High-dimensional Chaotic and Attractor Systems
A Comprehensive Introduction
(Sprache: Englisch)
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...
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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.
This is a graduate-level monographic textbook devoted to understanding, prediction and control of high-dimensional chaotic and attractor systems of real life. The objective of the book 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.
The book has nine Chapters. The first Chapter gives a textbook-like introduction into the low-dimensional attractors and chaos. This Chapter has an inspirational character, similar to other books on nonlinear dynamics and deterministic chaos. The second Chapter deals with Smale's topological transformations of stretching, squeezing and folding (of the system's phase-space), developed for the purpose of chaos theory. The third Chapter is devoted to Poincaré's 3-body problem and basic techniques of chaos control, mostly of Ott-Grebogi-Yorke type. The fourth Chapter is a review of both Landau's and topological phase transition theory, as well as Haken's synergetics. The fifth Chapter deals with phase synchronization in high-dimensional chaotic systems. The sixth Chapter presents high-tech Josephson junctions, the basic components for the future quantum computers. The seventh Chapter deals with fractals and fractional Hamiltonian dynamics. The 8th Chapter gives a review of modern techniques for dealing with turbulence, ranging from the parameter-space of the Lorenz attractor to the Lie symmetries. The last, 9th, Chapter attempts to give a brief on the cutting edge techniques of the high-dimensional nonlinear dynamics (including geometries, gauges and solitons, culminating into the chaos field theory).
The book has nine Chapters. The first Chapter gives a textbook-like introduction into the low-dimensional attractors and chaos. This Chapter has an inspirational character, similar to other books on nonlinear dynamics and deterministic chaos. The second Chapter deals with Smale's topological transformations of stretching, squeezing and folding (of the system's phase-space), developed for the purpose of chaos theory. The third Chapter is devoted to Poincaré's 3-body problem and basic techniques of chaos control, mostly of Ott-Grebogi-Yorke type. The fourth Chapter is a review of both Landau's and topological phase transition theory, as well as Haken's synergetics. The fifth Chapter deals with phase synchronization in high-dimensional chaotic systems. The sixth Chapter presents high-tech Josephson junctions, the basic components for the future quantum computers. The seventh Chapter deals with fractals and fractional Hamiltonian dynamics. The 8th Chapter gives a review of modern techniques for dealing with turbulence, ranging from the parameter-space of the Lorenz attractor to the Lie symmetries. The last, 9th, Chapter attempts to give a brief on the cutting edge techniques of the high-dimensional nonlinear dynamics (including geometries, gauges and solitons, culminating into the chaos field theory).
Inhaltsverzeichnis zu „High-dimensional Chaotic and Attractor Systems “
1. Introduction to Attractors and Chaos1.1 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 Folding1.2.3 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 Analysis1.4.1 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 Systems1.5 Continuous Chaotic Dynamics1.5.1 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 Chaos1.7 Chaotic Dynamics of Binary Systems1.7.1 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 Stability2.2.2 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
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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 Sensitivity3.2.3 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 Control3.2.7 Blind Chaos Control3.2.8 Jerk Functions of Simple Chaotic Flows3.2.9 Example: Chaos Control in Molecular Dynamics 4. Phase Transitions and Synergetics4.1 Phase Transitions, Partition Function and Noise4.1.1 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 Perception4.2.5 Kick Dynamics and Dissipation-Fluctuation Theorem4.3 Synergetics of Recurrent and Attractor Neural Networks4.3.1 Stochastic Dynamics of Neuronal Firing States4.3.2 Synaptic Symmetry and Lyapunov Functions4.3.3 Detailed Balance and Equilibrium Statistical Mechanics4.3.4 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
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Bibliographische Angaben
- Autoren: Vladimir G. Ivancevic , Tijana T. Ivancevic
- 2007, 697 Seiten, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer Netherlands
- ISBN-10: 1402054556
- ISBN-13: 9781402054556
- Erscheinungsdatum: 05.12.2006
Sprache:
Englisch
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