Dynamical Systems with Applications

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1 Stephen Lynch Dynamical Systems with Applications using MATLAB Birkhauser Boston Basel Berlin

2 Preface xi 0 A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox Tutorial One: The Basics and the Symbolic Math Toolbox (One Hour) Tutorial Two: Plots and Differential Equations (One Hour) MATLAB Program Files, or M-Files Hints for Programming MATLAB Exercises... ' Linear Discrete Dynamical Systems Recurrence Relations The Leslie Model Harvesting and Culling Policies MATLAB Commands Exercises 30 2 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 Henon Maps 55

3 vi 2.5 Applications MATLAB Commands Exercises 66 3 Complex Iterative Maps Julia Sets and the Mandelbrot Set Boundaries of Periodic Orbits MATLAB Commands Exercises 79 4 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 MATLAB Commands Exercises Fractals and Multifractals Construction of Simple Examples Calculating Fractal Dimensions A Multifractal Formalism Multifractals in the Real World and Some Simple Examples MATLAB Commands Exercises Controlling Chaos Historical Background Controlling Chaos in the Logistic Map Controlling Chaos in the Henon Map MATLAB Commands Exercises Differential Equations Simple Differential Equations and Applications Applications to Chemical Kinetics Applications to Electric Circuits Existence and Uniqueness Theorem MATLAB Commands Exercises 180

4 vii 8 Planar Systems Canonical Forms Eigenvectors Denning Stable and Unstable Manifolds Phase Portraits of Linear Systems in the Plane Linearization and Hartman's Theorem Constructing Phase Plane Diagrams MATLAB Commands Exercises Interacting Species Competing Species Predator-Prey Models Other Characteristics Affecting Interacting Species MATLAB Commands Exercises Limit Cycles Historical Background Existence and Uniqueness of Limit Cycles in the Plane Nonexistence of Limit Cycles in the Plane Exercises Hamiltonian Systems, Lyapunov Functions, and Stability Hamiltonian Systems in the Plane Lyapunov Functions and Stability Exercises Bifurcation Theory Bifurcations of Nonlinear Systems in the Plane Multistability and Bistability MATLAB Commands... I Exercises Three-Dimensional Autonomous Systems and Chaos Linear Systems and Canonical Forms Nonlinear Systems and Stability The Rossler System and Chaos The Lorenz Equations, Chua's Circuit, and the Belousov-Zhabotinski Reaction MATLAB Commands Exercises Poincare Maps and Nonautonomous Systems in the Plane Poincare Maps 298

5 viii 14.2 Hamiltonian Systems with Two Degrees of Freedom Nonautonomous Systems in the Plane MATLAB Commands Exercises Local and Global Bifurcations Small-Amplitude Limit Cycle Bifurcations Melnikov Integrals and Bifurcating Limit Cycles from a Center Homoclinic Bifurcations MATLAB Commands Exercises The Second Part of Hubert's Sixteenth Problem Statement of Problem and Main Results Poincare" Compactification Global Results for Lienard Systems Local Results for Lienard Systems Exercises Neural Networks Introduction The Delta Learning Rule and Backpropagation The Hopfield Network and Lyapunov Stability Neurodynamics MATLAB Commands Exercises Simulink Introduction Electric Circuits A Mechanical System Nonlinear Optics The Lorenz Equations and Chaos Synchronization Exercises Solutions to Exercises Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter7 416

6 ix 19.8 Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter References 429 Textbooks 429 Research Papers 434 MATLAB Program File Index 443 Simulink Model File Index 447 Index 449

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