Chaos: Concepts, Control and Constructive Use

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Springer Science & Business Media, Aug 6, 2009 - Science - 198 pages
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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 s- 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 simpli?cation 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 ?rst chapter discusses the concept of chaos. It attempts to describe the me- ing of chaos according to the current understanding of it in physics and mat- 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-Lof ̈ 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 de?nitions will be explained.
 

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Contents

Introduction
2
Paradigm for Chaos
5
21 Order and Disorder
6
22 Algorithms and the Turing Machine
8
23 Complexity and Randomness
11
24 Chaos in a Simple Dynamical System
14
Main Features of Chaotic Systems
19
32 Spectral Density and Correlation Functions
21
Synchronization of Chaotic Systems
100
61 Statement of Problem
102
62 Geometry and Dynamics of the Synchronization Process
103
63 General Definition of Dynamical System Synchronization
106
64 Chaotic Synchronization of Hamiltonian Systems
108
65 Realization of Chaotic Synchronization Using Control Methods
111
66 Synchronization Induced by Noise
117
67 Synchronization of SpaceTemporal Chaos
122

33 Lyapunovs Exponent
25
34 Invariant Measure
32
Reconstruction of Dynamical Systems
35
42 Embedding Dimension
38
43 Attractor Dimension
41
44 Finding the Embedding Dimension
47
45 Global Reconstruction of Dynamical Systems
50
Controlling Chaos
51
52 Discrete Parametric Control and Its Strategy
52
53 Main Equations for Chaos Control
56
54 Control of Chaos Without Motion Equations
61
55 Targeting Procedure in Dissipative Systems
65
56 Chaos Control in Hamiltonian Systems
67
57 Stabilization of the Chaotic Scattering
70
58 Control of HighPeriodic Orbits in Reversible Mapping
73
59 Controlling Chaos in a TimeDependant Irregular Environment
78
510 Continuous Control with Feedback
80
511 Can Quantum Dynamics Be Controlled?
91
68 Additive Noise and NonIdentity Systems Influence on Synchronization Effects
125
69 Synchronization of Chaotic Systems and Transmission of Information
129
Stochastic Resonance
135
72 The Interaction Between the Particle and Its Surrounding Environment Langevins Equation
138
73 The TwoStates Model
142
74 Stochastic Resonance in Chaotic Systems
149
75 Stochastic Resonance and Global Change in the Earths Climate
153
The Appearance of Regular Fluxes Without Gradients
158
82 Dynamical Model of the Ratchet
163
83 Ratchet Effect an Example of Real Realization
168
84 Principal Types of Ratchets
171
85 Nonlinear Friction as the Mechanism of Directed Motion Generation
176
86 Change of Current Direction in the Deterministic Ratchet
182
87 Bio or Molecular Motors
185
References
189
Index
195
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