ORDINARY NON-LINEAR DIFFERENTIAL EQUATIONS IN ENGINEERING AND PHYSICAL SCIENCES

By JOHN LEE

CONTENTS INTRODUCTION SOME SIMPLE ODES EXAMPLES DESCRIPTIVE CONCEPTS SOLUTIONS OF ODES EXAMPLES AND DISCUSSION GEOMETRIC INTERPRETATION OF SOLUTIONS SNIPPETS OF SOLUTION TECHNIQUES COMPUTER SOLUTIONS LINEAR EQUATIONS WITH CONSTANT COE?CIENTS FIRST-ORDER LINEAR EQUATIONS SEPARABLE EQUATIONS PHYSICALLY BASED SECOND-ORDER ODES LINEAR SPRING–MASS SYSTEMS NONLINEARITY, PART I: THE RESTORING FORCE ENERGY NONLINEARITY, PART II: THE FRICTIONAL FORCE SYSTEMS OF ODES TOPICS COVERED IN THIS BOOK GENERAL OVERVIEW A CASE STUDY IN THE QUALITATIVE THEORY OF ODES EXERCISES CORE EXERCISES COMPUTATIONAL EXERCISES ANTICIPATORY EXERCISES PHD EXERCISES PEARLS OF WISDOM MISCELLANEOUS THE CONCEPT “GENERIC” A COMPARISON THEOREM LINEAR SYSTEMS WITH CONSTANT COE?CIENTS PREVIEW DE?NITION AND PROPERTIES OF THE MATRIX EXPONENTIAL PRELIMINARIES ABOUT NORMS CONVERGENCE THE MAIN THEOREM AND ITS PROOF CALCULATION OF THE MATRIX EXPONENTIAL THE ROLE OF SIMILARITY TWO PROBLEMATIC CASES USE OF THE JORDAN FORM LARGE-TIME BEHAVIOR OF SOLUTIONS OF LINEAR SYSTEMS THE MAIN RESULTS TESTS FOR EIGENVALUES IN THE LEFT HALF-PLANE CLASSI?CATION AND PHASE PORTRAITS FOR 2 × 2 SYSTEMS SADDLES AND NODES FOCI AND CENTERS ADDITIONAL REMARKS SOLUTION OF INHOMOGENEOUS PROBLEMS PRACTICE WITH LINEAR ALGEBRA PRACTICE SKETCHING PHASE PORTRAITS ALTERNATIVE NORMS NONDI?ERENTIABLE LIMITS AND THE CANTOR SET MORE ON GENERIC BEHAVIOR NONLINEAR SYSTEMS: LOCAL THEORY TWO COUNTEREXAMPLES LOCAL EXISTENCE THEORY STATEMENT OF THE EXISTENCE THEOREM C1 IMPLIES LIPSCHITZ CONTINUITY REFORMULATION OF THE IVP AS AN INTEGRAL EQUATION THE CONTRACTION-MAPPING PRINCIPLE PROOF OF THE EXISTENCE THEOREM AN ILLUSTRATIVE EXAMPLE AND PICARD ITERATION CONCLUDING REMARKS UNIQUENESS THEORY GRONWALL’S LEMMA MORE ON LIPSCHITZ FUNCTIONS THE UNIQUENESS THEOREM GENERALIZATION TO NONAUTONOMOUS SYSTEMS NONLINEAR SYSTEMS LINEAR SYSTEMS LINEAR ODES WITH PERIODIC COE?CIENTS RESONANCE NONLINEAR SYSTEMS: GLOBAL THEORY THE MAXIMAL INTERVAL OF EXISTENCE TWO SU?CIENT CONDITIONS FOR GLOBAL EXISTENCE LINEAR GROWTH OF THE RHS TRAPPING REGIONS LEVEL SETS AND TRAPPING REGIONS INTRODUCTION VIA DU?NG’S EQUATION THE CHEMOSTAT THE TORQUED PENDULUM AND ODES ON MANIFOLDS NULLCLINES AND TRAPPING REGIONS NULLCLINES IN THE CHEMOSTAT AN ACTIVATOR–INHIBITOR SYSTEM SEL’KOV’S MODEL FOR GLYCOLYSIS VAN DER POL’S EQUATION MICHAELIS–MENTEN KINETICS CONTINUITY PROPERTIES OF THE SOLUTION THE MAIN ISSUE: CONTINUOUS DEPENDENCE ON INITIAL CONDITIONS SOME ASSOCIATED FORMALISM CONTINUITY WITH RESPECT TO PARAMETERS DI?ERENTIABILITY PROPERTIES OF THE SOLUTION DEPENDENCE ON INITIAL CONDITIONS THE PERSPECTIVE OF DI?ERENTIABILITY THE ORDER NOTATION PROOF OF THEOREM 4 6 1 TYING UP LOOSE ENDS GENERALIZATIONS APPLYING THE DI?ERENTIATION THEOREMS SOME MOPPING-UP EXERCISES COMPUTING EXERCISE APPENDIX: EULER’S METHOD THEORETICAL BASIS FOR THE APPROXIMATION CONVERGENCE OF THE NUMERICAL SOLUTION NONDIMENSIONALIZATION AND SCALING CLASSES OF ODES IN APPLICATIONS MECHANICAL MODELS ELECTRICAL MODELS “BATHTUB” MODELS SCALING EXAMPLE 0: TWO MODELS FROM ECOLOGY SCALING EXAMPLE 1: A NONLINEAR OSCILLATOR SCALING EXAMPLE 2: SEL’KOV’S MODEL FOR GLYCOLYSIS SCALING EXAMPLE 3: THE CHEMOSTAT ODES MODELING FLOW THROUGH A REACTOR SCALING EXAMPLE 4: AN ACTIVATOR–INHIBITOR SYSTEM SCALING EXAMPLE 5: MICHAELIS–MENTEN KINETICS MAKING SCALING WORK FOR YOU A NOD TO SCIENTI?C LITERACY TRAJECTORIES NEAR EQUILIBRIA STABILITY OF EQUILIBRIA THE MAIN THEOREM AN EASY APPLICATION TERMINOLOGY TO CLASSIFY EQUILIBRIA TERMS RELATED TO THEOREM 6 1 1 OTHER TERMS BASED ON EIGENVALUES SECTION 1 6 REVISITED, PART I TWO-DIMENSIONAL EQUILIBRIA AND SLOPES OF NULLCLINES ACTIVATOR–INHIBITOR SYSTEMS AND THE TURING INSTABILITY EQUILIBRIA OF THE ACTIVATOR–INHIBITOR SYSTEM THE TURING INSTABILITY: DESTABILIZATION BY DI?USION FEEDBACK STABILIZATION OF AN INVERTED PENDULUM LYAPUNOV FUNCTIONS LASALLE’S INVARIANCE PRINCIPLE CONSTRUCTION OF LYAPUNOV FUNCTIONS: AN EXAMPLE STABLE AND UNSTABLE MANIFOLDS A LINEAR EXAMPLE STATEMENT OF THE LOCAL THEOREM A NONLINEAR EXAMPLE GLOBAL BEHAVIOR OF STABLE/UNSTABLE MANIFOLDS DRAWING PHASE PORTRAITS EXAMPLE 1: THE CHEMOSTAT EXAMPLE 2: THE ACTIVATOR–INHIBITOR EXAMPLE 3: SECTION 1 6 REVISITED, PART II USES OF A LYAPUNOV FUNCTION PHASE-PORTRAIT EXERCISES THE HARTMAN–GROBMAN THEOREM AND TOPOLOGICAL CONJUGACY STRUCTURAL STABILITY 6 10 1 REFORMULATION OF THE IVP AS AN INTEGRAL EQUATION 6 10 2 FIXED-POINT ANALYSIS 6 10 3 THE STABLE MANIFOLD 6 10 4 STABLE MANIFOLDS AT NONHYPERBOLIC EQUILIBRIA 6 11 APPENDIX 2: CENTER MANIFOLDS AND NONHYPERBOLICITY 6 11 1 FIRST EXAMPLE 6 11 2 SECOND EXAMPLE OSCILLATIONS IN ODES PERIODIC SOLUTIONS BASIC ISSUES EXAMPLES OF PERIODIC SOLUTIONS A LEISURELY OVERVIEW OF THIS CHAPTER SPECIAL BEHAVIOR IN TWO DIMENSIONS MOSTLY THE POINCAR´E–BENDIXSON THEOREM: MINIMAL VERSION APPLICATIONS OF THE THEOREM LIMIT SETS (IN ANY DIMENSION) THE POINCAR´E–BENDIXSON THEOREM: STRONG VERSION NONEXISTENCE: DULAC’S THEOREM SECTION 1 6 REVISITED, PART III STABILITY OF PERIODIC ORBITS AND THE POINCAR´E MAP AN EIGENVALUE TEST FOR STABILITY BASICS OF THE POINCAR´E MAP DISCRETE-TIME DYNAMICS AND THE PROOF OF THEOREM 7 3 2 STABILITY OF THE LIMIT CYCLE IN THE TORQUED PENDULUM VAN DER POL WITH SMALL ?: WEAKLY NONLINEAR ANALYSIS TWO ILLUSTRATIVE EXAMPLES OF PERTURBATION THEORY APPLICATION TO THE VAN DER POL EQUATION VAN DER POL WITH LARGE ?: SINGULAR PERTURBATION THEORY TWO SOURCES OF GUIDANCE APPROXIMATION OF THE INITIAL DECAY IN (7 58) PHASE-PLANE ANALYSIS OF A RELATED EQUATION STABILITY OF THE VAN DER POL LIMIT CYCLES CASE 1: SMALL ? CASE 2: LARGE ? THE POINCAR´E–LINDSTEDT METHOD CHANGES OF VARIABLES AREA AND DULAC’S THEOREM POINCAR´E-LIKE MAPS IN CONSTRUCTING PERIODIC SOLUTIONS STABLE/UNSTABLE MANIFOLDS IN OTHER CONTEXTS 7 10 APPENDIX: STABILIZING AN INVERTED PENDULUM 7 10 1 A SMIDGEN OF FLOQUET THEORY 7 10 2 SOME STABLE SOLUTIONS OF MATHIEU’S EQUATION 7 10 3 APPLICATION TO THE INVERTED VIBRATED PENDULUM BIFURCATION FROM EQUILIBRIA EXAMPLES OF PITCHFORK BIFURCATION BEAD ON A ROTATING HOOP THE LORENZ EQUATIONS A LATERALLY SUPPORTED INVERTED PENDULUM PERSPECTIVES ON THIS CHAPTER AN OUTLINE OF THE CHAPTER A BIFURCATION THEOREM EXAMPLES OF TRANSCRITICAL BIFURCATION SECTION 1 6 REVISITED: PART IV EXAMPLES OF SADDLE-NODE BIFURCATION THE TORQUED PENDULUM ACTIVATOR–INHIBITOR SYSTEMS THE LYAPUNOV–SCHMIDT REDUCTION BARE BONES OF THE REDUCTION PROOF OF THEOREM 8 2 2 ONE-DIMENSIONAL BIFURCATION PROBLEMS EXCHANGE OF STABILITY SYMMETRY AND THE PITCHFORK BIFURCATION ADDITIONAL PARAMETERS IN BIFURCATION PROBLEMS STEADY-STATE BIFURCATION IN TWO APPLICATIONS THE TWO-CELL TURING INSTABILITY THE CSTR EXAMPLES OF HOPF BIFURCATION AN ACADEMIC EXAMPLE THE “REPRESSILATOR” SECTION 1 6 REVISITED: PART V THE “DENATURED” MORRIS–LECAR SYSTEM THEORETICAL DESCRIPTION OF HOPF BIFURCATION THE ACTIVATOR–INHIBITOR: EXTREME NONGENERIC BEHAVIOR SUB/SUPERCRITICALITY IN TWO DIMENSIONS APPLICATIONS OF BIFURCATION THEORY 8 10 PEARLS OF WISDOM 8 10 1 COMMENTS ON PROVING THE HOPF BIFURCATION THEOREM 8 10 2 HIGH-DIMENSIONAL BIFURCATION: SYMMETRY AND MODE COMPETITION 8 10 3 HOMEOSTASIS, OR “ANTIBIFURCATION” EXAMPLES OF GLOBAL BIFURCATION HOMOCLINIC BIFURCATION THE VAN DER POL EQUATION WITH A NONLINEAR RESTORING FORCE SECTION 1 6 REVISITED: PART VI OTHER EXAMPLES OF HOMOCLINIC AND HETEROCLINIC BIFURCATIONS SADDLE-NODE BIFURCATION OF LIMIT CYCLES THE DENATURED MORRIS–LECAR EQUATION THE OVERDAMPED TORQUED PENDULUM POINCAR´E MAPS AND STABILITY LOSS OF LIMIT CYCLES MUTUAL ANNIHILATION OF TWO LIMIT CYCLES PHASE-LOCKING IN COUPLED OSCILLATORS HOPF-LIKE BIFURCATION TO AN INVARIANT TORUS THE PERIODICALLY FORCED VAN DER POL EQUATION OTHER EXAMPLES OF BIFURCATION TO AN INVARIANT TORUS PERIOD-DOUBLING ACADEMIC EXAMPLE 1: MAPPINGS CARDIAC ALTERNANS ACADEMIC EXAMPLE 2: ODES A PERIODICALLY FORCED PENDULUM R¨OSSLER’S EQUATION OTHER EXAMPLES THE ONSET OF CHAOS IN THE LORENZ EQUATIONS BURSTING IN THE DENATURED MORRIS–LECAR EQUATIONS COMPUTATIONS TO SUPPORT CLAIMS IN THE TEXT BIFURCATION IN A QUADRATIC MAP 9 10 PEARLS OF WISDOM 9 10 1 REMARKS ON HETEROCLINIC ORBITS 9 10 2 BIFURCATION IN FLUID-MECHANICS PROBLEMS 9 10 3 ROUTES TO CHAOS 10 EPILOGUE 10 1 BOUNDARY VALUE PROBLEMS 10 1 1 AN OVERVIEW THROUGH EXAMPLES 10 1 2 EIGENVALUE PROBLEMS 10 2 STOCHASTIC POPULATION MODELS 10 3 NUMERICAL METHODS: TWO SOBERING EXAMPLES 10 3 1 STI?ODES 10 3 2 UNREASONABLE BEHAVIOR OF REASONABLE METHODS 10 4 ODES ON A TORUS: ENTRAINMENT 10 5 DELAY DI?ERENTIAL EQUATIONS 10 6 A PEEK AT CHAOS 10 6 1 A ONE-DIMENSIONAL MAPPING MODEL 10 6 2 THE LORENZ EQUATIONS 10 7 EXERCISES 10 7 1 CORE EXERCISES 10 7 2 PHD EXERCISES 10 8 PEARLS OF WISDOM 10 8 1 THE ELASTICA 10 8 2 A BIT MORE ON CHAOS