High-Dimensional Chaotic and Attractor Systems

by Ivancevic, Vladimir G.; Ivancevic, Tijana T.

• ISBN13: 9781402054556
• ISBN10: 1402054556
• Format: Hardcover
• Copyright: 2007-01-03
• Publisher: Springer Verlag

Summary

This is a graduate'level monographic textbook devoted to understanding, prediction and control of high'dimensional chaotic and attractor systems of real life. The objective of the book is to provide the serious reader with a serious scientific tool that will enable the actual performance of competitive research in high'dimensional chaotic and attractor dynamics.The book has nine Chapters. The first Chapter gives a textbook-like introduction into the low-dimensional attractors and chaos. This Chapter has an inspirational character, similar to other books on nonlinear dynamics and deterministic chaos. The second Chapter deals with Smale's topological transformations of stretching, squeezing and folding (of the system's phase'space), developed for the purpose of chaos theory. The third Chapter is devoted to Poincarés 3-body problem and basic techniques of chaos control, mostly of Ott-Grebogi-Yorke type. The fourth Chapter is a review of both Landau's and topological phase transition theory, as well as Haken's synergetics. The fifth Chapter deals with phase synchronization in high-dimensional chaotic systems. The sixth Chapter presents high-tech Josephson junctions, the basic components for the future quantum computers. The seventh Chapter deals with fractals and fractional Hamiltonian dynamics. The 8th Chapter gives a review of modern techniques for dealing with turbulence, ranging from theparameter'space of the Lorenz attractor to the Lie symmetries. The last, 9th, Chapter attempts to give a brief on the cutting edge techniques of the high-dimensional nonlinear dynamics (including geometries, gauges and solitons, culminating into the chaos field theory).

Table of Contents

Introduction to Attractors and Chaos

Basics of Attractor and Chaotic Dynamics

Brief History of Chaos Theory in 5 Steps

Henry PoincarA'e: Qualitative Dynamics, Topology and Chaos

Steve Smale: Topological Horseshoe and Chaos of Stretching and Folding

Ed Lorenz: Weather Prediction and Chaos

Mitchell Feigenbaum: Feigenbaum Constant and Universality

Lord Robert May: Population Modelling and Chaos

Michel HA'enon: A Special 2D Map and Its Strange Attractor

Some Classical Attractor and Chaotic Systems

Basics of Continuous Dynamical Analysis

A Motivating Example

Systems of ODEs

Linear Autonomous Dynamics: Attractors & Repellors

Conservative versus Dissipative Dynamics

Basics of Nonlinear Dynamics

Ergodic Systems

Continuous Chaotic Dynamics

Dynamics and Nona¬"equilibrium Statistical Mechanics

Statistical Mechanics of Nonlinear Oscillator Chains

Geometrical Modelling of Continuous Dynamics

Lagrangian Chaos

Standard Map and Hamiltonian Chaos

Chaotic Dynamics of Binary Systems

Examples of Dynamical Maps

Correlation Dimension of an Attractor

Basic Hamiltonian Model of Biodynamics

Smale Horseshoes and Homoclinic Dynamics

Smale Horseshoe Orbits and Symbolic Dynamics

Horseshoe Trellis

Trellisa¬"Forced Dynamics

Homoclinic Braid Type

Homoclinic Classes for Generic Vectorâ¼"Fields

Lyapunov Stability

Homoclinic Classes

Complexa¬"Valued HA'enon Maps and Horseshoes

Complex Henona¬"Like Maps

Complex Horseshoes

Chaos in Functional Delay Equations

PoincarA'e Maps and Homoclinic Solutions

Starting Value and Targets

Successive Modifications of g

Transversality

Transversally Homoclinic Solutions

3a¬"Body Problem and Chaos Control

Mechanical Origin of Chaos

Restricted 3a¬"Body Problem

Scaling and Reduction in the 3a¬"Body Problem

Periodic Solutions of the 3a¬"Body Problem

Bifurcating Periodic Solutions of the 3a¬"Body Problem

Bifurcations in Lagrangian Equilibria

Continuation of KAMa¬"Tori

Parametric Resonance and Chaos in Cosmology

Elements of Chaos Control

Feedback and Nona¬"Feedback Algorithms for Chaos Control

Exploiting Critical Sensitivity

Lyapunov Exponents and KYa¬"Dimension

Kolmogorovâ¼"Sinai Entropy

Classical Chaos Control by Ott, Grebogi and Yorke

Floquet Stability Analysis and OGY Control

Blind Chaos Control

Jerk Functions of Simple Chaotic Flows

Example: Chaos Control in Molecular Dynamics

Phase Transitions and Synergetics

Phase Transitions, Partition Function and Noise

Equilibrium Phase Transitions

Classification of Phase Transitions

Basic Properties of Phase Transitions

Landau's Theory of Phase Transitions

Partition Function

Noiseâ¼"Induced Nona¬"equilibrium Phase Transitions

Elements of Haken's Synergetics

Phase Transitions

Mezoscopic Derivation of Order Parameters

Example: Synergetic Control of Biodynamics

Example: Chaotic Psychodynamics of Perception

Kick Dynamics and Dissipationa¬"Fluctuation Theorem

Synergetics of Recurrent and Attractor Neural Networks

Stochastic Dynamics of Neuronal Firing States

Synaptic Symmetry and Lyapunov Functions

Detailed Balance and Equilibrium Statistical Mechanics

Simple Recurrent Networks with Binary Neurons

Simple Recurrent Networks of Coupled Oscillators

Attractor Neural Networks with Binary Neurons

Attractor Neural Networks with Continuous Neurons

Correlationa¬" and Responsea¬"Functions

Patha¬"Integral Approach for Complex Dynamics

Hierarchical Selfa¬"Programming in Neural Networks

Topological Phase Transitions and Hamiltonian Chaos

Phase Transitions in Hamiltonian Systems

Geometry of the Largest Lyapunov Exponent

Euler Characteristics of Hamiltonian Systems4.4.4

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