Robust Chaos and Its Applications

471 Pages·2011·24.675 MB·English

by  Elhadj Zeraoulia, Julien Clinton Sprott

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WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE
Editor: Leon O. Chua
University of California, Berkeley
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ce Volume 65: Chua’s Circuit Implementations: Yesterday, Today and Tomorrow
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cientifipermitt Volume 66: DJ.i-fMfe.r eGnintiaolu Gxeometry Applied to Dynamical Systems
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d Itn 0 Volume 73: 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach
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ChaRSI Volume 74: Physarum Machines: Computers from Slime Mould
st VE A. Adamatzky
RobuUNI Volume 75: Discrete Systems with Memory
A R. Alonso-Sanz
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SI L. O. Chua
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N E. D. Cooper
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by A. M. Samoilenko & O. Stanzhytskyi
Volume 79: Robust Chaos and Its Applications
Z. Elhadj & J. C. Sprott
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Lakshmi - Robust Chaos.pmd 1 9/7/2011, 4:09 PM
WORLD SCIENTIFIC SERIES ON
NONLINEAR SCIENCE Series A Vol. 79

Series Editor: Leon O. Chua
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World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
8296hc.9789814374071-tp.indd 2 9/16/11 4:21 PM
Published by
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RoA U For photocopying of material in this volume, please pay a copying fee through the Copyright
U Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
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Printed in Singapore.
Lakshmi - Robust Chaos.pmd 2 9/7/2011, 4:09 PM
August29,2011 11:29 WorldScienti(cid:12)cBook-9inx6in robust-main
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ntifimitt To my family: My wife Nadjette and my father Tayeb and my brothers
w.worldsciectly not per Zohir, Samra, Mourad, Saiwdar,itaentdhiBsabcoholikla. and to all who helped me
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September7,2011 15:49 WorldScienti(cid:12)cBook-9inx6in robust-main
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August29,2011 11:29 WorldScienti(cid:12)cBook-9inx6in robust-main
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A Preface
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d Its n 05/ chapter is a set of exercises and open problems intended to reinforce the
st Chaos anVERSITY o indoenaRlsinoabenuadrstpsccrihoeavnoicdseeaiannddgdeiitntiseornaaapllpalenixcdaptecirhoinaenossciesthfaoerotrbeyxottibnhorpoeakardtdieceruvsolaatrne.ddrteosetahrechuenrdseirn-
buNI standing and prediction of robust chaos in real-world dynamical systems.
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A This book contains 260 exercises in different topics of chaos theory and
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H dynamical systems.
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N Let us briefly characterize the content of each chapter:
TSI Chapter 1. Poincar´e map technique, Smale horseshoe, and
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A symbolic dynamics. In this chapter we give an overview of the Poincar´e
N
O map technique used to describe bifurcations and chaos in dynamical sys-
TI
A tems. For example, all the content of Chapter 7 is based on the construc-
N
y tion of a Poincar´e map to confirm the existence of robust chaos in such
b
a continuous-time dynamical system. We first define the Poincar´e map
and the generalized Poincar´e mappings for a continuous-time dynamical
vii
August29,2011 11:29 WorldScienti(cid:12)cBook-9inx6in robust-main
viii Robust Chaos and its Applications
system with a continuous, piecewise-linear vector field. Then we discuss
interval methods for calculating Poincar´e maps including several methods
s.
cle used to prove the existence of periodic orbits using a generalized Poincar´e
arti map associated with the continuous-time flow under consideration. Sec-
s
es ondly, we discuss the Smale horseshoe as a basic tool for studying chaos
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A and its properties such as symbolic dynamics. In particular, the so-called
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pe method of fixed point index is described as a way to prove the existence
O
or of such horseshoe-type mappings in a dynamical system. This method is
pt f based on the periodic points of the so-called TS-maps and the existence of
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c.comed, ex semCi-choanpjutegracy2.. Robustness of chaos. In this chapter we discuss the
ntifimitt conceptof robust chaos. First, wedescribethe notion ofstrange attractors
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ns Douse an etrragcotdoicrstshuecohrya.sTthheenrawteeopfrdeesceanyt oofthceorrrsetlaattiisotnicsa,lthperocpeenrttriaelsliomficthtahoetoicreamt-,
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s A5/0 characterizing hyperbolic (robust) systems.
d Itn 0 Chapter 4. Structural stability. Inthischapterwediscussthecon-
no
os aTY cept of structural stability as a criterion for robustness of invariant sets of
ChaRSI dynamicalsystems. Inparticular,wedefineandstateimportantproperties
st VE of this notion, along with its conditions. As an example, we state and dis-
buNI
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UA diffeomorphisms of a compact C∞ manifold M without boundary due to
H
G Robinson and Verjovsky, along with a weaker version called Ω-stability.
N
SI Chapter 5. Transversality, Invariant foliation, and the Shad-
T
L owing Lemma. In this chapter we first discuss some relevant properties
A
N andtheimportanceoftransversality,invariantfoliation,andtheshadowing
O
TI lemma in analyzing chaotic dynamical systems in general, and robust ones
A
N in particular. Indeed, transversality is the opposite of the tangency that
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b is responsible for the coexistence of attractors. Secondly, we present and
discuss the notion of the invariant foliation that is used to prove chaos and
its topological properties in the singular-hyperbolic attractors. Thirdly, we
August29,2011 11:29 WorldScienti(cid:12)cBook-9inx6in robust-main
Preface ix
present the shadowing lemma that is used to verify the existence of a true
chaotic attractor. In particular, a robust chaotic attractor is more impor-
s.
cle tantthanoneobtainedwithoutverification. Thepossiblerelationsbetween
arti homoclinic orbits and shadowing are given along with some Shilnikov the-
s
es orems concerning criteria for the existence of chaos.
c
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A Chapter 6. Chaotic attractors with hyperbolic structure. In
n
pe this chapter we discuss the most interesting results concerning chaotic at-
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or tractors with a hyperbolic structure. First, hyperbolic dynamics is dis-
pt f cussed along with some concepts and definitions. Then we present Anosov
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ntifimitt dynamical systems, which is based on rigorous mathematical analysis and
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d Itn 0 methods for generating hyperbolic attractors, in particular, the transition
no
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ChaRSI systems with hyperbolic attractors, are presented along with a discussion
st VE of the density of hyperbolicity and homoclinic bifurcations in arbitrary di-
buNI
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A
U H´enon map and to the forced damped pendulum.
H
G Chapter 7. Robust chaos in hyperbolic systems. In this chapter
N
SI we give several examples of realistic models describing hyperbolic (robust)
T
L chaos. Indeed, we present a method for the construction of the Smale–
A
N Williams attractor for a three-dimensional map along with a theoretical
O
TI andnumericalverificationofitshyperbolicity. Anexampleofaflowsystem
A
N is presented with an attractor concentrated mostly at the surface of a two-
y
b dimensional torus, the dynamics of which is governed by the Arnold cat
map. A model for the Bernoulli map is constructed using the Poincar´e
mapmethoddiscussedinChapter1withahyperchaoticattractor(onewith
August29,2011 11:29 WorldScienti(cid:12)cBook-9inx6in robust-main
x Robust Chaos and its Applications
two positive Lyapunov exponents) in a perturbed heteroclinic cycle. The
Poincar´emapdefinedbya3-Dflowconstructedasaburstingneuronmodel
s.
cle and exhibiting a Plykin-like attractor (a strange hyperbolic attractor) is
arti also given and discussed.
s
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e Chapter 8. Lorenz-type systems. In this chapter, we present some
c
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p the major properties of these types of systems along with one definition
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or resulting from observations of the dynamics of the standard Lorenz system
pt f is presented from the viewpoint of its existence and structure. Another
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ntifimitt sented in some detail. In particular, we discuss the essence of wild strange
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s A5/0 Chapter 10. No robust chaos in quasi-attractors. Inthischapter
d Itn 0 we discuss some relevant results and properties of quasi-attractors as the
no
os aTY third type in the classification of strange attractors of dynamical systems.
ChaRSI In particular, we give evidence for the existence of transversal homoclinic
st VE pointsthatprovethismapisaquasi-attractor,alsotheuniformhyperbolic-
buNI
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A
U UsingtheproceduredescribedinSec.6.9.1,wegiveaproofthattheH´enon
H
G attractor is a quasi-attractor. Other examples with quasi-attractors in-
N
SI cludetheso-calledStrelkova–Anishchenko map,theAnishchenko–Astakhov
T
L oscillator, and Chua circuit, which is a famous example of a system that
A
N displays quasi-attractors with homoclinic and heteroclinic orbits and dou-
O
TI ble scrolls. The so-called geometric model of this circuit is also introduced
A
N in some detail. The distinction of this model is that it is different from
y
b the Lorenz-type presented in Chapter 8 or the quasi-attractors discussed
in this chapter. This new type contains unstable points in addition to the
Cantor set structure of hyperbolic points. In particular, the corresponding