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Cover image for Complex nonlinearity : chaos, phase transitions, topology change and path integrals
Title:
Complex nonlinearity : chaos, phase transitions, topology change and path integrals
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Series:
Understanding complex systems,
Publication Information:
Berlin : Springer, 2008
Physical Description:
xv, 844 p. : ill. ; 24 cm.
ISBN:
9783540793564
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30000010210785 QA402.35 I92 2008 Open Access Book Book
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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 Dynamicsp. 1
1.1 Introduction to Chaos Theoryp. 1
1.2 Basics of Attractor and Chaotic Dynamicsp. 16
1.3 Brief History of Chaos Theoryp. 25
1.3.1 Poincare's Qualitative Dynamics, Topology and Chaosp. 26
1.3.2 Smale's Topological Horseshoe and Chaos of Stretching and Foldingp. 34
1.3.3 Lorenz' Weather Prediction and Chaosp. 44
1.3.4 Feigenbaum's Constant and Universalityp. 47
1.3.5 May's Population Modelling and Chaosp. 48
1.3.6 Henon's Special 2D Map and Its Strange Attractorp. 52
1.4 More Chaotic and Attractor Systemsp. 55
1.5 Continuous Chaotic Dynamicsp. 67
1.5.1 Dynamics and Non-Equilibrium Statistical Mechanicsp. 69
1.5.2 Statistical Mechanics of Nonlinear Oscillator Chainsp. 82
1.5.3 Geometrical Modelling of Continuous Dynamicsp. 84
1.5.4 Lagrangian Chaosp. 86
1.6 Standard Map and Hamiltonian Chaosp. 95
1.7 Chaotic Dynamics of Binary Systemsp. 101
1.7.1 Examples of Dynamical Mapsp. 103
1.7.2 Correlation Dimension of an Attractorp. 107
1.8 Spatio-Temporal Chaos and Turbulence in PDEsp. 108
1.8.1 Turbulencep. 108
1.8.2 Sine-Gordon Equationp. 113
1.8.3 Complex Ginzburg-Landau Equationp. 114
1.8.4 Kuramoto-Sivashinsky Systemp. 115
1.8.5 Burgers Dynamical Systemp. 116
1.8.6 2D Kuramoto-Sivashinsky Equationp. 118
1.9 Basics of Chaos Controlp. 124
1.9.1 Feedback and Non-Feedback Algorithms for Chaos Controlp. 124
1.9.2 Exploiting Critical Sensitivityp. 127
1.9.3 Lyapunov Exponents and Kaplan-Yorke Dimensionp. 129
1.9.4 Kolmogorov-Sinai Entropyp. 131
1.9.5 Chaos Control by Ott, Grebogi and Yorke (OGY)p. 132
1.9.6 Floquet Stability Analysis and OGY Controlp. 135
1.9.7 Blind Chaos Controlp. 139
1.9.8 Jerk Functions of Simple Chaotic Flowsp. 143
1.9.9 Example: Chaos Control in Molecular Dynamicsp. 146
1.10 Spatio-Temporal Chaos Controlp. 155
1.10.1 Models of Spatio-Temporal Chaos in Excitable Mediap. 158
1.10.2 Global Chaos Controlp. 160
1.10.3 Non-Global Spatially Extended Controlp. 163
1.10.4 Local Chaos Controlp. 165
1.10.5 Spatio-Temporal Chaos-Control in the Heartp. 166
2 Phase Transitions and Synergeticsp. 173
2.1 Introduction to Phase Transitionsp. 173
2.1.1 Equilibrium Phase Transitionsp. 173
2.1.2 Classification of Phase Transitionsp. 175
2.1.3 Basic Properties of Phase Transitionsp. 176
2.1.4 Landau's Theory of Phase Transitionsp. 179
2.1.5 Example: Order Parameters in Magnetite Phase Transitionp. 180
2.1.6 Universal Mandelbrot Set as a Phase-Transition Modelp. 183
2.1.7 Oscillatory Phase Transitionp. 187
2.1.8 Partition Function and Its Path-Integral Descriptionp. 192
2.1.9 Noise-Induced Non-Equilibrium Phase Transitionsp. 199
2.1.10 Noise-Driven Ferromagnetic Phase Transitionp. 206
2.1.11 Phase Transition in a Reaction-Diffusion Systemp. 218
2.1.12 Phase Transition in Negotiation Dynamicsp. 224
2.2 Elements of Haken's Synergeticsp. 229
2.2.1 Phase Transitions and Synergeticsp. 231
2.2.2 Order Parameters in Human/Humanoid Biodynamicsp. 233
2.2.3 Example: Synergetic Control of Biodynamicsp. 236
2.2.4 Example: Chaotic Psychodynamics of Perceptionp. 237
2.2.5 Kick Dynamics and Dissipation-Fluctuation Theoremp. 241
2.3 Synergetics of Recurrent and Attractor Neural Networksp. 244
2.3.1 Stochastic Dynamics of Neuronal Firing Statesp. 246
2.3.2 Synaptic Symmetry and Lyapunov Functionsp. 251
2.3.3 Detailed Balance and Equilibrium Statistical Mechanicsp. 253
2.3.4 Simple Recurrent Networks with Binary Neuronsp. 259
2.3.5 Simple Recurrent Networks of Coupled Oscillatorsp. 267
2.3.6 Attractor Neural Networks with Binary Neuronsp. 275
2.3.7 Attractor Neural Networks with Continuous Neuronsp. 287
2.3.8 Correlation- and Response-Functionsp. 293
3 Geometry and Topology Change in Complex Systemsp. 305
3.1 Riemannian Geometry of Smooth Manifoldsp. 305
3.1.1 Riemannian Manifolds: an Intuitive Picturep. 305
3.1.2 Smooth Manifolds and Their (Co)Tangent Bundlesp. 317
3.1.3 Local Riemannian Geometryp. 328
3.1.4 Global Riemannian Geometryp. 338
3.2 Riemannian Approach to Chaosp. 343
3.2.1 Geometrization of Newtonian Dynamicsp. 345
3.2.2 Geometric Description of Dynamical Instabilityp. 347
3.2.3 Examplesp. 361
3.3 Morse Topology of Smooth Manifoldsp. 368
3.3.1 Intro to Euler Characteristic and Morse Topologyp. 368
3.3.2 Sets and Topological Spacesp. 371
3.3.3 A Brief Intro to Morse Theoryp. 380
3.3.4 Morse Theory and Energy Functionalsp. 382
3.3.5 Morse Theory and Riemannian Geometryp. 384
3.3.6 Morse Topology in Human/Humanoid Biodynamicsp. 388
3.3.7 Cobordism Topology on Smooth Manifoldsp. 392
3.4 Topology Change in 3Dp. 394
3.4.1 Attaching Handlesp. 396
3.4.2 Oriented Cobordism and Surgery Theoryp. 402
3.5 Topology Change in Quantum Gravityp. 405
3.5.1 A Top-Down Framework for Topology Changep. 405
3.5.2 Morse Metrics and Elementary Topology Changesp. 406
3.5.3 'Good' and 'Bad' Topology Changep. 408
3.5.4 Borde-Sorkin Conjecturep. 410
3.6 A Handle-Body Calculus for Topology Changep. 411
3.6.1 Handle-body Decompositionsp. 414
3.6.2 Instantons in Quantum Gravityp. 418
4 Nonlinear Dynamics of Path Integralsp. 425
4.1 Sum over Historiesp. 425
4.1.1 Intuition Behind a Path Integralp. 426
4.1.2 Basic Path-Integral Calculationsp. 437
4.1.3 Brief History of Feynman's Path Integralp. 445
4.1.4 Path-Integral Quantizationp. 452
4.1.5 Statistical Mechanics via Path Integralsp. 460
4.1.6 Path Integrals and Green's Functionsp. 462
4.1.7 Monte Carlo Simulation of the Path Integralp. 468
4.2 Sum over Geometries and Topologiesp. 474
4.2.1 Simplicial Quantum Geometryp. 475
4.2.2 Discrete Gravitational Path Integralsp. 477
4.2.3 Regge Calculusp. 479
4.2.4 Lorentzian Path Integralp. 481
4.2.5 Non-Perturbative Quantum Gravityp. 486
4.3 Dynamics of Fields and Stringsp. 511
4.3.1 Topological Quantum Field Theoryp. 511
4.3.2 TQFT and Seiberg-Witten Theoryp. 515
4.3.3 Stringy Actions and Amplitudesp. 528
4.3.4 Transition Amplitudes for Stringsp. 532
4.3.5 Weyl Invariance and Vertex Operator Formulationp. 535
4.3.6 More General Stringy Actionsp. 535
4.3.7 Transition Amplitude for a Single Point Particlep. 536
4.3.8 Witten's Open String Field Theoryp. 537
4.3.9 Topological Stringsp. 554
4.3.10 Geometrical Transitionsp. 569
4.3.11 Topological Strings and Black Hole Attractorsp. 572
4.4 Chaos Field Theoryp. 578
4.5 Non-Physical Applications of Path Integralsp. 580
4.5.1 Stochastic Optimal Controlp. 580
4.5.2 Nonlinear Dynamics of Option Pricingp. 584
4.5.3 Dynamics of Complex Networksp. 594
4.5.4 Path-Integral Dynamics of Neural Networksp. 596
4.5.5 Cerebellum as a Neural Path-Integralp. 617
4.5.6 Dissipative Quantum Brain Modelp. 623
4.5.7 Action-Amplitude Psychodynamicsp. 637
4.5.8 Joint Action Psychodynamicsp. 651
4.5.9 General Adaptation Psychodynamicsp. 654
5 Complex Nonlinearity: Combining It All Togetherp. 657
5.1 Geometrical Dynamics, Hamiltonian Chaos, and Phase Transitionsp. 657
5.2 Topology and Phase Transitionsp. 664
5.2.1 Computation of the Euler Characteristicp. 666
5.2.2 Topological Hypothesisp. 668
5.3 A Theorem on Topological Origin of Phase Transitionsp. 670
5.4 Phase Transitions, Topology and the Spherical Modelp. 673
5.5 Topology Change and Causal Continuityp. 680
5.5.1 Morse Theory and Surgeryp. 682
5.5.2 Causal Discontinuityp. 687
5.5.3 General 4D Topology Changep. 689
5.5.4 A Black Hole Examplep. 690
5.5.5 Topology Change and Path Integralsp. 692
5.6 'Hard' vs. 'Soft' Complexity: A Bio-Mechanical Examplep. 693
5.6.1 Bio-Mechanical Complexityp. 694
5.6.2 Dynamical Complexity in Bio-Mechanicsp. 697
5.6.3 Control Complexity in Bio-Mechanicsp. 700
5.6.4 Computational Complexity in Bio-Mechanicsp. 705
5.6.5 Simplicity, Predictability and 'Macro-Entanglement'p. 706
5.6.6 Reduction of Mechanical DOF and Associated Controllersp. 707
5.6.7 Self-Assembly, Synchronization and Resolutionp. 709
Referencesp. 713
Indexp. 831
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