The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov

by Steeb, Willi-Hans; Hardy, Yorick (COL); Stoop, Ruedi (COL)

• ISBN13: 9789812818522
• ISBN10: 9812818529
• Edition: 4th
• Format: Hardcover
• Copyright: 2008-08-05
• Publisher: World Scientific Pub Co Inc

Summary

"The study of nonlinear dynamical systems has advanced tremendously in the last 20 years, making a big impact on science and technology. This book provides all the techniques and methods used in nonlinear dynamics. The concepts and underlying mathematics are discussed in detail." "The text has been designed for a one-year course at both the junior and senior levels in nonlinear dynamics. The topics discussed in the book are part of e-learning and distance learning courses conducted by the International School for Scientific Computing, University of Johannesburg."--BOOK JACKET.

Table of Contents

Nonlinear and Chaotic Maps	p. 1
One-Dimensional Maps	p. 1
Exact and Numerical Trajectories	p. 3
Fixed Points and Stability	p. 14
Invariant Density	p. 16
Liapunov Exponent	p. 20
Autocorrelation Function	p. 23
Discrete Fourier Transform	p. 25
Fast Fourier Transform	p. 28
Logistic Map and Liapunov Exponent for r ∈ [3, 4]	p. 33
Logistic Map and Bifurcation Diagram	p. 34
Random Number Map and Invariant Density	p. 36
Random Number Map and Random Integration	p. 38
Circle Map and Rotation Number	p. 40
Newton Method	p. 41
Feigenbaum's Constant	p. 43
Symbolic Dynamics	p. 45
Chaotic Repeller	p. 47
Chaos and Encoding	p. 48
Two-Dimensional Maps	p. 54
Introduction	p. 54
Phase Portrait	p. 57
Fixed Points and Stability	p. 64
Liapunov Exponents	p. 65
Correlation Integral	p. 67
Capacity	p. 68
Hyperchaos	p. 70
Domain of Attraction	p. 74
Newton Method in the Complex Domain	p. 75
Newton Method in Higher Dimensions	p. 77
Ruelle-Takens-Newhouse Scenario	p. 78
Periodic Orbits and Topological Degree	p. 80
JPEG file	p. 82
Time Series Analysis	p. 85
Introduction	p. 85
Correlation Coefficient	p. 86
Liapunov Exponent from Time Series	p. 87
Jacobian Matrix Estimation Algorithm	p. 88
Direct Method	p. 89
Hurst Exponent	p. 96
Introduction	p. 96
Implementation for the Hurst Exponent	p. 98
Random Walk	p. 102
Higuchi's Algorithm	p. 106
Complexity	p. 107
Autonomous Systems in the Plane	p. 111
Classification of Fixed Points	p. 111
Homoclinic Orbit	p. 113
Pendulum	p. 114
Limit Cycle Systems	p. 116
Lotka-Volterra Systems	p. 119
Nonlinear Hamilton Systems	p. 123
Hamilton Equations of Motion	p. 123
Hamilton System and Variational Equation	p. 126
Integrable Hamilton Systems	p. 127
Hamilton Systems and First Integrals	p. 127
Lax Pair and Hamilton Systems	p. 128
Floquet Theory	p. 130
Chaotic Hamilton Systems	p. 133
Henon-Heiles Hamilton Function and Trajectories	p. 133
Henon Heiles and Surface of Section Method	p. 135
Quartic Potential and Surface of Section Technique	p. 136
Nonlinear Dissipative Systems	p. 139
Fixed Points and Stability	p. 139
Trajectories	p. 144
Phase Portrait	p. 148
Liapunov Exponents	p. 150
Generalized Lotka-Volterra Model	p. 153
Hyperchaotic Systems	p. 155
Hopf Bifurcation	p. 158
Time-Dependent First Integrals	p. 161
Nonlinear Driven Systems	p. 163
Introduction	p. 163
Driven Anharmonic Systems	p. 166
Phase Portrait	p. 166
Poincare Section	p. 167
Liapunov Exponent	p. 169
Autocorrelation Function	p. 170
Power Spectral Density	p. 173
Driven Pendulum	p. 174
Phase Portrait	p. 174
Poincare Section	p. 176
Parametrically Driven Pendulum	p. 178
Phase Portrait	p. 178
Poincare Section	p. 179
Driven Van der Pol Equation	p. 181
Phase Portrait	p. 181
Liapunov Exponent	p. 183
Parametrically and Externally Driven Pendulum	p. 185
Torsion Numbers	p. 187
Controlling of Chaos	p. 191
Introduction	p. 191
Ott-Yorke-Grebogi Method	p. 191
One-Dimensional Maps	p. 191
Systems of Difference Equations	p. 195
Small Periodic Perturbation	p. 199
Resonant Perturbation and Control	p. 201
Synchronization of Chaos	p. 203
Introduction	p. 203
Synchronization of Chaos	p. 203
Synchronization Using Control	p. 203
Synchronizing Subsystems	p. 206
Synchronization of Coupled Dynamos	p. 209
Phase Coupled Systems	p. 211
Fractals	p. 217
Introduction	p. 217
Iterated Function System	p. 219
Introduction	p. 219
Cantor Set	p. 220
Heighway's Dragon	p. 223
Sierpinski Gasket	p. 225
Koch Curve	p. 227
Fern	p. 229
Grey Level Maps	p. 231
Mandelbrot Set	p. 232
Julia Set	p. 234
Fractals and Kronecker Product	p. 236
Lindenmayer Systems and Fractals	p. 240
Weierstrass Function	p. 243
Cellular Automata	p. 245
Introduction	p. 245
One-Dimensional Cellular Automata	p. 248
Sznajd Model	p. 249
Conservation Laws	p. 252
Two-Dimensional Cellular Automata	p. 253
Button Game	p. 257
Solving Differential Equations	p. 261
Introduction	p. 261
Euler Method	p. 262
Lie Series Technique	p. 264
Runge-Kutta-Fehlberg Technique	p. 268
Ghost Solutions	p. 269
Symplectic Integration	p. 272
Verlet Method	p. 277
Stormer Method	p. 279
Invisible Chaos	p. 280
First Integrals and Numerical Integration	p. 281
Neural Networks	p. 283
Introduction	p. 283
Hopfield Model	p. 287
Introduction	p. 287
Synchronous Operations	p. 289
Energy Function	p. 291
Basins and Radii of Attraction	p. 293
Spurious Attractors	p. 293
Hebb's Law	p. 294
Hopfield Example	p. 296
Hopfield C++ Program	p. 298
Asynchronous Operation	p. 302
Translation Invariant Pattern Recognition	p. 303
Similarity Metrics	p. 305
Kohonen Network	p. 309
Introduction	p. 309
Kohonen Algorithm	p. 310
Kohonen Example	p. 312
Traveling Salesman Problem	p. 318
Perceptron	p. 322
Introduction	p. 322
Boolean Functions	p. 324
Linearly Separable Sets	p. 325
Perceptron Learning	p. 326
Perceptron Learning Algorithm	p. 330
One and Two Layered Networks	p. 333
XOR Problem and Two-Layered Networks	p. 335
Multilayer Perceptrons	p. 339
Introduction	p. 339
Cybenko's Theorem	p. 340
Back-Propagation Algorithm	p. 340
Recursive Deterministic Perceptron Neural Networks	p. 348
Chaotic Neural Networks	p. 350
Neuronal-Oscillator Models	p. 351
Radial Basis Function Networks	p. 353
Neural Network, Matrices and Eigenvalues	p. 355
Genetic Algorithms	p. 357
Introduction	p. 357
Sequential Genetic Algorithm	p. 358
Schemata Theorem	p. 362
Bitwise Operations	p. 364
Introduction	p. 364
Assembly Language	p. 367
Floating Point Numbers and Bitwise Operations	p. 369
Java Bitset Class	p. 370
C++ bitset Class	p. 371
Bit Vector Class	p. 373
Penna Bit-String Model	p. 376
Maximum of One-Dimensional Maps	p. 378
Maximum of Two-Dimensional Maps	p. 384
Finding a Fitness Function	p. 392
Problems with Constraints	p. 398
Introduction	p. 398
Knapsack Problem	p. 399
Traveling Salesman Problem	p. 404
Simulated Annealing	p. 412
Gene Expression Programming	p. 415
Introduction	p. 415
Example	p. 418
Numerical-Symbolic Manipulation	p. 430
Multi Expression Programming	p. 435
Optimization	p. 441
Lagrange Multiplier Method	p. 441
Karush-Kuhn-Tucker Conditions	p. 449
Support Vector Machine	p. 453
Introduction	p. 453
Linear Decision Boundaries	p. 453
Nonlinear Decision Boundaries	p. 457
Kernel Fisher Discriminant	p. 461
Discrete Wavelets	p. 465
Introduction	p. 465
Multiresolution Analysis	p. 468
Pyramid Algorithm	p. 470
Biorthogonal Wavelets	p. 475
Two-Dimensional Wavelets	p. 480
Discrete Hidden Markov Processes	p. 483
Introduction	p. 483
Markov Chains	p. 485
Discrete Hidden Markov Processes	p. 489
Forward-Backward Algorithm	p. 493
Viterbi Algorithm	p. 496
Baum-Welch Algorithm	p. 497
Distances between HMMs	p. 498
Application of HMMs	p. 499
C++ Program	p. 502
Fuzzy Sets and Fuzzy Logic	p. 513
Introduction	p. 513
Operators for Fuzzy Sets	p. 521
Logical Operators	p. 521
Algebraic Operators	p. 524
Defuzzification Operators	p. 525
Fuzzy Concepts as Fuzzy Sets	p. 527
Hedging	p. 528
Quantifying Fuzzyness	p. 529
C++ Implementation of Discrete Fuzzy Sets	p. 530
Applications: Simple Decision-Making Problems	p. 549
Fuzzy Numbers and Fuzzy Arithmetic	p. 555
Introduction	p. 555
Algebraic Operations	p. 556
LR-Representations	p. 559
Algebraic Operations on Fuzzy Numbers	p. 562
C++ Implementation of Fuzzy Numbers	p. 563
Applications	p. 570
Fuzzy Rule-Based Systems	p. 571
Introduction	p. 571
Fuzzy If-Then Rules	p. 574
Inverted Pendulum Control System	p. 575
Fuzzy Controllers with B-Spline Models	p. 577
Application	p. 580
Fuzzy C-Means Clustering	p. 582
fXOR Fuzzy Logic Networks	p. 586
Fuzzy Hamming Distance	p. 588
Fuzzy Truth Values and Probabilities	p. 591
Bibliography	p. 593
Index	p. 601
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