Dynamical Systems with Applications using Mathematica

Transcription

1 Stephen Lynch Dynamical Systems with Applications using Mathematica Birkhäuser Boston Basel Berlin

2 Contents Preface xi 0 A Tutorial Introduction to Mathematica A Quick Tour of Mathematica Tutorial One: The Basics (One Hour) Tutorial Two: Plots and Differential Equations (One Hour) The Manipulate Command and Simple Mathematica Programs Hints for Programming Mathematica Exercises 12 1 Differential Equations Simple Differential Equations and Applications Applications to Chemical Kinetics Applications to Electric Circuits Existence and Uniqueness Theorem Mathematica Commands in Text Format Exercises 36 2 Planar Systems Canonical Forms Eigenvectors Defining Stahle and Unstable Manifolds Phase Portraits of Linear Systems in the Plane 49

3 VI Contents 2.4 Linearization and Hartman's Theorem Constructing Phase Plane Diagrams Mathematica Commands in Text Format Exercises 65 3 Interacting Species Competing Species Predator-Prey Models Other Characteristics Affecting Interacting Species Mathematica Commands in Text Format Exercises 81 4 Limit Cycles Historical Background Existence and Uniqueness of Limit Cycles in the Plane Nonexistence of Limit Cycles in the Plane Perturbation Methods Mathematica Commands in Text Format Exercises Hamiltonian Systems, Lyapunov Functions, and Stability Hamiltonian Systems in the Plane Lyapunov Functions and Stability Mathematica Commands in Text Format Exercises Bifurcation Theory Bifurcations of Nonlinear Systems in the Plane Normal Forms Multistability and Bistability Mathematica Commands in Text Format Exercises Three-Dimensional Autonomous Systems and Chaos Linear Systems and Canonical Forms Nonlinear Systems and Stability The Rössler System and Chaos The Lorenz Equations, Chua's Circuit, and the Belousov-Zhabotinski Reaction Mathematica Commands in Text Format Exercises Poincare Maps and Nonautonomous Systems in the Plane Poincare Maps 172

4 Contents vii 8.2 Hamiltonian Systems with Two Degrees of Freedom Nonautonomous Systems in the Plane Mathematica Commands in Text Format Exercises Local and Global Bifurcations Small-Amplitude Limit Cycle Bifurcations Gröbner Bases Melnikov Integrals and Bifurcating Limit Cycles from a Center Bifurcations Involving Homoclinic Loops Mathematica Commands in Text Format Exercises The Second Part of Hilbert's Sixteenth Problem Statement of Problem and Main Results Poincare Compactification Global Results for Lienard Systems Local Results for Lienard Systems Exercises Linear Discrete Dynamical Systems Recurrence Relations The Leslie Model Harvesting and Culling Policies Mathematica Commands in Text Format Exercises Nonlinear Discrete Dynamical Systems The Tent Map and Graphical Iterations Fixed Points and Periodic Orbits The Logistic Map, Bifurcation Diagram, and Feigenbaum Number Gaussian and He"non Maps Applications Mathematica Commands in Text Format Exercises Complex Iterative Maps Julia Sets and the Mandelbrot Set Boundaries of Periodic Orbits Mathematica Commands in Text Format Exercises 302

5 viii Contents 14 Electromagnetic Waves and Optical Resonators Maxwell's Equations and Electromagnetic Waves Historical Background The Nonlinear SFR Resonator Chaotic Attractors and Bistability Linear Stability Analysis Instabilities and Bistability Mathematica Commands in Text Format Exercises Fractals and Multifractals Construction of Simple Examples Calculating Fractal Dimensions A Multifractal Formalism Multifractals in the Real World and Some Simple Examples Mathematica Commands in Text Format Exercises Chaos Control and Synchronization Historical Background Controlling Chaos in the Logistic Map Controlling Chaos in the Henon Map Chaos Synchronization Mathematica Commands in Text Format Exercises Neural Networks Introduction The Delta Learning Rule and Backpropagation The Hopneld Network and Lyapunov Stability Neurodynamics Mathematica Commands in Text Format Exercises Examination-Type Questions Dynamical Systems with Applications Dynamical Systems with Mathematica Solutions to Exercises ChapterO Chapterl Chapter Chapter Chapter4 435

6 Contents ix 19.5 Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter References 451 Textbooks 451 Research Papers 459 Mathematica Program Index 469 Index 473