I NONLINEAR EWORKBOOK

Transcription

1 I NONLINEAR EWORKBOOK Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and SymbolicC++ Programs Willi-Hans Steeb International School for Scientific Computing, South Africa steeb_ wh yahoo, com in collaboration with Yorick Hardy International School for Scientific Computing, South Africa Ruedi Stoop Institute for Neuroinformatics, University of Ziirich/ETHZ \\b World Scientific New Jersey London Singapore Hong Kong

2 Contents 1 Nonlinear and Chaotic Maps One-Dimensional Maps Exact and Numerical Trajectories Fixed Points and Stability Invariant Density Liapunov Exponent Autocorrelation Function Discrete Fourier Transform Fast Fourier Transform Logistic Map and Liapunov Exponent for r [3,4] Logistic Map and Bifurcation Diagram Random Number Map and Invariant Density Random Number Map and Random Integration Circle Map and Rotation Number Newton Method Feigenbaum's Constant Symbolic Dynamics Two-Dimensional Maps Introduction ' Phase Portrait Fixed Points and Stability Liapunov Exponents Correlation Integral Capacity Hyperchaos Domain of Attraction Newton Method in the Complex Domain Newton Method in Higher Dimensions Ruelle-Takens-Newhouse Scenario JPEG file Time Series Analysis Introduction Correlation Coefficient Liapunov Exponent from Time Series 112 ix

3 CONTENTS Jacobian Matrix Estimation Algorithm Direct Method Hurst Exponent Introduction Implementation for the Hurst Exponent Complexity 134 Autonomous Systems in the Plane Classification of Fixed Points Homoclinic Orbit Pendulum Limit Cycle Systems Lotka-Volterra Systems 151 Nonlinear Hamilton Systems Hamilton Equations of Motion Hamilton System and Variational Equation Integrable Hamilton Systems Hamilton Systems and First Integrals Lax Pair and Hamilton Systems Floquet Theory Chaotic Hamilton Systems Henon-Heiles Hamilton Function and Trajectories Henon Heiles and Surface of Section Method Quartic Potential and Surface of Section Technique 176 Nonlinear Dissipative Systems Fixed Points and Stability Trajectories Phase Portrait Liapunov Exponents Generalized Lotka-Volterra Model Hyperchaotic Systems Hopf Bifurcation Time-Dependent First Integrals 209 Nonlinear Driven Systems Introduction Driven Anharmonic Systems Phase Portrait Poincare Section Liapunov Exponent Autocorrelation Function Power Spectral Density Driven Pendulum 229

4 CONTENTS xi Phase Portrait Poincare Section Parametrically Driven Pendulum Phase Portrait Poincare Section Driven Van der Pol Equation Phase Portrait Liapunov Exponent Parametrically and Externally Driven Pendulum Controlling and Synchronization of Chaos Introduction Ott-Yorke-Grebogi Method One-Dimensional Maps Systems of Difference Equations Small Periodic Perturbation Resonant Perturbation and Control Synchronization of Chaos Synchronization Using Control Synchronizing Subsystems Phase Coupled Systems Fractals Introduction Iterated Function System Introduction Cantor Set Heighway's Dragon Sierpinski Gasket Koch Curve Mandelbrot Set Julia Set Weierstrass Function Cellular Automata Introduction One-Dimensional Cellular Automata Two-Dimensional Cellular Automata Button Game Solving Differential Equations Introduction Euler Method Lie Series Technique Runge-Kutta-Fehlberg Technique 325

5 xii CONTENTS 10.5 Ghost Solutions Symplectic Integration Invisible Chaos Stormer Method Neural Networks Introduction Hopfield Model Introduction Synchronous Operations Energy Function Basins and Radii of Attraction Spurious Attractors Hebb's Law Example Program Asynchronous Operation Translation Invariant Pattern Recognition Similarity Metrics Kohonen Network Introduction Algorithm Example Traveling Salesman Problem Perceptron Introduction Boolean Functions Linearly Separable Sets Perceptron Learning One and Two Layered Networks Perceptron Learning Algorithm The XOR Problem and Two-Layered Networks Multilayer Perceptrons Introduction Cybenko's Theorem Back-Propagation Algorithm Genetic Algorithms Introduction The Sequential Genetic Algorithm Schemata Theorem Bitwise Operations A Bit Vector Class Maximum of One-Dimensional Maps Maximum of Two-Dimensional Maps 455

6 CONTENTS xiii 12.8 Problems with Constraints Introduction Knapsack Problem Traveling Salesman Problem Simulated Annealing Parallel Genetic Algorithms Gene Expression Programming Introduction Example ; Discrete Wavelets Introduction Example Two-Dimensional Wavelets Fuzzy Sets and Fuzzy Logic Introduction Operators for Fuzzy Sets Logical Operators Algebraic Operators Denazification Operators Fuzzy Concepts as Fuzzy Sets Hedging Quantifying Fuzzyness C++ Implementation of Discrete Fuzzy Sets "Applications: Simple Decision-Making Problems Fuzzy Numbers and Fuzzy Arithmetic Introduction Algebraic Operations LR-Representations Algebraic Operations on Fuzzy Numbers C++ Implementation of Fuzzy Numbers Applications Fuzzy Rule-Based Systems Introduction Fuzzy If-Then Rules Inverted Pendulum Control System Application Fuzzy Truth Values and Probabilities 612 Bibliography 613 Index 619