If we try to describe real world in mathematical terms, we will see that real life is very often a highdimensional 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 highdimensional 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 highdimensional nonlinear dynamics. In the present text, we consider the extreme cases of nonlinear dynamics, the highdimensional 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 lowdimensional chaos, thus providing only an inspiration for the readers to actually throw themselves into the reallife chaotic dynamics.
This book details prediction and control of highdimensional chaotic and attractor systems of real life. It provides a scientific tool that will enable the actual performance of competitive research in highdimensional chaotic and attractor dynamics. Coverage details Smale's topological transformations of stretching, squeezing and folding and Poincaré's 3-body problem and basic techniques of chaos control. It offers a review of both Landau's and topological phase transition theory as well as Haken's synergetics and deals with phase synchronization in high-dimensional chaotic systems. In addition, the book presents high-tech Josephson junctions, deals with fractals and fractional Hamiltonian dynamics, and offers a review of modern techniques for dealing with turbulence. It also offers a brief on the cutting edge techniques of the high-dimensional nonlinear dynamics (including geometries, gauges and solitons, culminating into the chaos field theory).Inhalt
1. Introduction to Attractors and Chaos 1.1 Basics of Attractor and Chaotic Dynamics 1.2 Brief History of Chaos Theory in 5 Steps 1.2.1 Henry Poincar´e: Qualitative Dynamics, Topology and Chaos 1.2.2 Steve Smale: Topological Horseshoe and Chaos of Stretching and Folding 1.2.3 Ed Lorenz: Weather Prediction and Chaos 1.2.4 Mitchell Feigenbaum: Feigenbaum Constant and Universality 1.2.5 Lord Robert May: Population Modelling and Chaos 1.2.6 Michel H´enon: A Special 2D Map and Its Strange Attractor 1.3 Some Classical Attractor and Chaotic Systems 1.4 Basics of Continuous Dynamical Analysis 1.4.1 A Motivating Example 1.4.2 Systems of ODEs 1.4.3 Linear Autonomous Dynamics: Attractors & Repellors 1.4.4 Conservative versus Dissipative Dynamics 1.4.5 Basics of Nonlinear Dynamics 1.4.6 Ergodic Systems 1.5 Continuous Chaotic Dynamics 1.5.1 Dynamics and Nonequilibrium Statistical Mechanics 1.5.2 Statistical Mechanics of Nonlinear Oscillator Chains 1.5.3 Geometrical Modelling of Continuous Dynamics 1.5.4 Lagrangian Chaos 1.6 Standard Map and Hamiltonian Chaos 1.7 Chaotic Dynamics of Binary Systems 1.7.1 Examples of Dynamical Maps 1.7.2 Correlation Dimension of an Attractor 1.8 Basic Hamiltonian Model of Biodynamics 2. Smale Horseshoes and Homoclinic Dynamics 2.1 Smale Horseshoe Orbits and Symbolic Dynamics 2.1.1 Horseshoe Trellis 2.1.2 TrellisForced Dynamics 2.1.3 Homoclinic Braid Type 2.2 Homoclinic Classes for Generic VectorFields 2.2.1 Lyapunov Stability 2.2.2 Homoclinic Classes 2.3 ComplexValued H´enon Maps and Horseshoes 2.3.1 Complex HenonLike Maps 2.3.2 Complex Horseshoes 2.4 Chaos in Functional Delay Equations 2.4.1 Poincar´e Maps and Homoclinic Solutions 2.4.2 Starting Value and Targets 2.4.3 Successive Modifications of g 2.4.4 Transversality 2.4.5 Transversally Homoclinic Solutions 3. 3BodyProblem and Chaos Control 3.1 Mechanical Origin of Chaos 3.1.1 Restricted 3Body Problem 3.1.2 Scaling and Reduction in the 3Body Problem 3.1.3 Periodic Solutions of the 3Body Problem 3.1.4 Bifurcating Periodic Solutions of the 3Body Problem 3.1.5 Bifurcations in Lagrangian Equilibria 3.1.6 Continuation of KAMTori 3.1.7 Parametric Resonance and Chaos in Cosmology 3.2 Elements of Chaos Control 3.2.1 Feedback and NonFeedback Algorithms for Chaos Control 3.2.2 Exploiting Critical Sensitivity 3.2.3 Lyapunov Exponents and KYDimension 3.2.4 KolmogorovSinai Entropy 3.2.5 Classical Chaos Control by Ott, Grebogi and Yorke 3.2.6 Floquet Stability Analysis and OGY Control 3.2.7 Blind Chaos Control 3.2.8 Jerk Functions of Simple Chaotic Flows 3.2.9 Example: Chaos Control in Molecular Dynamics 4. Phase Transitions and Synergetics 4.1 Phase Transitions, Partition Function and Noise 4.1.1 Equilibrium Phase Transitions 4.1.2 Classification of Phase Transitions 4.1.3 Basic Properties of Phase Transitions 4.1.4 Landau's Theory of Phase Transitions 4.1.5 Partition Function 4.1.6 NoiseInduced Nonequilibrium Phase Transitions 4.2 Elements of Haken's Synergetics 4.2.1 Phase Transitions 4.2.2 Mezoscopic Derivation of Order Parameters 4.2.3 Example: Synergetic Control of Biodynamics 4.2.4 Example: Chaotic Psychodynamics of Perception 4.2.5 Kick Dynamics and DissipationFluctuation Theorem 4.3 Synergetics of Recurrent and Attractor Neural Networks 4.3.1 Stochastic Dynamics of Neuronal Firing States 4.3.2 Synaptic Symmetry and Lyapunov Functions 4.3.3 Detailed Balance and Equilibrium Statistical Mechanics 4.3.4 Simple Recurrent Networks with Binary Neurons 4.3.5 Simple Recurrent Networks of Coupled Oscillators 4.3.6 Attractor Neural Networks with Binary Neurons 4.3.7 Attractor Neural Networks with Continuous Neurons 4.3.8