Synopses & Reviews
This graduate-level textbook is devoted to understanding, prediction and control of high-dimensional chaotic and attractor systems of real life. The objective 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. From introductory material on low-dimensional attractors and chaos, the text explores concepts including Poincaré's 3-body problem, high-tech Josephson junctions, and more.
Review
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)
Synopsis
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."
Synopsis
This book details prediction and control of high-dimensional chaotic and attractor systems of real life. It provides a scientific tool that will enable the actual performance of competitive research in high-dimensional chaotic and attractor dynamics. Coverage details Smale's topological transformations of stretching, squeezing and folding and Poincar '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).
Table of Contents
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 Non-equilibrium 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 Trellis-Forced Dynamics 2.1.3 Homoclinic Braid Type 2.2 Homoclinic Classes for Generic Vector-Fields 2.2.1 Lyapunov Stability 2.2.2 Homoclinic Classes 2.3 Complex-Valued H´enon Maps and Horseshoes 2.3.1 Complex Henon-Like 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. 3-Body Problem and Chaos Control 3.1 Mechanical Origin of Chaos 3.1.1 Restricted 3-Body Problem 3.1.2 Scaling and Reduction in the 3-Body Problem 3.1.3 Periodic Solutions of the 3-Body Problem 3.1.4 Bifurcating Periodic Solutions of the 3-Body Problem 3.1.5 Bifurcations in Lagrangian Equilibria 3.1.6 Continuation of KAM-Tori 3.1.7 Parametric Resonance and Chaos in Cosmology 3.2 Elements of Chaos Control 3.2.1 Feedback and Non-Feedback Algorithms for Chaos Control 3.2.2 Exploiting Critical Sensitivity 3.2.3 Lyapunov Exponents and KY-Dimension 3.2.4 Kolmogorov-Sinai 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 Noise-Induced Non-equilibrium 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 Dissipation-Fluctuation 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 Correlation- and Response-Functions 4.3.9 Path-Integral Approach for Complex Dynamics 4.3.10 Hierarchical Self-Programming in Neural Networks 4.4 Topological Phase Transitions and Hamiltonian Chaos 4.4.1 Phase Transitions in Hamiltonian Systems 4.4.2 Geometry of the Largest Lyapunov Exponent 4.4.3 Euler Characteristics of Hamiltonian Systems 4.4.4 Pathways to Self-Organization in Human Biodynamics 5. Phase Synchronization in Chaotic Systems 5.1 Lyapunov vectors and Lyapunov exponents: a general approach 5.1.1 Forced Rossler Oscillator 5.1.2 A Perturbative Calculation of the Second Lyapunov Exponent 5.2 Phase Synchronization in Coupled Chaotic Oscillators 5.3 Oscillatory Phase Neurodynamics 5.3.1 Kuramoto Synchronization Model 5.3.2 Lyapunov Chaotic Synchronization 5.4 Synchronization Geometry 5.4.1 Geometry of Coupled Nonlinear Oscillators 5.4.2 Noisy Coupled Nonlinear Oscillators 5.4.3 Synchronization Condition 5.5 Complex Networks and Chaotic Transients 6. Josephson Junctions and Quantum Engineering 6.0.1 Josephson Effect 6.0.2 Pendulum Analog 6.1 Dissipative Josephson Junction 6.1.1 Junction Hamiltonian and Its Eigenstates 6.1.2 Transition Rate 6.2 Josephson Junction Ladder 6.2.1 Underdamped JJL 6.3 Synchronization in Arrays of Josephson Junctions 6.3.1 Phase Model for Underdamped Junction Ladder 6.3.2 Comparison of LKM2 and RCSJ Models 6.3.3 'Small-world' Connections in Junction Ladder Arrays 7. Fractals and Fractional Dynamics 7.1 Fractals 7.1.1 Mandelbrot Set 7.2 Robust Strange Non-Chaotic Attractors 7.2.1 Quasi-Periodically Forced Maps 7.2.2 2D Map on a Torus 7.2.3 High Dimensional Maps 7.3 Effective Dynamics in Hamiltonian Systems 7.3.1 Effective Dynamical Invariants 7.4 Formation of Fractal Structure in Many-Body Systems 7.4.1 A Many-Body Hamiltonian 7.4.2 Linear Perturbation Analysis 7.5 Fractional Calculus and Chaos Control 7.5.1 Fractional calculus 7.5.2 Fractional-Order Chua's Circuit 7.5.3 Feedback Control of Chaos 7.6 Fractional Gradient and Hamiltonian Dynamics 7.6.1 Gradient Systems 7.6.2 Fractional Differential Forms 7.6.3 Fractional Gradient Systems 7.6.4 Hamiltonian Systems 7.6.5 Fractional Hamiltonian Systems 8. Turbulence 8.1 Parameter-Space Analysis of the Lorenz Attractor 8.1.1 Structure of the Parameter-Space 8.1.2 Attractors and Bifurcations 8.2 Periodically-Driven Lorenz Dynamics 8.2.1 Illustration by Means of a Toy Model 8.3 Lorenzian Diffusion 8.4 Turbulence 8.4.1 Turbulent Flow 8.4.2 The Governing Equations of Turbulence 8.4.3 Global Well-Posedness of the Navier-Stokes Equations 8.4.4 Spatio-Temporal Chaos and Turbulence in PDEs 8.4.5 General Fluid Dynamics 8.4.6 Computational Fluid Dynamics 8.5 Turbulence Kinetics 8.5.1 Kinetic Theory 8.5.2 Filtered Kinetic Theory 8.5.3 Hydrodynamic Limit 8.5.4 Hydrodynamic Equations 8.6 Lie Symmetries in the Models of Turbulence 8.6.1 Lie Symmetries and Prolongations on Manifolds 8.6.2 Noether Theorem and Navier-Stokes Equations 8.6.3 Large-Eddy Simulation 8.6.4 Model Analysis 8.6.5 Thermodynamic Consistence 8.6.6 Stability of Turbulence Models 8.7 Advection of Vector-Fields by Chaotic Flows 8.7.1 Advective Fluid Flow 8.7.2 Chaotic Flows 8.8 Brownian Motion and Diffusion 8.8.1 Random Walk Model 8.8.2 More Complicated Transport Processes 8.8.3 Advection-Diffusion 8.8.4 Beyond the Diffusion Coefficient 9. Geometry, Solitons and Chaos Field Theory 9.1 Chaotic Dynamics and Riemannian Geometry 9.2 Chaos in Physical Gauge Fields 9.3 Solitions 9.3.1 History of Solitons in Brief 9.3.2 The Fermi-Pasta-Ulam Experiments 9.3.3 The Kruskal-Zabusky Experiments 9.3.4 A First Look at KdV 9.3.5 Split-Stepping KdV 9.3.6 Solitons from a Pendulum Chain 9.3.7 1D Crystal Soliton 9.3.8 Solitons and Chaotic Systems 9.4 Chaos Field Theory References Index