Chaos : concepts, control and constructive use / Yurii Bolotin, Anatoli Tur, Vladimir Yanovsky.

Format:

Book

Author/Creator:

Bolotin, Yurii.

Contributor:

Tur, Anatoli.

Yanovsky, Vladimir.

Series:

Understanding complex systems

Language:

English

Subjects (All):

Chaotic behavior in systems.

Physical Description:

viii, 198 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; New York : Springer, [2009]

Summary:

The study of chaotic behaviour in nonlinear, dynamical systems is now a well established research domain with ramifications into all fields of sciences, spanning a vast range of applications, from celestial mechanics, via climate change, to the functioning of brownian motors in cells.

A more recent discovery is that chaos can be controlled and, under appropriate conditions, can actually be constructive in the sense of becoming a control parameter itself for the system under investigation, stochastic resonance being a prime example.

The Present work is putting emphasis on the latter aspects, and after recalling the paradigm changes introduced by the concept of chaos, leads the reader skillfully through the basics of chaos control by detailing relevant algorithms for both Hamiltonian and dissipative systems amongst others.

The main part of the book is then devoted to the issue of synchronization in chaotic systems, and introduction to stochastic resonance and a survey of ratchet models.

This short and concise primer is particularly suitable for postgraduate students and non-specialist scientists from related areas, wishing to enter the field quickly and efficiently.

Contents:

1 Introduction 1

2 Paradigm for Chaos 5

2.1 Order and Disorder 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 Poincare Sections 19

3.2 Spectral Density and Correlation Functions 21

3.3 Lyapunov's Exponent 25

3.4 Invariant Measure 32

4 Reconstruction of Dynamical Systems 35

4.1 What Is Reconstruction? 35

4.2 Embedding Dimension 38

4.3 Attractor Dimension 41

4.4 Finding the Embedding Dimension 47

4.5 Global Reconstruction of Dynamical Systems 50

5 Controlling Chaos 51

5.1 Statement of the Problem 51

5.2 Discrete Parametric Control and Its Strategy 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 Stabilization of the Chaotic Scattering 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

6 Synchronization of Chaotic Systems 101

6.1 Statement of Problem 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.

ISBN:

3642009360

9783642009365

OCLC:

429183618