PARTIAL DIFFERENTIAL EQUATIONS

By JAMES JACKSON

CONTENTS PART 1 CLASSICAL THEORY CHAPTER 1 INTRODUCTION §1 1 NEWTON’S EQUATIONS §1 2 CLASSI?CATION OF DI?ERENTIAL EQUATIONS §1 3 FIRST ORDER AUTONOMOUS EQUATIONS §1 4 FINDING EXPLICIT SOLUTIONS §1 5 QUALITATIVE ANALYSIS OF ?RST ORDER EQUATIONS CHAPTER 2 INITIAL VALUE PROBLEMS §2 1 FIXED POINT THEOREMS §2 2 THE BASIC EXISTENCE AND UNIQUENESS RESULT §2 3 DEPENDENCE ON THE INITIAL CONDITION §2 4 EXTENSIBILITY OF SOLUTIONS §2 5 EULER’S METHOD AND THE PEANO THEOREM §2 6 APPENDIX: VOLTERRA INTEGRAL EQUATIONS CHAPTER 3 LINEAR EQUATIONS §3 1 PRELIMINARIES FROM LINEAR ALGEBRA §3 2 LINEAR AUTONOMOUS ?RST ORDER SYSTEMS §3 3 GENERAL LINEAR ?RST ORDER SYSTEMS §3 4 PERIODIC LINEAR SYSTEMS CHAPTER 4 DI?ERENTIAL EQUATIONS IN THE COMPLEX DOMAIN §4 1 THE BASIC EXISTENCE AND UNIQUENESS RESULT §4 2 LINEAR EQUATIONS §4 3 THE FROBENIUS METHOD §4 4 SECOND ORDER EQUATIONS CHAPTER 5 BOUNDARY VALUE PROBLEMS §5 1 INTRODUCTION §5 2 SYMMETRIC COMPACT OPERATORS §5 3 REGULAR STURM-LIOUVILLE PROBLEMS §5 4 OSCILLATION THEORY PART 2 DYNAMICAL SYSTEMS CHAPTER 6 DYNAMICAL SYSTEMS §6 1 DYNAMICAL SYSTEMS §6 2 THE ?OW OF AN AUTONOMOUS EQUATION §6 3 ORBITS AND INVARIANT SETS §6 4 STABILITY OF ?XED POINTS §6 5 STABILITY VIA LIAPUNOV’S METHOD §6 6 NEWTON’S EQUATION IN ONE DIMENSION CHAPTER 7 LOCAL BEHAVIOR NEAR ?XED POINTS §7 1 STABILITY OF LINEAR SYSTEMS §7 2 STABLE AND UNSTABLE MANIFOLDS §7 3 THE HARTMAN-GROBMAN THEOREM §7 4 APPENDIX: HAMMERSTEIN INTEGRAL EQUATIONS CHAPTER 8 PLANAR DYNAMICAL SYSTEMS §8 1 THE POINCAR´E–BENDIXSON THEOREM §8 2 EXAMPLES FROM ECOLOGY §8 3 EXAMPLES FROM ELECTRICAL ENGINEERING CHAPTER 9 HIGHER DIMENSIONAL DYNAMICAL SYSTEMS §9 1 ATTRACTING SETS §9 2 THE LORENZ EQUATION §9 3 HAMILTONIAN MECHANICS §9 4 COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS §9 5 THE KEPLER PROBLEM §9 6 THE KAM THEOREM PART 3 CHAOS CHAPTER 10 DISCRETE DYNAMICAL SYSTEMS §10 1 THE LOGISTIC EQUATION §10 2 FIXED AND PERIODIC POINTS §10 3 LINEAR DI?ERENCE EQUATIONS §10 4 LOCAL BEHAVIOR NEAR ?XED POINTS CHAPTER 11 PERIODIC SOLUTIONS §11 1 STABILITY OF PERIODIC SOLUTIONS §11 2 THE POINCAR´E MAP §11 3 STABLE AND UNSTABLE MANIFOLDS §11 4 MELNIKOV’S METHOD FOR AUTONOMOUS PERTURBATIONS §11 5 MELNIKOV’S METHOD FOR NONAUTONOMOUS PERTURBATIONS CHAPTER 12 DISCRETE DYNAMICAL SYSTEMS IN ONE DIMENSION §12 1 PERIOD DOUBLING §12 2 SARKOVSKII’S THEOREM §12 3 ON THE DE?NITION OF CHAOS §12 4 CANTOR SETS AND THE TENT MAP §12 5 SYMBOLIC DYNAMICS §12 6 STRANGE ATTRACTORS/REPELLORS AND FRACTAL SETS §12 7 HOMOCLINIC ORBITS AS SOURCE FOR CHAOS CHAPTER 13 CHAOS IN HIGHER DIMENSIONAL SYSTEMS §13 1 THE SMALE HORSESHOE §13 2 THE SMALE-BIRKHO?HOMOCLINIC THEOREM §13 3 MELNIKOV’S METHOD FOR HOMOCLINIC ORBITS