DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

Transcription

1 DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University ELSEVIER ACADEMIC PRESS Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Academic Press is an imprint of Elsevier

2 Contents Preface x CHAPTER 1 First-Order Equations The Simplest Example The Logistic Population Model Constant Harvesting and Bifurcations Periodic Harvesting and Periodic Solutions Computing the Poincare Map Exploration: A Two-Parameter Family 15 CHAPTER 2 Planar Linear Systems Second-Order Differential Equations Planar Systems Preliminaries from Algebra Planar Linear Systems Eigenvalues and Eigenvectors Solving Linear Systems The Linearity Principle 36 CHAPTER 3 Phase Portraits for Planar Systems Real Distinct Eigenvalues Complex Eigenvalues 44

3 vi Contents 3.3 Repeated Eigenvalues Changing Coordinates 49 CHAPTER 4 Classification of Planar Systems The Trace-Determinant Plane Dynamical Classification Exploration: A 3D Parameter Space 71 CHAPTER 5 Higher Dimensional Linear Algebra Preliminaries from Linear Algebra Eigenvalues and Eigenvectors Complex Eigenvalues Bases and Subspaces Repeated Eigenvalues Genericity 101 CHAPTER 6 Higher Dimensional Linear Systems Distinct Eigenvalues Harmonic Oscillators Repeated Eigenvalues The Exponential of a Matrix Nonautonomous Linear Systems 130 CHAPTER 7 Nonlinear Systems Dynamical Systems The Existence and Uniqueness Theorem Continuous Dependence of Solutions The Variational Equation Exploration: Numerical Methods 153 CHAPTER 8 Equilibria in Nonlinear Systems Some Illustrative Examples Nonlinear Sinks and Sources Saddles Stability Bifurcations Exploration: Complex Vector Fields 182

4 Contents vii CHAPTER 9 Global Nonlinear Techniques Nullclines Stability of Equilibria Gradient Systems Hamiltonian Systems Exploration: The Pendulum with Constant Forcing 210 CHAPTER 10 Closed Orbits and Limit Sets Limit Sets Local Sections and Flow Boxes The Poincare Map Monotone Sequences in Planar Dynamical Systems The Poincare-Bendixson Theorem Applications of Poincare-Bendixson Exploration: Chemical Reactions That Oscillate 230 CHAPTER 11 Applications in Biology Infectious Diseases Predator/Prey Systems Competitive Species Exploration: Competition and Harvesting 252 CHAPTER 12 Applications in Circuit Theory An RLC Circuit The Lienard Equation The van der Pol Equation A Hopf Bifurcation Exploration: Neurodynamics 272 CHAPTER 13 Applications in Mechanics Newton's Second Law Conservative Systems Central Force Fields The Newtonian Central Force System 285

5 viii Contents 13.5 Kepler's First Law The Two-Body Problem Blowing Up the Singularity Exploration: Other Central Force Problems Exploration: Classical Limits of Quantum Mechanical Systems 298 CHAPTER 14 The Lorenz System Introduction to the Lorenz System Elementary Properties of the Lorenz System The Lorenz Attract or A Model for the Lorenz Attractor The Chaotic Attractor Exploration: The Rossler Attractor 324 CHAPTER 15 Discrete Dynamical Systems Introduction to Discrete Dynamical Systems Bifurcations The Discrete Logistic Model Chaos Symbolic Dynamics The Shift Map The Cantor Middle-Thirds Set Exploration: Cubic Chaos Exploration: The Orbit Diagram 353 CHAPTER 16 Homoclinic Phenomena The Shil'nikov System The Horseshoe Map The Double Scroll Attractor Homoclinic Bifurcations Exploration: The Chua Circuit 379 CHAPTER 17 Existence and Uniqueness Revisited The Existence and Uniqueness Theorem Proof of Existence and Uniqueness 385

6 Contents ix Bibliography 407 Index Continuous Dependence on Initial Conditions Extending Solutions Nonautonomous Systems Differentiability of the Flow 400