High-Dimensional Chaotic and Attractor Systems

A Comprehensive Introduction

By: Vladimir G. Ivancevic, Tijana T. Ivancevic

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Hardcover | 5 December 2006

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712 Pages

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If we try to describe real world in mathematical terms, we will see that real life is very often a high-dimensional chaos. Sometimes, by 'pushing hard', we manage to make order out of it; yet sometimes, we need simply to accept our life as it is. To be able to still live successfully, we need tounderstand, predict, and ultimately control this high-dimensional chaotic dynamics of life. This is the main theme of the present book. In our previous book, Geometrical - namics of Complex Systems, Vol. 31 in Springer book series Microprocessor- Based and Intelligent Systems Engineering, we developed the most powerful mathematical machinery to deal with high-dimensional nonlinear dynamics. In the present text, we consider the extreme cases of nonlinear dynamics, the high-dimensional chaotic and other attractor systems. Although they might look as examples of complete disorder - they still represent control systems, with their inputs, outputs, states, feedbacks, and stability. Today, we can see a number of nice books devoted to nonlinear dyn- ics and chaos theory (see our reference list). However, all these books are only undergraduate, introductory texts, that are concerned exclusively with oversimpli?ed low-dimensional chaos, thus providing only an inspiration for the readers to actually throw themselves into the real-life chaotic dynamics.

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From the reviews:

"This is an ambitious book that ... is devoted to the understanding, prediction and control of high-dimensional chaotic and attractor systems in real life. ... Finally, and most usefully, the book has a substantial list of references (over 30 pages of them), meaning that the book can be used as a guide to literature in a diverse range of topics related to high- (and indeed low-) dimensional chaotic and nonlinear systems." (Peter Ashwin, Mathematical Reviews, Issue 2008 h)

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Published: 19th November 2010

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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
Table of Contents provided by Publisher. All Rights Reserved.

High-Dimensional Chaotic and Attractor Systems

A Comprehensive Introduction

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Hardcover

Industry Reviews

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