Bolotin Y., Tur A., Yanovsky V. Chaos: Concepts, Control and Constructive Use

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Yurii Bolotin · Anatoli Tur · Vladimir Yanovsky

Chaos: Concepts, Control

and Constructive Use

123

Yurii Bolotin

Kharkov Institute of Physics &

Technology

National Science Center

Academicheskaya St., 1

Kharkov 61108

Ukraine

Vladimir Yanovsky

National Academy of Sciences

of Ukraine

SSI Institute for Single

Crystals

Lenin Ave., 60

Kharkov 61001

Ukraine

Anatoli Tur

Centre d’

Etude Spatiale des

Rayonnements (CESR)

9 avenue du Colonel Roche

31028 Toulouse CX 04

France

anatoly.tour@cesr.fr

ISBN 978-3-642-00936-5 e-ISBN 978-3-642-00937-2

DOI 10.1007/978-3-642-00937-2

Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2009927027

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Preface

The study of physics has changed in character, mainly due to the passage from

the analyses of linear systems to the analyses of nonlinear systems. Such a change

began, it goes without saying, a long time ago but the qualitative change took place

and boldly evolved after the understanding of the nature of chaos in nonlinear sys-

tems. The importance of these systems is due to the fact that the major part of

physical reality is nonlinear. Linearity appears as a result of the simpliﬁcation of

real systems, and often, is hardly achievable during the experimental studies. In this

book, we focus our attention on some general phenomena, naturally linked with

nonlinearity where chaos plays a constructive part.

The ﬁrst chapter discusses the concept of chaos. It attempts to describe the mean-

ing of chaos according to the current understanding of it in physics and mathe-

matics. The content of this chapter is essential to understand the nature of chaos

and its appearance in deterministic physical systems. Using the Turing machine,

we formulate the concept of complexity according to Kolmogorov. Further, we

state the algorithmic theory of Kolmogorov–Martin-L

of randomness, which gives

a deep understanding of the nature of deterministic chaos. Readers will not need

any advanced knowledge to understand it and all the necessary facts and deﬁnitions

will be explained.

The second chapter is concerned with the description of the main properties of

chaos and of its numerous qualitative characteristics. The features of chaos which

are presented in this chapter are broadly used for research of various nonlinear pro-

cesses and objects, as well as for theoretical and experimental researches. In this

regard, it deals with the principal tools used for the study of chaos, such as the

Poincar

e Section, Lyapunov Index, etc.

In the third chapter we brieﬂy consider the problem of the restoration of a dynam-

ical system with an attractor based on the observation of temporal data for some

generalized coordinates. We discuss the attractor fractional dimension and the fun-

damental Takens theory.

Chapter four deals with one of the essential ﬁelds in nonlinear dynamics con-

trolling chaos, which was discovered in the pioneer work of Ott, Grebogi, Yorke

“Controlling chaos” Phys. Rev. Lett. 64: 1196 (1990). If a system is chaotic, then

small perturbations increase exponentially in time and change the behavior of a sys-

tem entirely. On one hand, this feature complicates work with a chaotic system, but

vi Preface

on the other hand, it allows purposeful control of the systems behavior using these

small perturbations. In the same chapter, we consider the different methods of the

transformation of chaos into periodical motion (both discreet and continuous) based

on the main concept: the reconstruction of the global system at the expense of small

perturbation. We describe algorithms for dissipative, reversible, and Hamiltonian

systems as well as for scattering problems. The methods developed are applied to

control chaos both in abstract models and in real operational systems. At the end of

this chapter we discuss the possibility of quantum dynamic control.

Chapter ﬁve is about the synchronization of chaotic systems. Two identical

chaotic systems with almost the same initial conditions can diverge exponentially

in the phase space. This is the major difﬁculty to be mastered if we want to solve

this problem. We outline the methods for its solution, both for the dissipative sys-

tems and for the Hamiltonian ones. We demonstrate the possibility of solving the

synchronization problem with the control methods explained in the previous chapter.

We are also interested in the inﬂuence of noise on the process of synchronization.

Also, we brieﬂy examine the principles of the transfer of coded information, which

are based on the effects of chaos synchronization.

In the sixth chapter we investigate in detail the stochastic resonance effect. This

effect makes it possible to set up the stochastic system for the maximum ampliﬁ-

cation of the modulation signal by means of noise intensity variations. Stochastic

resonance can be realized in any nonlinear system which has some characteristic

time scale and one scale can be controlled with the help of the noise. We examine

the different generalizations of the initial problem such as the stochastic resonance

in the chaotic systems. The eventual relation of this effect to the global changes of

Earth’s climate is also discussed in this chapter.

The main topic of Chapter seven is the transport phenomenon in space periodical

systems without macroscopic forces (gradients). We discuss in detail the necessary

conditions for the existence of a macroscopic current in such a situation. At the

same time, we give the classiﬁcation of the system (ratchets) in which currents

without gradient are possible. As one possible realization of the effect, we consider

the generation of regular motion in media with nonlinear friction. In this chapter we

also expound the theory of biological motors (molecular motors), which is a very

important application of the ratchet theory.

We wish to express our deepest gratitude to Hermann Haken for his interest in

our work and his support of this book.

We are very grateful to Christian Caron and Gabriele Hakuba for their invaluable

help in editing this book.

We would like to thank CNRS and particularly Philippe Louarn for his support

of our project.

Kharkov, Toulouse. Yurii Bolotin

June 2009 Anatoli Tur

Vladimir Yanovsky

Contents

1 Introduction.................................................... 1

2 Paradigm for Chaos ............................................. 5

2.1 OrderandDisorder ........................................ 6

2.2 Algorithms and the Turing Machine . . ........................ 8

2.3 Complexity and Randomness . ............................... 11

2.4 Chaos in a Simple Dynamical System ......................... 14

3 Main Features of Chaotic Systems ................................ 19

3.1 Poincar

eSections.......................................... 19

3.2 Spectral Density and Correlation Functions .................... 21

3.3 Lyapunov’s Exponent . . . ................................... 25

3.4 InvariantMeasure ......................................... 32

4 Reconstruction of Dynamical Systems ............................. 35

4.1 What Is Reconstruction? . ................................... 35

4.2 Embedding Dimension . . ................................... 38

4.3 AttractorDimension ....................................... 41

4.4 Finding the Embedding Dimension . . . ........................ 47

4.5 Global Reconstruction of Dynamical Systems . . . . . ............. 50

5 Controlling Chaos .............................................. 51

5.1 StatementoftheProblem ................................... 51

5.2 DiscreteParametricControlandItsStrategy ................... 52

5.3 Main Equations for Chaos Control . . . ........................ 56

5.4 Control of Chaos Without Motion Equations . . . . . . ............. 61

5.5 Targeting Procedure in Dissipative Systems .................... 65

5.6 Chaos Control in Hamiltonian Systems ........................ 67

5.7 StabilizationoftheChaoticScattering ........................ 70

5.8 Control of High-Periodic Orbits in Reversible Mapping . . . . ...... 73

5.9 Controlling Chaos in a Time-Dependant Irregular Environment . . . 78

5.10 Continuous Control with Feedback . . . ........................ 80

5.11 Can Quantum Dynamics Be Controlled? . . .................... 91

vii

viii Contents

6 Synchronization of Chaotic Systems...............................101

6.1 StatementofProblem ......................................102

6.2 Geometry and Dynamics of the Synchronization Process . . . ......103

6.3 General Deﬁnition of Dynamical System Synchronization . . ......106

6.4 Chaotic Synchronization of Hamiltonian Systems . . .............108

6.5 Realization of Chaotic Synchronization Using Control Methods . . . 111

6.6 Synchronization Induced by Noise . . . ........................117

6.7 Synchronization of Space-Temporal Chaos ....................122

6.8 Additive Noise and NonIdentity Systems Inﬂuence on

Synchronization Effects .................................... 125

6.9 Synchronization of Chaotic Systems and Transmission of

Information.............................................. 129

7 Stochastic Resonance ............................................135

7.1 Qualitative Description of the Effect . . ........................135

7.2 The Interaction Between the Particle and its Surrounding

Environment ............................................. 138

7.3 The Two-States Model . . ...................................142

7.4 Stochastic Resonance in Chaotic Systems . ....................149

7.5 Stochastic Resonance and Global Change in the Earth’s Climate . . 153

8 The Appearance of Regular Fluxes Without Gradients ..............159

8.1 Introduction ..............................................159

8.2 Dynamical Model of the Ratchet . . . . . ........................163

8.3 Ratchet Effect – an Example of Real Realization . . . .............168

8.4 Principal Types of Ratchets . . ...............................171

8.5 Nonlinear Friction as the Mechanism of Directed Motion

Generation . .............................................. 176

8.6 Change of Current Direction in the Deterministic Ratchet . . ......182

8.7 Bio or Molecular Motors. ...................................185

References .........................................................189

Index .............................................................195

Chapter 1

Introduction

The notion of chaos is so deeply integrated into all ﬁelds of science, culture and

other human activities, that it has become fundamental. It is quite probable that each

person has his or her own intuitive concept of what is chaos. The ﬁrst cosmogonical

views which explained the origin of the world contained the notion of chaos. For

example, the ancient Greeks believed that chaos appeared before everything and

then only afterwards the world appeared.

For example, Hesiod says: “In the beginning there was only chaos. Then out of

the void appeared Erebus, the unknowable place where death dwells, and Night. All

else was empty, silent, endless, darkness. Then somehow Love was born bringing a

start of order. From Love came Light and Day. Once there was Light and Day, Gaea,

the earth appeared.”

The origin of the word chaos itself – χαoς comes from ancient Greek χαινω,

which means to open wide.

We can ﬁnd a similar concept concerning the origins of the world in ancient

Chinese myth: “In the beginning, the heavens and earth were still one and all was

chaos ...”

In ancient Indian literature the origins of the world are also associated with chaos:

“A time is envisioned when the world was not, only a watery chaos (the dark,

“indistinguishable sea”) and a warm cosmic breath, which could give an impetus

of life”[1].

An ancient Egyptian origin myth holds that in the beginning, the universe was

ﬁlled with the primeval waters of chaos, which was the god Nun.

Such unanimity of such distant civilizations (in space as well as in time) on the

idea of the origination of the world from chaos is quite striking. It seems that this

notion is one of the most important to come from ancient times. What is also striking

is the fact that the problem of chaos is still topical after some hundreds of years of

its study in mathematics and physics. Even a simple list of ﬁelds where chaos plays

a fundamental part in modern science, would take up most of this book. However, a

new understanding of the origins of chaos was reached quite recently in connection

with the discovery of deterministic chaos.

Before the discovery of this phenomenon, all studies of random processes and

of chaos were usually conducted within the frame of classical theory of probability,

Y. Bolotin et al., Chaos:Concepts, Control and Constructive Use, Understanding

Complex Systems, DOI 10.1007/978-3-642-00937-2



Springer-Verlag Berlin Heidelberg 2009

2 1 Introduction

which requires one to deﬁne a set of random events or a set of random process

realizations or a set of other statistical ensembles. After that, probability itself is

assigned and studied as a measure on this set, which satisﬁes Kolmogorov’s axioms

[2]. The discovery of deterministic chaos radically changed this situation.

Chaos was found in dynamical systems, which do not contain elements of ran-

domness at all, i.e. they do not have any statistical ensembles. On the contrary, the

dynamic of such systems is completely predictable, the trajectory, assigned precisely

by its initial conditions, reproduces itself precisely, but nevertheless its behavior is

chaotic.

At ﬁrst sight, this situation does not correspond to our intuitive understanding,

according to which chaotic behavior by its very nature cannot be reproduced. A

simpliﬁed and pragmatic explanation is frequently used to explain this phenomenon.

Dynamical chaos appears in nonlinear systems with trajectories utterly sensible to

minor modiﬁcations of initial conditions. In that case any person calculating a tra-

jectory using a computer, observes that the small uncertainty of initial conditions

engenders chaotic behavior.

This answer does leave some feeling of dissatisfaction. In fact, we know that,

for instance, the number

√

2 exists accurately, without any uncertainty. What would

happen if the trajectory began precisely from

√

2? The usual answer to this question

is that the behavior of trajectory will become more and more complex, because the

number

√

2 is irrational and ultimately will be practically indistinguishable from the

chaotic, although remaining determined.

However, two questions persist: what do we mean by “complex” and what does

“practically indistinguishable from the chaotic” mean? For example, genetic code

is complex but not chaotic, while a coin toss is a simple, but chaotic process. Even

from the above we can see that the phenomenon of deterministic chaos requires

a deeper understanding of randomness, not based on the notion of a statistical

ensemble.

Such a theory was developed by Kolmogorov and his disciples even before the

discovery of the phenomenon of deterministic chaos. The main principle of this

theory will be stated in the next chapter, where we will introduce all its necessary

components: algorithms, Turing machine, Kolmogorov’s complexity, etc. It is sig-

niﬁcant that Kolmogorov came to his theory when discussing in articles [3, 4] the

limited nature of Shannon’s theory of information [5].

As an example, let us question how to understand what is the genetic information,

for example, of a tiger or of Mr. Smith. It seems that, since the notion of information

is based on the introduction of probabilities, we have to examine a set of tigers with

assigned probability. Only after that, one can calculate Shannon information in the

tiger’s genes. It is clear that something in these considerations provokes anxiety.

Above all, the dissatisfaction is caused by the introduction of the set of Smiths,

let’s say. Obviously, it is more pleasant to consider that Mr. Smith is unique and

has individual genetic information like a particular tiger. The limited nature of the

probabilistic approach becomes even clearer if you consider how much information

is contained in the book “War and Peace” by Leo Tolstoy. Then the problem with

the introduction of the set “War and Peace” becomes perfectly evident.