DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

By M. MIONK

CONTENTS CHAPTER 1 FIRST-ORDER EQUATIONS THE SIMPLEST EXAMPLE THE LOGISTIC POPULATION MODEL CONSTANT HARVESTING AND BIFURCATIONS PERIODIC HARVESTING AND PERIODIC SOLUTIONS COMPUTING THE POINCARÉ MAP EXPLORATION: A TWO-PARAMETER FAMILY 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 CHAPTER 3 PHASE PORTRAITS FOR PLANAR SYSTEMS REAL DISTINCT EIGENVALUES COMPLEX EIGENVALUES REPEATED EIGENVALUES CHANGING COORDINATES CHAPTER 4 CLASSIFICATION OF PLANAR SYSTEMS THE TRACE-DETERMINANT PLANE DYNAMICAL CLASSIFICATION EXPLORATION: A 3D PARAMETER SPACE CHAPTER 5 HIGHER DIMENSIONAL LINEAR ALGEBRA PRELIMINARIES FROM LINEAR ALGEBRA BASES AND SUBSPACES GENERICITY CHAPTER 6 HIGHER DIMENSIONAL LINEAR SYSTEMS DISTINCT EIGENVALUES HARMONIC OSCILLATORS THE EXPONENTIAL OF A MATRIX NONAUTONOMOUS LINEAR SYSTEMS CHAPTER 7 NONLINEAR SYSTEMS DYNAMICAL SYSTEMS THE EXISTENCE AND UNIQUENESS THEOREM CONTINUOUS DEPENDENCE OF SOLUTIONS THE VARIATIONAL EQUATION EXPLORATION: NUMERICAL METHODS CHAPTER 8 EQUILIBRIA IN NONLINEAR SYSTEMS SOME ILLUSTRATIVE EXAMPLES NONLINEAR SINKS AND SOURCES SADDLES STABILITY BIFURCATIONS EXPLORATION: COMPLEX VECTOR FIELDS CONTENTS CHAPTER 9 GLOBAL NONLINEAR TECHNIQUES NULLCLINES STABILITY OF EQUILIBRIA GRADIENT SYSTEMS HAMILTONIAN SYSTEMS EXPLORATION: THE PENDULUM WITH CONSTANT FORCING CHAPTER 10 CLOSED ORBITS AND LIMIT SETS LIMIT SETS LOCAL SECTIONS AND FLOW BOXES THE POINCARÉ MAP MONOTONE SEQUENCES IN PLANAR DYNAMICAL SYSTEMS THE POINCARÉ-BENDIXSON THEOREM APPLICATIONS OF POINCARÉ-BENDIXSON EXPLORATION: CHEMICAL REACTIONS THAT OSCILLATE CHAPTER 11 APPLICATIONS IN BIOLOGY INFECTIOUS DISEASES PREDATOR/PREY SYSTEMS COMPETITIVE SPECIES EXPLORATION: COMPETITION AND HARVESTING CHAPTER 12 APPLICATIONS IN CIRCUIT THEORY AN RLC CIRCUIT THE LIENARD EQUATION THE VAN DER POL EQUATION A HOPF BIFURCATION EXPLORATION: NEURODYNAMICS CHAPTER 13 APPLICATIONS IN MECHANICS NEWTON’S SECOND LAW CONSERVATIVE SYSTEMS CENTRAL FORCE FIELDS THE NEWTONIAN CENTRAL FORCE SYSTEM KEPLER’S FIRST LAW THE TWO-BODY PROBLEM BLOWING UP THE SINGULARITY EXPLORATION: OTHER CENTRAL FORCE PROBLEMS EXPLORATION: CLASSICAL LIMITS OF QUANTUM MECHANICAL SYSTEMS CHAPTER 14 THE LORENZ SYSTEM INTRODUCTION TO THE LORENZ SYSTEM ELEMENTARY PROPERTIES OF THE LORENZ SYSTEM THE LORENZ ATTRACTOR A MODEL FOR THE LORENZ ATTRACTOR THE CHAOTIC ATTRACTOR EXPLORATION: THE RÖSSLER ATTRACTOR CHAPTER 15 DISCRETE DYNAMICAL SYSTEMS INTRODUCTION TO DISCRETE DYNAMICAL SYSTEMS THE DISCRETE LOGISTIC MODEL CHAOS SYMBOLIC DYNAMICS THE SHIFT MAP THE CANTOR MIDDLE-THIRDS SET EXPLORATION: CUBIC CHAOS EXPLORATION: THE ORBIT DIAGRAM CHAPTER 16 HOMOCLINIC PHENOMENA THE SHIL’NIKOV SYSTEM THE HORSESHOE MAP THE DOUBLE SCROLL ATTRACTOR HOMOCLINIC BIFURCATIONS EXPLORATION: THE CHUA CIRCUIT CHAPTER 17 EXISTENCE AND UNIQUENESS REVISITED PROOF OF EXISTENCE AND UNIQUENESS CONTINUOUS DEPENDENCE ON INITIAL CONDITIONS EXTENDING SOLUTIONS NONAUTONOMOUS SYSTEMS DIFFERENTIABILITY OF THE FLOW