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
  • Sarkovskii's Theorem
  • Discrete One-Dimensional 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
  • One-Dimensional Newton Method
  • Feigenbaum's Constant
  • Symbolic Dynamics
  • Chaotic Repeller
  • Chaos and Encoding
  • Chaotic Data Communication
  • Exercises
• 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
  • Melnikov Analysis for Maps
  • Periodic Orbits and Topological Degree
  • JPEG file
  • Baker's Map and Ergodic Theorems
  • Fixed Point Theorems
• Exercises

The following sections are included:

• Introduction
• Correlation Coefficient
• Liapunov Exponent from Time Series
  • Jacobian Matrix Estimation Algorithm
  • Direct Method
• Hurst Exponent
  • Introduction
  • Implementation of the Hurst Exponent
• Random Walk
• Brownian Motion
• Higuchi's Algorithm
• Complexity
• Exercises

The following sections are included:

• Classification of Fixed Points
• Homoclinic Orbit
• One-Dimensional Pendulum
• Limit Cycle Systems
• Lotka-Volterra Systems
• Biochemical Equations
• Exercises

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
  • Trajectories and Hénon-Heiles Hamilton Function
  • Surface of Section Method
• Exercises

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
• Exercises

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
• Exercises

The following sections are included:

• Introduction
• Ott-Yorke-Grebogi Method
  • One-Dimensional Maps
  • Systems of Difference Equations
• Time-Delayed Feedback Control
• Small Periodic Perturbation
• Resonant Perturbation and Control
• Exercises

The following sections are included:

• Introduction
• Synchronization of Chaos
  • Synchronization Using Control
  • Synchronizing Subsystems
  • Synchronization of Coupled Dynamos
  • Phase Coupled Systems
  • Exercises

The following sections are included:

• Introduction
• Fractal Dimensions
• 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
• Lévy-Flight Random Walk
• Exercises

The following sections are included:

• Introduction
• A Spin System and Cellular Automata
• Sznajd Model
• Conservation Laws
• Two-Dimensional Cellular Automata
• Button Game
• Langton's Ant
• Quantum Cellular Automata
• Exercises

The following sections are included:

• Introduction
• Euler Method
• Lie Series Technique
• Runge-Kutta-Fehlberg Technique
• Ghost Solutions
• Symplectic Integration
• Verlet Method
• Störmer Method
• Invisible Chaos
• First Integrals and Numerical Integration
• Exercises

The following sections are included:

• Lagrange Multiplier Method
• Examples
• Coordinate Systems
• Differential Forms and Cartan Forms
• Karush-Kuhn-Tucker Conditions
• Support Vector Machine
  • Introduction
  • Linear Decision Boundaries
  • Nonlinear Decision Boundaries
  • Kernel Fisher Discriminant
• Particle Swarm Optimization
• Exercises

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
  • Perceptron Learning
  • One and Two Layered Networks
  • XOR Problem and Two Layered Networks
• Multilayer Perceptrons
  • Introduction
  • Cybenko's Theorem
  • Back-Propagation Algorithm
• Radial Basis Function Networks
• Recursive Perceptron Neural Networks
• Chaotic Neural Networks
• Neuronal-Oscillator Models
• Neural Network, Matrices and Eigenvalues
• Exercises

The following sections are included:

• Introduction
• Sequential Genetic Algorithm
• Schemata Theorem
• Bitwise Operations
  • Introduction
  • Assembly Language
  • Floating Point Numbers and Bitwise Operations
  • C++ bitset Class
  • Java Bitset 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
  • Exercises

The following sections are included:

• Introduction
• Examples
  • Maps
  • Boolean Expression
• Numerical-Symbolic Manipulation
• Multi Expression Programming
• Exercises

The following sections are included:

• Introduction
• Haar Wavelet
• Other Wavelets
• Multiresolution Analysis
• Biorthogonal Wavelets
• Two-Dimensional Wavelets
• Pyramid Algorithm and Discrete Wavelets
• Exercises

The following sections are included:

• Introduction
• Stochastic Matrices
• Markov Chains
• Discrete Hidden Markov Models
  • Definitions
  • The Three Problems
  • Forward-Backward Algorithm
  • Viterbi Algorithm
  • Baum-Welch Algorithm
  • Distances between Hidden Markov Models
• Applications of Hidden Markov Models
  • Speech Recognition
  • Biological Systems
• C++ Program
• Exercises

The following sections are included:

• Introduction
• Definitions and Examples
• Operators for Fuzzy Sets
  • Logical Operators
  • Algebraic Operators
  • Defuzzification Operators
  • Fuzzy Concepts as Fuzzy Sets
  • Hedging
  • Quantifying Fuzzyness
  • C++ Implementation of Discrete Fuzzy Sets
• 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
• T-Norms and T-Conorms
• Fuzzy Logic Networks
• Fuzzy Hamming Distance
• Fuzzy Truth Values and Probabilities
• Exercises

The following sections are included:

• Bibliography
• Index