From Instability to Intelligence - Complexity and Predictability in Nonlinear Dynamics

Cover ......Page 1
Lecture Notes in Physics: Monographs 49 ......Page 3
From Instability to Intelligence: Complexity and Predictability in Nonlinear Dynamics ......Page 5
Dedication ......Page 7
1 Preface ......Page 9
Contents ......Page 11
2 Introduction ......Page 17
I Predictability in Classical Dynamics ......Page 25
3.1.1 Mathematical Formulation and Dynamical Invariants ......Page 27
3.1.2 Ignorable Coordinates and Orbital Instability ......Page 28
3.2.1 Static Attractors ......Page 29
3.2.2 Periodic Attractors ......Page 31
3.2.3 Chaotic Attractors ......Page 32
3.2.4 Distributed Systems ......Page 35
3.2.5 Dynamical Systems and Information Processing ......Page 37
3.3.1 Introduction ......Page 40
3.3.2 Neural Net as a Dynamical System ......Page 42
3.3.4 Unsupervised Learning ......Page 44
3.4.1 Introduction ......Page 50
3.4.2 Neurodynamics with Diffusion ......Page 51
3.4.3 Neurodynamics with Dispersion ......Page 54
3.4.4 Convective Neurodynamics ......Page 56
3.4.5 Coupled Phase-Space Neurodynamics ......Page 59
3.4.6 General Model ......Page 62
3.4.7 Limitations of the Classical Approach ......Page 63
4.1 Basic Concepts ......Page 71
4.2.1 Ignorable Coordinates ......Page 72
4.2.2 Orbital Instability of Inertial Motion ......Page 75
4.2.3 Orbital Instability of Potential Motion ......Page 79
4.2.4 General Case ......Page 80
4.3.1 General Remarks ......Page 84
4.3.2 Failure of Hyperbolicity in Distributed Systems ......Page 90
4.3.3 The Criteria of Hadamard\'s Instability ......Page 93
4.3.4 Boundaries of Applicability of Classical Models ......Page 97
4.4.1 Degenerating Hyperbolic Equations ......Page 127
4.4.2 Uniqueness of the Solution ......Page 128
4.4.3 Stability of the Solution ......Page 130
4.4.4 Snap of a Whip ......Page 131
4.4.5 General Case ......Page 132
4.4.6 Failure of Lipschitz Conditions ......Page 135
4.5.1 Measure of Lagrangian Turbulence ......Page 136
4.5.2 Governing Equations ......Page 138
4.5.3 Criteria for Lagrangian Turbulence ......Page 139
4.5.4 The Intensification of Vorticity by Extension of Vortex Lines ......Page 142
4.6.1 Thermal Instability ......Page 145
4.6.5 Mathematical Approach to Analysis of Instabilities ......Page 146
5.1.1 General Remarks ......Page 149
5.1.2 Instability Dependence Upon Metrics of Configuration Space ......Page 150
5.1.3 Instability Dependence Upon Frame of Reference ......Page 151
5.1.4 Instability Dependence Upon Class of Functions ......Page 156
5.2.1 Concept of Multivaluedness of Displacements ......Page 157
5.2.2 Generalized Models of Continua ......Page 158
5.2.3 Applications to the Model of an Inviscid Fluid ......Page 164
5.3.1 General Remarks ......Page 172
5.4.1 Postinstability Motion of Flexible Bodies ......Page 174
5.4.2 Postinstability Motion of Films ......Page 179
5.4.3 Prediction of Chaotic Motions ......Page 217
5.4.4 Predictions of Turbulent Motion ......Page 232
5.5 Stabilization Using Minimum Entropy ......Page 248
5.5.1 Numerical Example ......Page 253
II Terminal (Non-Lipsschitz) Dynamics in Biology and Physics ......Page 255
6.1 Introduction ......Page 257
6.1.1 Adaptability ......Page 259
6.2.1 Review of Preliminary Data ......Page 261
6.2.2 Examples ......Page 263
6.2.3 Conclusions ......Page 265
6.4 Detecting Non-Lipschitz Dynamics ......Page 267
6.4.1 Divergence of the Second Derivative ......Page 268
6.4.3 Phase Plane Portraits ......Page 271
6.4.4 Wavelet Singularities ......Page 272
6.4.5 Noise Scaling ......Page 274
6.4.6 Recurrence Analysis ......Page 276
6.5 Discussion ......Page 278
7.1 Basic Concepts ......Page 281
7.2.2 Terminal Attractors and Repellers ......Page 289
7.2.3 Static Terminal Attractors and Their Trajectories ......Page 292
7.2.4 Physical Interpretation of Terminal Attractors ......Page 295
7.2.5 Periodic Terminal Limit Sets ......Page 296
7.2.6 Unpredictability in Terminal Dynamics ......Page 297
7.2.7 Irreversibility of Terminal Dynamics ......Page 300
7.3.1 Terminal Version of the Liouville-Gibbs Theorem ......Page 301
7.3.2 Terminal Dynamics Model of Random Walk ......Page 303
7.3.3 Numerical Implementation ......Page 307
7.3.4 Multidimensional Systems ......Page 308
7.4 Stochastic Attractors in Terminal Dynamics ......Page 310
7.4.1 One-Dimensional Restricted Random Walk ......Page 311
7.4.2 Multi-Dimensional Restricted Random Walk ......Page 314
7.4.3 Examples ......Page 316
7.5 Self-Organization in Terminal Dynamics ......Page 318
7.6 Guided Systems ......Page 320
7.7 Relevance to Chaos ......Page 322
7.8 Relevance to Classical Thermodynamics ......Page 324
8.1 Introduction ......Page 327
8.2.1 Terminal Attractors: Content Addressable Memory ......Page 330
8.2.2 Terminal Attractors: Pattern Recognition ......Page 334
8.2.3 Models with Hierarchy of Terminal Attractors ......Page 337
8.2.4 Spontaneously Activated Neural Nets ......Page 348
8.2.5 Discussion ......Page 366
8.3 Weakly Connected Neural Nets ......Page 368
8.4.1 Irreversibility and Local Time ......Page 372
8.4.2 Terminal Chaos ......Page 375
8.5 Spatial Coherent Structures ......Page 386
8.6 Neurodynamics with a Fuzzy Objective Function ......Page 389
8.7 Discussion and Conclusions ......Page 393
9.1 Introduction ......Page 395
9.2 Stochastic Attractor as a Tool for Generalization ......Page 396
9.3 Collective Brain Paradigm ......Page 400
9.3.2 Model of Collective Brain ......Page 401
9.3.3 Collective Brain with Fuzzy Objective ......Page 405
9.4 Open Systems in Terminal Neurodynamics ......Page 409
9.5 Neurodynamical Model of Information Fusion ......Page 412
9.6 Conclusion ......Page 416
10.1 Introduction ......Page 419
10.2 Time-Driven Discrete Systems ......Page 420
10.3 Event-Driven Discrete Systems ......Page 425
10.4 Systems Driven by Temporal Events ......Page 428
10.5 Systems Driven by State-Dependent Events ......Page 431
10.6 Events Depending upon State Variable Properties ......Page 433
10.7 Multi-Scale Chains of Events ......Page 436
10.8 Multi-Dimensional Systems ......Page 437
10.9 Synthesis of Discrete-Event Systems ......Page 441
10.9.1 System Identification ......Page 442
10.9.3 Optimization Based upon Local Rules ......Page 443
11 Modeling Heartbeats Using Terminal Dynamics ......Page 445
11.1 Theory ......Page 446
11.2 Theoretical Model ......Page 447
11.3 Discussion and Conclusions ......Page 449
12.1 Introduction ......Page 451
12.2 Mechanical Model of Random Walk ......Page 452
12.3 Phenomenological Force ......Page 454
12.4 Non-Lipschitz Macroscopic Effects ......Page 456
12.5 Microscopic View ......Page 459
12.6 Discussion and Conclusion ......Page 461
13.1 Introduction ......Page 463
13.2 Constitutive Equations ......Page 464
13.3 Governing Equations ......Page 467
13.4 Large-Scale Effects ......Page 469
13.5 Behavior Around Equilibria ......Page 471
13.6 Attraction to Equilibrium After Sudden Move of Boundaries ......Page 474
13.7 Sudden Start from Rest ......Page 477
13.8 Phenomenological Approach ......Page 482
13.9 Application to Acoustics ......Page 484
13.10 Application to Elastic Bodies ......Page 487
13.11 Discussion and Conclusions ......Page 488
14.1 Introduction ......Page 491
14.2 Proof of Concept ......Page 493
14.3 Attractors and Nonlinear Waves of Probability ......Page 500
14.4 Simulation of Conditional Probabilities ......Page 503
14.5 Simulations of Probabilistic Turing Machine ......Page 508
14.6 Simulation of Intelligent Systems ......Page 513
14.7 Quantum Intelligence ......Page 515
14.8 Simulation of Schrodinger Equation ......Page 520
15.1 Introduction ......Page 523
15.2.1 The Navier-Stokes Equation ......Page 524
15.2.2 Computational Techniques ......Page 531
15.2.3 Discussion ......Page 536
15.3 Terminal Dynamics Schrodinger Equation Simulation ......Page 537
15.3.1 Numerical Simulation ......Page 538
Epilogue ......Page 543
References ......Page 545
Index ......Page 557