Complexity And Control: Towards A Rigorous Behavioral Theory Of Complex Dynamical Systems

by Vladimir G Ivancevic, Darryn J Reid

Overview

The book Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems is a graduate-level monographic textbook, intended to be a novel and rigorous contribution to modern Complexity Theory.This book contains 11 chapters and is designed as a one-semester course for engineers, applied and pure mathematicians, theoretical and experimental physicists, computer and economic scientists, theoretical chemists and biologists, as well as all mathematically educated scientists and students, both in industry and academia, interested in predicting and controlling complex dynamical systems of arbitrary nature.

Product Details

ISBN-13:	9789814635868
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/30/2014
Series:	Series On Stability, Vibration And Control Of Systems, Series A , #20
Pages:	612
Product dimensions:	6.10(w) x 9.10(h) x 1.40(d)

Table of Contents

1 Introduction 1

2 Local Geometrical Machinery for Complexity and Control 5

2.1 Introduction: Why Kähler Manifolds? 5

2.1.1 Dynamical Prologue: Complex Hamiltonian Dynamics 6

2.1.2 Unified Behavioral Picture: Kähler Geometrodynamics 8

2.2 Complex Manifolds and Their Vector Bundles 10

2.2.1 Behavioral Dynamics on Complex Plane 10

2.2.2 3D Rotations: Spinors 14

2.2.3 Attractor Dynamics on Riemarm Surfaces 16

2.2.4 Complex Manifolds and Vector Bundles 17

2.2.5 (Co)Taugent Spaces to the Complex Manifold 20

2.2.6 Complex Vector Bundles 21

2.3 From Kahnan Systems to Riemaim Manifolds 22

2.4 Basic Kähler Geometry 23

2.4.1 Essential Kähler Tensors and Cohomology Groups 23

2.4.2 Global Kähler Geometry 27

2.4.3 Local Kähler Geometry 30

2.4.4 Invariant Hamiltonian Dynamics on Kähler Manifolds 32

2.5 Dynamics on Quaternion-Kähler Manifolds 33

2.5.1 Quaternion-Kähler Manifolds 33

2.5.2 Hamiltonian Dynamics on Quaternion-Kähler Manifolds 36

2.5.3 Lagrangian Dynamics on Quaternion-Kähler Manifolds 37

2.6 Kähler-Ricci-Flow Framework 38

2.6.1 Motivation for the Kähler-Ricci Flow 38

2.6.2 Ricci Flow on a Kähler Manifold 39

2.6.3 Definition of the Kähler-Ricci Flow 40

2.6.4 Evolution of the Kähler-Perclman Entropy 41

2.7 Summary of Kähler-Ricci Geometrical Dynamics 42

2.8 Appendix 49

2.8.1 Real Banach and Complex Hilbert Spaces 49

2.8.2 From Linear to Nonlinear MIMO Control Theory 51

2.8.3 Basic Complexity Geometrodynamics 60

2.8.4 Advanced Complexity Geometrodynamics 87

3 Global Categorical Framework for Complexity and Control 117

3.1 Introduction 117

3.2 Categories, Functors and Naturality 119

3.2.1 Categories 120

3.2.2 Functors 122

3.2.3 Natural Transformations 124

3.3 Adjunctions 126

3.3.1 Crowd/Team Dynamics Adjunction 127

3.3.2 Neurophysiological Sensory-Motor Adjunction 128

3.3.3 Quantum Teleportation Example 129

3.4 Hierarchical Recursive Categories 131

3.4.1 Topological Structure of n-Categories 134

3.4.2 Multicategorical Team/Group Dynamics 136

3.5 Crowd Symplectic Machine in a Category 137

3.6 Quantum Categorical Structures 139

3.6.1 Monoidal Tensor Product 139

3.6.2 Snake Lemma and Tensor Products 139

3.6.3 Product of Hilbert Spaces and Quantum Entanglement 141

3.7 Quantum Protocols 143

3.7.1 Gate Teleportation Protocol 143

3.7.2 Entanglement Swapping Protocol 144

3.7.3 Quantum Gambling Protocol 145

3.8 Appendix 147

3.8.1 Abelian Category of Chain and Cochain Complexes 147

3.8.2 A Brief on Categorical Logic 149

3.8.3 Natural Geometrical Operations on Kähler Manifolds 151

3.8.4 Tensor-Product State-Space for n Quantum Particles 170

3.8.5 Complex-Valued Neural Networks 172

4 Dynamics of Crowd Behaviors: From Complex Plane to Quantum Random Fields 175

4.1 Complex Plane Dynamics of Crowds and Groups 175

4.1.1 Introduction 175

4.1.2 Complex-Valued Dynamics of Crowd and Group Behaviors 179

4.4.3 Kähler Geometry of Crowd and Group Dynamics 180

4.1.4 Computer Simulations of Crowds and Croups Dynamics 182

4.1.5 Braids of Agents' Behaviors in the Complex Plane 183

4.4.6 Hilbert-Space Control of Crowds and Groups Dynamics 187

4.2 Quantum Random Fields: A Unique Framework for Simulation. Optimization, Control and Learning 189

4.2.1 Introduction 190

4.2.2 Adaptive Quantum Oscillator 191

4.2.3 Optimization and Learning on Banach and Hilbert Spaces 197

4.3 Appendix 203

4.3.1 Complex-Valued Image Processing 203

4.3.2 Linear Integral Equations 213

4.3.3 Riemann-Liouville Fractional Calculus 234

4.3.4 Rigorous Geometric Quantization 244

4.3.5 Supervised Machine-Learning Methods 246

4.3.6 First-Order Logic and Quantum Random Fields 247

5 Hierarchical Self-Similarity in Group and Crowd Behaviors 251

5.1 Introduction 251

5.1.1 From Correlation Dynamics to Quantum Dynamics 252

5.2 Modeling Framework: Open Liouville Equation 255

5.2.1 Hamiltonian Formalism 256

5.2.2 Conservative Classical Dynamics 256

5.2.3 Conservative Quantum Dynamics 257

5.2.4 Open Classical Dynamics: Hamiltonian Crowd Model 257

5.2.5 Neural Crowd System 258

5.2.6 Open Quantum System 259

5.2.7 Equivalence of Hierarchical Models 261

5.3 Computational Compositions for Crowd Actions 261

5.3.1 Haskell Example Code 261

6 Hybrid Topological Lie-Hamiltonian Learning in Evolving Energy Landscapes 263

6.1 Introduction 263

6.2 The Hybrid Evolution Model 264

6.2.1 Hamiltonian Cost-Function Model 264

6.2.2 Synergetics Interpretation of the Hybrid Model 266

6.2.3 Geometry and Topology of the Hybrid Model 267

6.3 Appendix 270

6.3.1 Haken's Synergetics and Hopfield's Overlaps 270

6.3.2 The Hybrid-Evolution Algorithm Design 273

6.3.3 Topological Analysis of the Hybrid Evolution 275

7 Complexity and Control in Solitary Conductive PDEs 279

7.1 Introduction 279

7.2 Neural Action-Potential Solitons 280

7.2.1 Hodgkin-Huxley Theory 280

7.2.2 Wave Equation Alternative 283

7.2.3 Sine-Gordon Alternative 286

7.3 Fiber-Optics Solitons 289

7.3.1 NLS-Maxwell-Bloch System 290

7.3.2 Hirota-Maxwell-Bloch System 290

7.4 Appendix 294

7.4.1 A 'Zoo' of Sine-Gordon Solitons 294

7.4.2 The Emperor's New Clothes: From Tesla's 'Æther' to Modern Quantum Turbulence 315

8 Quantum-Com put at ion for Perceptual Control Architecture 321

8.1 Introduction 321

8.1.1 From Brain Research to Perceptual Control Architecture 321

8.1.2 Josephson Junctions 322

8.2 Effective Josephson-Junction Networks 323

8.2.1 Qubits, Loops, Ladders, Arrays and Networks 323

8.2.2 Complexity: Breathers and Chaos Synchronization 325

8.2.3 Formalism of Effective JNN Hamiltonians 328

8.2.4 Effective JJN Hamiltonian as Perturbative Sum 330

8.3 Commutative JJN Hierarchies 331

8.3.1 Fuzzy Associative Functors 331

8.3.1 Hierarchy for JJN Architectures 333

8.4 Appendix 336

8.4.1 Hardware for Quantum Computers 336

8.4.2 Adaptive Fuzzy Inference Systems 339

9 Complexity and Control in Entropic and Stochastic Self-Organization 345

9.1 A Path-Integral Model for Entropic Self-Organization 345

9.1.1 Physical Perspective 349

9.1.2 Global Functional Perspective 351

9.1.3 Local Geometric Perspective 353

9.1.4 Computational Perspective 355

9.2 Self-Organization and Stochastic Delay Differential Equations 359

9.2.1 Mean-Field Neurodynamics 362

9.2.2 Stochastic Neural DDEs 365

9.3 Appendix 368

9.3.1 Adaptive Path-Integral Computation in Python/Cython 368

9.3.2 Main Continuous Probability Distributions 379

10 Crash Simulator: Brain-and-Spine Injury Mechanics 385

10.1 Introduction 385

10.2 Brain-and-Spine Injury 390

10.2.1 Traumatic Brain Injury Mechanics 390

10.2.2 Spinal Injury Mechanics 401

10.3 Rigorous Crash Simulator Toolbox for Matlab® 403

10.3.1 Rigid Body Motion and ODEs on Smooth Manifolds 403

10.3.2 Computational Newton-Euler Dynamics 407

10.3.3 Full Spine Crash Simulator 413

10.3.4 Road-Vehicle Crash Simulation 416

10.4 Appendix: Biodynamics and Control of Hmnanoid Robots 417

10.4.1 Basics of Human Biodynamics 418

10.4.2 Spinal Control Level 418

10.4.3 Cerebellum-Like Velocity and Jerk Control 420

10.4.4 Cortical-Like Fuzzy Topological Control 421

11 Conclusion 425

12 Code Samples Used for Complexity and Control 429

12.1 Mathematica® Code 429

12.1.1 Generic Chaotic Simulator 429

12.1.2 Vector Differential Operators 432

12.1.3 NLS Explorer 433

12.2 C++ Code 436

12.2.1 C++ Lambda Functions for Real Calculus 436

12.2.2 Accclerometer Data Processor 437

12.2.3 Simple Predictor-Corrector Integrator 439

12.2.4 Solving the BVP with the Shooting Method 441

12.2.5 Linear Hyperbolic PDE Solver 443

12.2.6 Linear Elliptic PDE Solver 447

12.2.7 Method of Lines for a Set of the NLS Equations 452

12.3 C# Code 460

12.3.1 Iterative Equation Solver 460

12.3.2 Simulated Annealing: A Function Minimum 461

12.3.3 Simple Nonlinear Dynamics 462

12.3.4 Nonlinear Pendulum Simulator 464

12.3.5 Lagrangian Dynamics Simulator 466

12.3.6 Complex-Valued Crowd Attractor Dynamics 474

12.4 Freeform Fortran Code 482

12.4.1 Lorenz Attractor Simulator 482

12.4.2 Complex Lorenz Attractor 484

12.4.3 Simple SGE Solitou 486

12.4.4 Complex Signal Presentation 487

12.4.5 Gaussian Wave Packet 188

12.4.6 Hermitian Matrices 490

12.4.7 Euclidean L2-Norm 492

12.4.8 Vector/Matrix Operations 492

12.5 Plain C-Code: Lovenberg-Marquardt Optimizer 495

12.6 Free Basic Code: 2D Crowd Dynamics with 3000 Agents 522

References 535

Index 581