The Nonlinear Workbook

The following sections are included:

• Preface

• Contents

• Symbol Index

• Comments to the Programs

The following sections are included:

• 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
  • Chaotic Repeller
  • Chaos and Encoding

• 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
  • Periodic Orbits and Topological Degree
  • JPEG file

The following sections are included:

• Introduction

• Correlation Coefficient

• Liapunov Exponent from Time Series
  • Jacobian Matrix Estimation Algorithm
  • Direct Method

• Hurst Exponent
  • Introduction
  • Implementation for the Hurst Exponent
  • Random Walk

• Higuchi's Algorithm

• Complexity

The following sections are included:

• Classification of Fixed Points

• Homoclinic Orbit

• Pendulum

• Limit Cycle Systems

• Lotka-Volterra Systems

The following sections are included:

• 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
  • Hénon-Heiles Hamilton Function and Trajectories
  • Hénon Heiles and Surface of Section Method
  • Quartic Potential and Surface of Section Technique

The following sections are included:

• Fixed Points and Stability

• Trajectories

• Phase Portrait

• Liapunov Exponents

• Generalized Lotka-Volterra Model

• Hyperchaotic Systems

• Hopf Bifurcation

• Time-Dependent First Integrals

The following sections are included:

• Introduction

• Driven Anharmonic Systems
  • Phase Portrait
  • Poincaré Section
  • Liapunov Exponent
  • Autocorrelation Function
  • Power Spectral Density

• Driven Pendulum
  • Phase Portrait
  • Poincaré Section

• Parametrically Driven Pendulum
  • Phase Portrait
  • Poincaré Section

• Driven Van der Pol Equation
  • Phase Portrait
  • Liapunov Exponent

• Parametrically and Externally Driven Pendulum

• Torsion Numbers

The following sections are included:

• Introduction

• Ott-Yorke-Grebogi Method
  • One-Dimensional Maps
  • Systems of Difference Equations

• Small Periodic Perturbation

• Resonant Perturbation and Control

The following sections are included:

• Introduction

• Synchronization of Chaos
  • Synchronization Using Control
  • Synchronizing Subsystems

• Synchronization of Coupled Dynamos

• Phase Coupled Systems

The following sections are included:

• Introduction

• Iterated Function System
  • Introduction
  • Cantor Set
  • Heighway's Dragon
  • Sierpinski Gasket
  • Koch Curve
  • Fern
  • Grey Level Maps

• Mandelbrot Set

• Julia Set

• Fractals and Kronecker Product

• Lindenmayer Systems and Fractals

• Weierstrass Function

The following sections are included:

• Introduction

• One-Dimensional Cellular Automata

• Sznajd Model

• Conservation Laws

• Two-Dimensional Cellular Automata

• Button Game

The following sections are included:

• Introduction

• Euler Method

• Lie Series Technique

• Runge-Kutta-Fehlberg Technique

• Ghost Solutions

• Symplectic Integration

• Verlet Method

• Stürmer Method

• Invisible Chaos

• First Integrals and Numerical Integration

The following sections are included:

• Introduction

• Hopfield Model
  • Introduction
  • Synchronous Operations
  • Energy Function
  • Basins and Radii of Attraction
  • Spurious Attractors
  • Hebb's Law
  • Hopfield Example
  • Hopfield C++ Program
  • Asynchronous Operation
  • Translation Invariant Pattern Recognition

• Similarity Metrics

• Kohonen Network
  • Introduction
  • Kohonen Algorithm
  • Kohonen Example
  • Traveling Salesman Problem

• Perceptron
  • Introduction
  • Boolean Functions
  • Linearly Separable Sets
  • Perceptron Learning
  • Perceptron Learning Algorithm
  • One and Two Layered Networks
  • XOR Problem and Two-Layered Networks

• Multilayer Perceptrons
  • Introduction
  • Cybenko's Theorem
  • Back-Propagation Algorithm
  • Recursive Deterministic Perceptron Neural Networks

• Chaotic Neural Networks

• Neuronal-Oscillator Models

• Radial Basis Function Networks

• Neural Network, Matrices and Eigenvalues

The following sections are included:

• Introduction

• Sequential Genetic Algorithm

• Schemata Theorem

• Bitwise Operations
  • Introduction
  • Assembly Language
  • Floating Point Numbers and Bitwise Operations
  • Java Bitset Class
  • C++ bitset Class

• Bit Vector Class

• Penna Bit-String Model

• Maximum of One-Dimensional Maps

• Maximum of Two-Dimensional Maps

• Finding a Fitness Function

• Problems with Constraints
  • Introduction
  • Knapsack Problem
  • Traveling Salesman Problem

• Simulated Annealing

The following sections are included:

• Introduction

• Example

• Numerical-Symbolic Manipulation

• Multi Expression Programming

The following sections are included:

• Lagrange Multiplier Method

• Karush-Kuhn-Tucker Conditions

• Support Vector Machine
  • Introduction
  • Linear Decision Boundaries
  • Nonlinear Decision Boundaries
  • Kernel Fisher Discriminant

The following sections are included:

• Introduction

• Multiresolution Analysis

• Pyramid Algorithm

• Biorthogonal Wavelets

• Two-Dimensional Wavelets

The following sections are included:

• Introduction

• Markov Chains

• Discrete Hidden Markov Processes

• Forward-Backward Algorithm

• Viterbi Algorithm

• Baum-Welch Algorithm

• Distances between HMMs

• Application of HMMs

• C++ Program

The following sections are included:

• Introduction

• Operators for Fuzzy Sets
  • Logical Operators
  • Algebraic Operators
  • Defuzzification 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
  • Fuzzy Controllers with B-Spline Models
  • Application

• Fuzzy C-Means Clustering

• fXOR Fuzzy Logic Networks

• Fuzzy Hamming Distance

• Fuzzy Truth Values and Probabilities

The following sections are included:

• Bibliography

• Index