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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010210785 | QA402.35 I92 2008 | Open Access Book | Book | Searching... |
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Summary
Summary
Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals is a book about prediction & control of general nonlinear and chaotic dynamics of high-dimensional complex systems of various physical and non-physical nature and their underpinning geometro-topological change.
The book starts with a textbook-like expose on nonlinear dynamics, attractors and chaos, both temporal and spatio-temporal, including modern techniques of chaos-control. Chapter 2 turns to the edge of chaos, in the form of phase transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), as well as the related field of synergetics. While the natural stage for linear dynamics comprises of flat, Euclidean geometry (with the corresponding calculation tools from linear algebra and analysis), the natural stage for nonlinear dynamics is curved, Riemannian geometry (with the corresponding tools from nonlinear, tensor algebra and analysis). The extreme nonlinearity - chaos - corresponds to the topology change of this curved geometrical stage, usually called configuration manifold. Chapter 3 elaborates on geometry and topology change in relation with complex nonlinearity and chaos. Chapter 4 develops general nonlinear dynamics, continuous and discrete, deterministic and stochastic, in the unique form of path integrals and their action-amplitude formalism. This most natural framework for representing both phase transitions and topology change starts with Feynman's sum over histories, to be quickly generalized into the sum over geometries and topologies. The last Chapter puts all the previously developed techniques together and presents the unified form of complex nonlinearity. Here we have chaos, phase transitions, geometrical dynamics and topology change, all working together in the form of path integrals.
The objective of this book is to provide a serious reader with a serious scientific tool that willenable them to actually perform a competitive research in modern complex nonlinearity. It includes a comprehensive bibliography on the subject and a detailed index. Target readership includes all researchers and students of complex nonlinear systems (in physics, mathematics, engineering, chemistry, biology, psychology, sociology, economics, medicine, etc.), working both in industry/clinics and academia.
Table of Contents
1 Basics of Nonlinear and Chaotic Dynamics | p. 1 |
1.1 Introduction to Chaos Theory | p. 1 |
1.2 Basics of Attractor and Chaotic Dynamics | p. 16 |
1.3 Brief History of Chaos Theory | p. 25 |
1.3.1 Poincare's Qualitative Dynamics, Topology and Chaos | p. 26 |
1.3.2 Smale's Topological Horseshoe and Chaos of Stretching and Folding | p. 34 |
1.3.3 Lorenz' Weather Prediction and Chaos | p. 44 |
1.3.4 Feigenbaum's Constant and Universality | p. 47 |
1.3.5 May's Population Modelling and Chaos | p. 48 |
1.3.6 Henon's Special 2D Map and Its Strange Attractor | p. 52 |
1.4 More Chaotic and Attractor Systems | p. 55 |
1.5 Continuous Chaotic Dynamics | p. 67 |
1.5.1 Dynamics and Non-Equilibrium Statistical Mechanics | p. 69 |
1.5.2 Statistical Mechanics of Nonlinear Oscillator Chains | p. 82 |
1.5.3 Geometrical Modelling of Continuous Dynamics | p. 84 |
1.5.4 Lagrangian Chaos | p. 86 |
1.6 Standard Map and Hamiltonian Chaos | p. 95 |
1.7 Chaotic Dynamics of Binary Systems | p. 101 |
1.7.1 Examples of Dynamical Maps | p. 103 |
1.7.2 Correlation Dimension of an Attractor | p. 107 |
1.8 Spatio-Temporal Chaos and Turbulence in PDEs | p. 108 |
1.8.1 Turbulence | p. 108 |
1.8.2 Sine-Gordon Equation | p. 113 |
1.8.3 Complex Ginzburg-Landau Equation | p. 114 |
1.8.4 Kuramoto-Sivashinsky System | p. 115 |
1.8.5 Burgers Dynamical System | p. 116 |
1.8.6 2D Kuramoto-Sivashinsky Equation | p. 118 |
1.9 Basics of Chaos Control | p. 124 |
1.9.1 Feedback and Non-Feedback Algorithms for Chaos Control | p. 124 |
1.9.2 Exploiting Critical Sensitivity | p. 127 |
1.9.3 Lyapunov Exponents and Kaplan-Yorke Dimension | p. 129 |
1.9.4 Kolmogorov-Sinai Entropy | p. 131 |
1.9.5 Chaos Control by Ott, Grebogi and Yorke (OGY) | p. 132 |
1.9.6 Floquet Stability Analysis and OGY Control | p. 135 |
1.9.7 Blind Chaos Control | p. 139 |
1.9.8 Jerk Functions of Simple Chaotic Flows | p. 143 |
1.9.9 Example: Chaos Control in Molecular Dynamics | p. 146 |
1.10 Spatio-Temporal Chaos Control | p. 155 |
1.10.1 Models of Spatio-Temporal Chaos in Excitable Media | p. 158 |
1.10.2 Global Chaos Control | p. 160 |
1.10.3 Non-Global Spatially Extended Control | p. 163 |
1.10.4 Local Chaos Control | p. 165 |
1.10.5 Spatio-Temporal Chaos-Control in the Heart | p. 166 |
2 Phase Transitions and Synergetics | p. 173 |
2.1 Introduction to Phase Transitions | p. 173 |
2.1.1 Equilibrium Phase Transitions | p. 173 |
2.1.2 Classification of Phase Transitions | p. 175 |
2.1.3 Basic Properties of Phase Transitions | p. 176 |
2.1.4 Landau's Theory of Phase Transitions | p. 179 |
2.1.5 Example: Order Parameters in Magnetite Phase Transition | p. 180 |
2.1.6 Universal Mandelbrot Set as a Phase-Transition Model | p. 183 |
2.1.7 Oscillatory Phase Transition | p. 187 |
2.1.8 Partition Function and Its Path-Integral Description | p. 192 |
2.1.9 Noise-Induced Non-Equilibrium Phase Transitions | p. 199 |
2.1.10 Noise-Driven Ferromagnetic Phase Transition | p. 206 |
2.1.11 Phase Transition in a Reaction-Diffusion System | p. 218 |
2.1.12 Phase Transition in Negotiation Dynamics | p. 224 |
2.2 Elements of Haken's Synergetics | p. 229 |
2.2.1 Phase Transitions and Synergetics | p. 231 |
2.2.2 Order Parameters in Human/Humanoid Biodynamics | p. 233 |
2.2.3 Example: Synergetic Control of Biodynamics | p. 236 |
2.2.4 Example: Chaotic Psychodynamics of Perception | p. 237 |
2.2.5 Kick Dynamics and Dissipation-Fluctuation Theorem | p. 241 |
2.3 Synergetics of Recurrent and Attractor Neural Networks | p. 244 |
2.3.1 Stochastic Dynamics of Neuronal Firing States | p. 246 |
2.3.2 Synaptic Symmetry and Lyapunov Functions | p. 251 |
2.3.3 Detailed Balance and Equilibrium Statistical Mechanics | p. 253 |
2.3.4 Simple Recurrent Networks with Binary Neurons | p. 259 |
2.3.5 Simple Recurrent Networks of Coupled Oscillators | p. 267 |
2.3.6 Attractor Neural Networks with Binary Neurons | p. 275 |
2.3.7 Attractor Neural Networks with Continuous Neurons | p. 287 |
2.3.8 Correlation- and Response-Functions | p. 293 |
3 Geometry and Topology Change in Complex Systems | p. 305 |
3.1 Riemannian Geometry of Smooth Manifolds | p. 305 |
3.1.1 Riemannian Manifolds: an Intuitive Picture | p. 305 |
3.1.2 Smooth Manifolds and Their (Co)Tangent Bundles | p. 317 |
3.1.3 Local Riemannian Geometry | p. 328 |
3.1.4 Global Riemannian Geometry | p. 338 |
3.2 Riemannian Approach to Chaos | p. 343 |
3.2.1 Geometrization of Newtonian Dynamics | p. 345 |
3.2.2 Geometric Description of Dynamical Instability | p. 347 |
3.2.3 Examples | p. 361 |
3.3 Morse Topology of Smooth Manifolds | p. 368 |
3.3.1 Intro to Euler Characteristic and Morse Topology | p. 368 |
3.3.2 Sets and Topological Spaces | p. 371 |
3.3.3 A Brief Intro to Morse Theory | p. 380 |
3.3.4 Morse Theory and Energy Functionals | p. 382 |
3.3.5 Morse Theory and Riemannian Geometry | p. 384 |
3.3.6 Morse Topology in Human/Humanoid Biodynamics | p. 388 |
3.3.7 Cobordism Topology on Smooth Manifolds | p. 392 |
3.4 Topology Change in 3D | p. 394 |
3.4.1 Attaching Handles | p. 396 |
3.4.2 Oriented Cobordism and Surgery Theory | p. 402 |
3.5 Topology Change in Quantum Gravity | p. 405 |
3.5.1 A Top-Down Framework for Topology Change | p. 405 |
3.5.2 Morse Metrics and Elementary Topology Changes | p. 406 |
3.5.3 'Good' and 'Bad' Topology Change | p. 408 |
3.5.4 Borde-Sorkin Conjecture | p. 410 |
3.6 A Handle-Body Calculus for Topology Change | p. 411 |
3.6.1 Handle-body Decompositions | p. 414 |
3.6.2 Instantons in Quantum Gravity | p. 418 |
4 Nonlinear Dynamics of Path Integrals | p. 425 |
4.1 Sum over Histories | p. 425 |
4.1.1 Intuition Behind a Path Integral | p. 426 |
4.1.2 Basic Path-Integral Calculations | p. 437 |
4.1.3 Brief History of Feynman's Path Integral | p. 445 |
4.1.4 Path-Integral Quantization | p. 452 |
4.1.5 Statistical Mechanics via Path Integrals | p. 460 |
4.1.6 Path Integrals and Green's Functions | p. 462 |
4.1.7 Monte Carlo Simulation of the Path Integral | p. 468 |
4.2 Sum over Geometries and Topologies | p. 474 |
4.2.1 Simplicial Quantum Geometry | p. 475 |
4.2.2 Discrete Gravitational Path Integrals | p. 477 |
4.2.3 Regge Calculus | p. 479 |
4.2.4 Lorentzian Path Integral | p. 481 |
4.2.5 Non-Perturbative Quantum Gravity | p. 486 |
4.3 Dynamics of Fields and Strings | p. 511 |
4.3.1 Topological Quantum Field Theory | p. 511 |
4.3.2 TQFT and Seiberg-Witten Theory | p. 515 |
4.3.3 Stringy Actions and Amplitudes | p. 528 |
4.3.4 Transition Amplitudes for Strings | p. 532 |
4.3.5 Weyl Invariance and Vertex Operator Formulation | p. 535 |
4.3.6 More General Stringy Actions | p. 535 |
4.3.7 Transition Amplitude for a Single Point Particle | p. 536 |
4.3.8 Witten's Open String Field Theory | p. 537 |
4.3.9 Topological Strings | p. 554 |
4.3.10 Geometrical Transitions | p. 569 |
4.3.11 Topological Strings and Black Hole Attractors | p. 572 |
4.4 Chaos Field Theory | p. 578 |
4.5 Non-Physical Applications of Path Integrals | p. 580 |
4.5.1 Stochastic Optimal Control | p. 580 |
4.5.2 Nonlinear Dynamics of Option Pricing | p. 584 |
4.5.3 Dynamics of Complex Networks | p. 594 |
4.5.4 Path-Integral Dynamics of Neural Networks | p. 596 |
4.5.5 Cerebellum as a Neural Path-Integral | p. 617 |
4.5.6 Dissipative Quantum Brain Model | p. 623 |
4.5.7 Action-Amplitude Psychodynamics | p. 637 |
4.5.8 Joint Action Psychodynamics | p. 651 |
4.5.9 General Adaptation Psychodynamics | p. 654 |
5 Complex Nonlinearity: Combining It All Together | p. 657 |
5.1 Geometrical Dynamics, Hamiltonian Chaos, and Phase Transitions | p. 657 |
5.2 Topology and Phase Transitions | p. 664 |
5.2.1 Computation of the Euler Characteristic | p. 666 |
5.2.2 Topological Hypothesis | p. 668 |
5.3 A Theorem on Topological Origin of Phase Transitions | p. 670 |
5.4 Phase Transitions, Topology and the Spherical Model | p. 673 |
5.5 Topology Change and Causal Continuity | p. 680 |
5.5.1 Morse Theory and Surgery | p. 682 |
5.5.2 Causal Discontinuity | p. 687 |
5.5.3 General 4D Topology Change | p. 689 |
5.5.4 A Black Hole Example | p. 690 |
5.5.5 Topology Change and Path Integrals | p. 692 |
5.6 'Hard' vs. 'Soft' Complexity: A Bio-Mechanical Example | p. 693 |
5.6.1 Bio-Mechanical Complexity | p. 694 |
5.6.2 Dynamical Complexity in Bio-Mechanics | p. 697 |
5.6.3 Control Complexity in Bio-Mechanics | p. 700 |
5.6.4 Computational Complexity in Bio-Mechanics | p. 705 |
5.6.5 Simplicity, Predictability and 'Macro-Entanglement' | p. 706 |
5.6.6 Reduction of Mechanical DOF and Associated Controllers | p. 707 |
5.6.7 Self-Assembly, Synchronization and Resolution | p. 709 |
References | p. 713 |
Index | p. 831 |