High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction by Vladimir G. IvancevicHigh-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction by Vladimir G. Ivancevic

High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction

byVladimir G. Ivancevic, Tijana T. Ivancevic

Paperback | November 19, 2010

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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.
Title:High-Dimensional Chaotic and Attractor Systems: A Comprehensive IntroductionFormat:PaperbackDimensions:697 pagesPublished:November 19, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048173728

ISBN - 13:9789048173723

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Table of Contents

1. Introduction to Attractors and Chaos 1.1 Basics of Attractor and Chaotic Dynamics1.2 Brief History of Chaos Theory in 5 Steps1.2.1 Henry Poincar´e: Qualitative Dynamics, Topology and Chaos1.2.2 Steve Smale: Topological Horseshoe and Chaos of Stretching and Folding 1.2.3 Ed Lorenz: Weather Prediction and Chaos1.2.4 Mitchell Feigenbaum: Feigenbaum Constant and Universality1.2.5 Lord Robert May: Population Modelling and Chaos1.2.6 Michel H´enon: A Special 2D Map and Its Strange Attractor1.3 Some Classical Attractor and Chaotic Systems1.4 Basics of Continuous Dynamical Analysis 1.4.1 A Motivating Example1.4.2 Systems of ODEs1.4.3 Linear Autonomous Dynamics: Attractors & Repellors1.4.4 Conservative versus Dissipative Dynamics1.4.5 Basics of Nonlinear Dynamics1.4.6 Ergodic Systems 1.5 Continuous Chaotic Dynamics 1.5.1 Dynamics and Non-equilibrium Statistical Mechanics1.5.2 Statistical Mechanics of Nonlinear Oscillator Chains1.5.3 Geometrical Modelling of Continuous Dynamics1.5.4 Lagrangian Chaos1.6 Standard Map and Hamiltonian Chaos 1.7 Chaotic Dynamics of Binary Systems 1.7.1 Examples of Dynamical Maps1.7.2 Correlation Dimension of an Attractor1.8 Basic Hamiltonian Model of Biodynamics 2. Smale Horseshoes and Homoclinic Dynamics2.1 Smale Horseshoe Orbits and Symbolic Dynamics2.1.1 Horseshoe Trellis2.1.2 Trellis-Forced Dynamics2.1.3 Homoclinic Braid Type2.2 Homoclinic Classes for Generic Vector-Fields2.2.1 Lyapunov Stability 2.2.2 Homoclinic Classes2.3 Complex-Valued H´enon Maps and Horseshoes2.3.1 Complex Henon-Like Maps2.3.2 Complex Horseshoes2.4 Chaos in Functional Delay Equations2.4.1 Poincar´e Maps and Homoclinic Solutions2.4.2 Starting Value and Targets2.4.3 Successive Modifications of g2.4.4 Transversality2.4.5 Transversally Homoclinic Solutions 3. 3-BodyProblem and Chaos Control3.1 Mechanical Origin of Chaos3.1.1 Restricted 3-Body Problem3.1.2 Scaling and Reduction in the 3-Body Problem3.1.3 Periodic Solutions of the 3-Body Problem3.1.4 Bifurcating Periodic Solutions of the 3-Body Problem3.1.5 Bifurcations in Lagrangian Equilibria3.1.6 Continuation of KAM-Tori3.1.7 Parametric Resonance and Chaos in Cosmology3.2 Elements of Chaos Control3.2.1 Feedback and Non-Feedback Algorithms for Chaos Control3.2.2 Exploiting Critical Sensitivity 3.2.3 Lyapunov Exponents and KY-Dimension3.2.4 Kolmogorov-Sinai Entropy3.2.5 Classical Chaos Control by Ott, Grebogi and Yorke3.2.6 Floquet Stability Analysis and OGY Control 3.2.7 Blind Chaos Control3.2.8 Jerk Functions of Simple Chaotic Flows3.2.9 Example: Chaos Control in Molecular Dynamics 4. Phase Transitions and Synergetics 4.1 Phase Transitions, Partition Function and Noise 4.1.1 Equilibrium Phase Transitions4.1.2 Classification of Phase Transitions4.1.3 Basic Properties of Phase Transitions4.1.4 Landau's Theory of Phase Transitions4.1.5 Partition Function4.1.6 Noise-Induced Non-equilibrium Phase Transitions4.2 Elements of Haken's Synergetics4.2.1 Phase Transitions4.2.2 Mezoscopic Derivation of Order Parameters4.2.3 Example: Synergetic Control of Biodynamics4.2.4 Example: Chaotic Psychodynamics of Perception 4.2.5 Kick Dynamics and Dissipation-Fluctuation Theorem 4.3 Synergetics of Recurrent and Attractor Neural Networks4.3.1 Stochastic Dynamics of Neuronal Firing States4.3.2 Synaptic Symmetry and Lyapunov Functions 4.3.3 Detailed Balance and Equilibrium Statistical Mechanics 4.3.4 Simple Recurrent Networks with Binary Neurons4.3.5 Simple Recurrent Networks of Coupled Oscillators4.3.6 Attractor Neural Networks with Binary Neurons4.3.7 Attractor Neural Networks with Continuous Neurons4.3.8

Editorial Reviews

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)