MATHEMATICAL MODELING AND ORDINARY DIFFERENTIAL EQUATIONS I-Liang Chern Department of Mathematics National Taiwan University 2007, 2015 January 6, 2016 2 Contents 1 First-Order Single Differential Equations 1 1.1 What is mathematical modeling? . .1 1.2 Relaxation and Equilibria . .2 1.3 Modeling population dynamics of single species . .6 1.4 Techniques to solve single first-order equations . 17 1.4.1 Linear first-order equation . 17 1.4.2 Separation of variables . 20 1.4.3 Other special classes that are solvable . 21 1.5 Vector Fields and Family of Curves . 22 1.5.1 Vector Fields . 22 1.5.2 Family of curves and Orthogonal trajectories . 27 1.5.3 *Envelop. 29 1.5.4 *An example from thermodynamics – existence of entropy . 31 1.6 Existence and Uniqueness . 33 1.7 *Numerical Methods: First Order Difference Equations . 37 1.7.1 Euler method . 37 1.7.2 First-order difference equation . 38 1.8 Historical Note . 39 2 Second Order Linear Equations 41 2.1 Models for linear oscillators . 41 2.1.1 The spring-mass system . 41 2.1.2 Electric circuit system . 42 2.2 Methods to solve homogeneous equations . 43 2.2.1 Homogeneous equations (complex case) . 43 2.2.2 Homogeneous equation (real case) . 45 2.3 Methods to solve Inhomogeneous equations . 47 2.3.1 Method of underdetermined coefficients . 48 2.3.2 Method of Variation of Constants . 49 2.4 Linear oscillators . 52 2.4.1 Harmonic oscillators . 52 2.4.2 Damping . 53 3 4 CONTENTS 2.4.3 Forcing and Resonance . 54 2.5 2 × 2 linear systems . 57 2.5.1 Independence and Wronskian . 58 2.5.2 Finding the fundamental solutions and Phase Portrait . 60 3 Nonlinear systems in two dimensions 71 3.1 Three kinds of physical models . 71 3.1.1 Lotka-Volterra system . 71 3.1.2 Conservative mechanical system . 72 3.1.3 Dissipative systems . 73 3.2 Autonomous systems . 74 3.3 Equilibria and linearization . 75 3.3.1 Hyperbolic equilibria . 75 3.3.2 The equilibria in the competition model . 79 3.4 Phase plane analysis . 81 3.5 Periodic solutions . 85 3.5.1 Predator-Prey system . 85 3.5.2 van der Pol oscillator . 86 3.6 Heteroclinic and Homoclinic and orbits . 87 4 Linear Systems with Constant Coefficients 93 4.1 Initial value problems for n × n linear systems . 93 4.2 Physical Models . 93 4.2.1 Coupled spring-mass systems . 93 4.2.2 Coupled Circuit Systems . 95 4.3 Linearity and solution space . 97 4.4 Decouping the systems . 99 4.4.1 Linear systems in three dimensions . 99 4.4.2 Rotation in three dimensions . 100 4.4.3 Decoupling the spring-mass systems . 102 4.5 Jordan canonical form . 104 4.5.1 Jordan matrix . 104 4.5.2 Outline of Spectral Theory . 109 4.6 Fundamental Matrices and exp(tA) ......................... 112 4.6.1 Fundamental matrices . 112 4.6.2 Computing exp(A) ............................. 112 4.6.3 Linear Stability Analysis . 117 4.7 Non-homogeneous Linear Systems . 118 5 Methods of Laplace Transforms 121 5.1 Laplace transform . 121 5.1.1 Examples . 122 5.1.2 Properties of Laplace transform . 123 CONTENTS 5 5.2 Laplace transform for differential equations . 125 5.2.1 General linear equations with constant coefficients . 125 5.2.2 Laplace transform applied to differential equations . 126 5.2.3 Generalized functions and Delta function . 129 5.2.4 Green’s function . 131 6 Calculus of Variations 135 6.1 A short story about Calculus of Variations . 135 6.2 Problems from Geometry . 135 6.3 Euler-Lagrange Equation . 136 6.4 Problems from Mechanics . 139 6.5 Method of Lagrange Multiplier . 143 6.6 Examples . 146 6.6.1 The hanging rope problem . 146 6.6.2 Isoperimetric inequality . 147 6.6.3 The Brachistochrone . 149 6.6.4 Phase field model . 151 7 Examples of Nonlinear Systems 155 7.1 Hamiltonian systems . 155 7.1.1 Motivation . 155 7.1.2 Trajectories on Phase Plane . 157 7.1.3 Equilibria of a Hamiltonian system . 159 7.2 Gradient Flows . 161 7.3 Simple pendulum . 164 7.3.1 global structure of phase plane . 165 7.3.2 Period . 169 7.4 Cycloidal Pendulum – Tautochrone Problem . 169 7.4.1 The Tautochrone problem . 169 7.4.2 Construction of a cycloidal pendulum . 171 7.5 The orbits of planets and stars . 173 7.5.1 Centrally directed force and conservation of angular momentum . 173 7.6 General Hamiltonian flows . 180 7.6.1 Noether Theorem . 182 n 8 General Theory for ODE in R 183 8.1 Well-postness . 183 8.1.1 Local existence . 183 8.1.2 Uniqueness . 185 8.1.3 Continuous dependence on initial data . 186 8.1.4 A priori estimate and global existence . 188 8.2 Supplementary . 193 8.2.1 Uniform continuity . 193 CONTENTS 1 8.2.2 C(I) is a normed linear space . 194 8.2.3 C(I) is a complete space . 194 9 Stability Analysis 197 9.1 Introduction . 197 9.2 Damping systems . 198 9.3 Local stability . 201 9.4 Lyapunov function . 203 9.5 Poincare-Bendixson´ Theorem . ..