhances existing chaos or gives rise to chaos in a dynamical system when it is useful. This book is not meant either to compete with the findings of other authors or to repeat known mathematical tools. Except in a very few places, results published by other authors are not reviewed.This monograph begins with an introduction where the method of controlling chaos by weak periodic excitations is approached from the general idea of the control of nonlinear dynamical systems. Some relevant aspects of the technique, such as its flexibility, robustness, scope, and experimental applicability are also discussed. Emphasis is put on the comparison between harmonic and non-harmonic excitations.Chapter 2 presents an intuitive argument to illustrate how added periodic excitations modify the stability of perturbed generic limit cycles. The class of chaotic, dissipative, and non-autonomous dynamical systems to be controlled is described as well as Melnikov's method, which is the analytical technique used to obtain the vii PREFACE viii theoretical results. For the sake of clarity, the cases with and without noise are studied separately. Also, it is shown that the maximum survival of the symmetries of solutions from a wide class of dynamical systems, subjected to both a primary chaos-inducing and a chaos-controlling excitation, corresponds to the optimal choice of the control parameters. For the purely deterministic case, the theory considers separately th...

Dissipative dynamics of a particle in a vibrating periodic potential: Chaos and control

“…This unreasonable effectiveness of MM predictions beyond the perturbative regime has previously been reported in diverse contexts (see, e.g., Ref. [8]). An illustrative example is shown in Fig.…”

“…A case is that of repulsively, power-law interacting particles in one dimension [7]. When the presence of dissipation is unavoidable, the consideration of chaos-suppressing (CS) excitations F (t) is especially pertinent since it means the possibility of complete regularization of the dynamics in the whole phase space over finite regions of parameter space [8]. This is the case, for example, of the aforementioned nanoscale noncontact rack-and-pinion setups [3] as well as the formation of vortices in a Bose-Einstein condensate [9].…”

Synchronization criteria of discrete-time complex networks with time-varying delays and parameter uncertainties

“…(1) discrete-time version of multiple Lorenz chaos systems (Lorenz 1963) coupled via complex networks. This chaos system has quite complex and abundant property, such as homoclinic bifurcation, period doubling phenomena, preturbulence, intermittent chaos (Chacon 1998;Fradkov and Pogromsky 1998;Sparrow 1982). The dynamic equation of such networks is described by (52) with the following parameters The nonlinear function is defined as The outer coupling matrices are of the form…”

Influence of a locally-tailored external feedback control on the overall dynamics of a non-contact AFM model

“…Starting from the pioneering works of Ott et al [1] and Pyragas [2], several books (e.g., [3][4][5][6]), surveys (e.g., [7]) and journal issues (e.g., [8,9]) have been devoted to the topic of chaos control, together with various attempts to classify control techniques and goals to be attained with them. Recent classifications focused on the phenomenological aspects of chaos control [10] distinguish among techniques aimed at stabilizing an unstable zone in parameter space or moving away from previously known chaotic zones, techniques which stabilize a given, erratic solution (which the pioneering OGY and Pyragas methods can be referred to, along with their revisions and enhancements) and methods able to regularize the overall system dynamics, irrespective of a single solution behavior.…”