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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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.
Title:High-Dimensional Chaotic and Attractor Systems: A Comprehensive IntroductionFormat:PaperbackDimensions:720 pages, 9.25 × 6.1 × 0 inPublished:November 19, 2010Publisher:Springer NetherlandsLanguage: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 Solutions3. 3-Body Problem 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 Dynamics4. 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 Correlation- and Response-Functions4.3.9 Path-Integral Approach for Complex Dynamics4.3.10 Hierarchical Self-Programming in Neural Networks4.4 Topological Phase Transitions and Hamiltonian Chaos4.4.1 Phase Transitions in Hamiltonian Systems 4.4.2 Geometry of the Largest Lyapunov Exponent4.4.3 Euler Characteristics of Hamiltonian Systems4.4.4 Pathways to Self-Organization in Human Biodynamics5. Phase Synchronization in Chaotic Systems5.1 Lyapunov vectors and Lyapunov exponents: a general approach5.1.1 Forced Rossler Oscillator5.1.2 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 Oscillators5.4.2 Noisy Coupled Nonlinear Oscillators 5.4.3 Synchronization Condition 5.5 Complex Networks and Chaotic Transients6. Josephson Junctions and Quantum Engineering6.0.1 Josephson Effect6.0.2 Pendulum Analog 6.1 Dissipative Josephson Junction6.1.1 Junction Hamiltonian and Its Eigenstates6.1.2 Transition Rate 6.2 Josephson Junction Ladder6.2.1 Underdamped JJL6.3 Synchronization in Arrays of Josephson Junctions6.3.1 Phase Model for Underdamped Junction Ladder6.3.2 Comparison of LKM2 and RCSJ Models 6.3.3 'Small-world' Connections in Junction Ladder Arrays7. Fractals and Fractional Dynamics7.1 Fractals7.1.1 Mandelbrot Set7.2 Robust Strange Non-Chaotic Attractors7.2.1 Quasi-Periodically Forced Maps7.2.2 2D Map on a Torus 7.2.3 High Dimensional Maps7.3 Effective Dynamics in Hamiltonian Systems7.3.1 Effective Dynamical Invariants7.4 Formation of Fractal Structure in Many-Body Systems7.4.1 A Many-Body Hamiltonian7.4.2 Linear Perturbation Analysis7.5 Fractional Calculus and Chaos Control7.5.1 Fractional calculus7.5.2 Fractional-Order Chua's Circuit 7.5.3 Feedback Control of Chaos7.6 Fractional Gradient and Hamiltonian Dynamics7.6.1 Gradient Systems 7.6.2 Fractional Differential Forms7.6.3 Fractional Gradient Systems7.6.4 Hamiltonian Systems7.6.5 Fractional Hamiltonian Systems8. Turbulence8.1 Parameter-Space Analysis of the Lorenz Attractor8.1.1 Structure of the Parameter-Space8.1.2 Attractors and Bifurcations8.2 Periodically-Driven Lorenz Dynamics8.2.1 Illustration by Means of a Toy Model8.3 Lorenzian Diffusion8.4 Turbulence8.4.1 Turbulent Flow8.4.2 The Governing Equations of Turbulence8.4.3 Global Well-Posedness of the Navier-Stokes Equations8.4.4 Spatio-Temporal Chaos and Turbulence in PDEs8.4.5 General Fluid Dynamics8.4.6 Computational Fluid Dynamics8.5 Turbulence Kinetics8.5.1 Kinetic Theory8.5.2 Filtered Kinetic Theory8.5.3 Hydrodynamic Limit8.5.4 Hydrodynamic Equations8.6 Lie Symmetries in the Models of Turbulence8.6.1 Lie Symmetries and Prolongations on Manifolds8.6.2 Noether Theorem and Navier-Stokes Equations8.6.3 Large-Eddy Simulation8.6.4 Model Analysis8.6.5 Thermodynamic Consistence8.6.6 Stability of Turbulence Models8.7 Advection of Vector-Fields by Chaotic Flows8.7.1 Advective Fluid Flow8.7.2 Chaotic Flows8.8 Brownian Motion and Diffusion8.8.1 Random Walk Model8.8.2 More Complicated Transport Processes8.8.3 Advection-Diffusion8.8.4 Beyond the Diffusion Coefficient9. Geometry, Solitons and Chaos Field Theory9.1 Chaotic Dynamics and Riemannian Geometry9.2 Chaos in Physical Gauge Fields9.3 Solitions9.3.1 History of Solitons in Brief9.3.2 The Fermi-Pasta-Ulam Experiments9.3.3 The Kruskal-Zabusky Experiments9.3.4 A First Look at KdV9.3.5 Split-Stepping KdV 9.3.6 Solitons from a Pendulum Chain9.3.7 1D Crystal Soliton9.3.8 Solitons and Chaotic Systems9.4 Chaos Field Theory ReferencesIndex

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)