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Ordinary Differential Equations Chee Han Tan Last Modified Math 6410 : Ordinary Differential Equations Chee Han Tan Last modified : January 13, 2018 2 Contents Preface 5 1 Initial Value Problems7 1.1 Introduction......................................7 1.2 Planar Dynamics...................................9 1.3 Existence and Uniqueness..............................9 1.3.1 Picard's Method of Successive Approximation............... 10 1.3.2 Dependence on Initial Conditions...................... 13 1.4 Manifolds....................................... 15 1.5 Limit Cycle and Poincar´e-BendixsonTheorem................... 15 1.5.1 Orbits, Invariant Sets and !-Limit Sets................... 16 1.5.2 Local Transversals.............................. 18 1.5.3 Poincar´e-Bendixson Theorem........................ 18 1.6 Problems........................................ 21 2 Linear Systems and Stability of Nonlinear Systems 27 2.1 Autonomous Linear Systems............................. 27 2.1.1 Matrix Exponential.............................. 27 2.1.2 Normal Forms................................. 28 2.2 Non-Autonomous Linear Systems.......................... 31 2.2.1 Homogeneous Equation............................ 31 2.2.2 Inhomogeneous equation............................ 33 2.2.3 Stability and Bounded Sets......................... 35 2.2.4 Equations With Coefficients That Have A Limit.............. 37 2.3 Floquet Theory.................................... 39 2.3.1 Floquet Multipliers.............................. 39 2.3.2 Stability of Limit Cycles........................... 42 2.3.3 Mathieu Equation............................... 45 2.3.4 Transition Curves............................... 47 2.3.5 Perturbation Analysis of Transition Curves................. 48 2.4 Stability of Periodic Solutions............................ 51 2.5 Stable Manifold Theorem............................... 53 2.6 Centre Manifolds................................... 57 2.7 Problems........................................ 60 3 4 Contents 3 Perturbation Theory 63 3.1 Basics of Perturbation Theory............................ 63 3.1.1 Asymptotic Expansion............................ 63 3.1.2 Naive Expansions............................... 65 3.2 Method of Multiple Scales.............................. 66 3.3 Averaging Theorem: Periodic Case......................... 68 3.4 Phase Oscillators and Isochrones.......................... 68 3.5 Problems........................................ 69 4 Boundary Value Problems 71 4.1 Compact Symmetric Linear Operators....................... 71 4.2 Linear Differential Operators............................. 76 4.2.1 Formal Adjoint................................ 77 4.2.2 A Simple Eigenvalue Problem........................ 78 4.2.3 Adjoint Boundary Conditions........................ 79 4.2.4 Self-Adjoint Boundary Conditions...................... 80 4.3 Eigenvalue Problems and Spectral Theory..................... 81 4.3.1 Discrete Spectrum.............................. 82 4.3.2 Continuous Spectrum............................. 82 4.3.3 Mixed Spectrum............................... 83 4.3.4 Rayleigh-Ritz Variational Principle..................... 84 4.4 Distribution...................................... 89 4.4.1 Distributions and Test Functions...................... 90 4.4.2 Weak Derivatives............................... 91 4.5 Green's Function................................... 93 4.5.1 Fredholm Alternative............................. 93 4.5.2 Green's Functions for Homogeneous Boundary Conditions........ 94 4.5.3 Modified Green's Function.......................... 97 4.5.4 Eigenfunction Expansions.......................... 98 4.5.5 Inhomogeneous Boundary Conditions.................... 99 4.6 Problems........................................ 101 Preface These notes are largely based on Math 6410: Ordinary Differential Equations course, taught by Paul Bressloff in Fall 2015, at the University of Utah. Additional examples or remarks or results from other sources are added as we see fit, mainly to facilitate our understanding. These notes are by no means accurate or applicable, and any mistakes here are of course our own. Please report any typographical errors or mathematical fallacy to me by email [email protected]. 5 6 Chapter 1 Initial Value Problems 1.1 Introduction We begin by studying the initial value problem (IVP) ( x_ = f(x); x 2 U; f : U −! n R (IVP) T x(0) = x0 2 U; x = (x1; : : : ; xn) where U is an open subset of Rn. Unless stated otherwise, f will be as smooth as we need them to be. First order ODEs are of fundamental importance, due to the fact that any nth order ODE can be written as a system of n first order ODEs. Example 1.1.1 (Newtonian Dynamics). Consider mx¨ = −γx_ + F (x), where −γx_ is the damping term and F (x) some external force. This can be written as a system of first order ODEs: 8 < x_ = y = f1(x; y); γ F (x) : y_ = − y + = f2(x; y): m m • Observe that we have a conservative system if γ = 0. Indeed, let V (x) = − R x F (s) ds be the potential function. Multiplying the ODE byx _ yields dV d 1 d mx_x¨ = − x_ =) mx_ 2 = − V (x(t)): dx dt 2 dt 1 Thus, we obtain the well-known conservation of energy: mx_ 2 + V (x) = E. 2 • Let p = mx_ be the momentum, and define the Hamiltonian as 1 H = H(x; p) = p2 + V (x): 2m We have the following Hamilton's equations: @H @H x_ = ; p_ = − : @p @x 7 8 1.1. Introduction An important property of Hamiltonian is they are conserved over time, i.e. dH @H @H = x_ + p_ = 0: dt @x @p Example 1.1.2 (Classical Predator-Prey Model). Let N(t) be the amount of bacteria, and assume that N is large enough so that we can treat it as a continuous variable. One simple model is the following: N _ N = NK 1 − − d; N(0) = N0: Nmax Non-dimensionalising the system with the dimensionless parameters N d x = ; τ = Kt; D = Nmax KNmax yields the dimensionless equationx _ = (1 − x)x − D. This is a bifurcation problem, with bi- furcation paramter D. If 0 < D < 1=4, then there eixsts 2 fixed points; otherwise there is no fixed points. From a geometric perspective, it seems rather natural to think of solution to (IVP) as a function of initial value x(0) = x0 and time t. This motivates the definition of a flow. n 1 Definition 1.1.3. Let U ⊂ R and f 2 C (U). Given x0 2 U, let φt(x0) = φ(x0; t) be the solution of (IVP) with t 2 I(x0), where I(x0) is the maximal time interval over which a unique solution exists. Such set of mappings φt is called the flow of the vector field f. • For a fixed x0 2 U, φ(x0; ·): I(x0) −! U defines a curve or trajectory, Γ(x0) of the system through x0. Specifically, n o Γ(x0) = x 2 U : x = φt(x0); t 2 I(x0) : • If x0 varies over a set M ⊂ U, then the flow φt : M −! U defines the motion of a cloud of points in M. Theorem 1.1.4 (Time-shift invariance). For any x0 2 U, if t 2 I(x0) and s 2 I(φt(x0)), s > 0, then s + t 2 I(x0) and φs+t(x0) can be expressed as φs+t(x0) = φs(φt(x0)). Proof. Let I(x0) = (α; β) be the maximal time interval over which there exists a unique solution. Consider the function x:(α; s + t) −! U defined by ( φ(r; x ) if r 2 (α; t]; x(r) = 0 φ(r − t; φt(x0)) if r 2 [t; s + t]: We see that x(r) is a solution of the IVP on (α; s + t). Hence, s + t 2 I(x0) and uniqueness of solution gives φs+t(x0) = x(s + t) = φ(s; φt(x0)) = φs(φt(x0)): Initial Value Problems 9 Remark 1.1.5. The result is trivial if s = 0. The result holds for s < 0 by evolving backwards in time. A corollary of the above theorem is that φ−t(φt(x0)) = φt(φ−t(x0)) = x0, i.e. we have an abelian group structure here! Remark 1.1.6. There exists ODEs in which solution does not exist globally, even if f 2 C1(U). 1 Consider the ODEx _ = f(x) = , where f 2 C1(U) with U = fx 2 : x > 0g. For an x R p 2 initial condition x(0) = x0 2 U, a solution to this IVP is given by φt(x0) = 2t + x0, with −x2 I(x ) = 0 ; 1 . 0 2 1.2 Planar Dynamics Consider the general autonomous second order system ( x_ = X(x; y) y_ = Y (x; y) Assuming existence and uniqueness, then paths only cross at fixed points. 1.3 Existence and Uniqueness In this section, we focus on establishing sufficient conditions such that there exists a unique solution to (IVP). Not suprisingly, regularity of f : U −! Rn plays an important role, as we shall see below. (a) Continuity of f is not sufficient to guarantee uniqueness of a solution. • Considerx _ = 3x2=3, x(0) = 0. One solution is the trivial solution x(t) ≡ 0 for all t ≥ 0. Another solution is obtained by using separation of variable: Z x 1 1 3 2=3 ds = t =) x(t) = t : 3 0 s • Observe that f is continuous but not differentiable at the origin. (b) A solution can become unbounded at some finite time t = T , i.e. finite-time blow up. 1 • Considerx _ = x2, x(0) = 1. We have two branches of solution, x(t) = , depending 1 − t on the initial condition. • In this case, our solution is only defined on t 2 (−∞; 1), and lim x(t) = 1. t!1− • The other branch is defined on t 2 (1; 1), but is not reached by the given initial condition. Definition 1.3.1. 10 1.3. Existence and Uniqueness n n n (a) We say that f : R −! R is differentiable at x0 2 R if there exists a linear transfor- n mation Df(x0) 2 L(R ), called the n × n Jacobian matrix such that jf(x + h) − f(x ) − Df(x )hj lim 0 0 0 = 0: jhj!0 jhj (b) Let U be an open subset of Rn. A function f : U −! Rn is said to satisfy a Lipschitz condition on U if there exists K > 0 such jf(x) − f(y)j ≤ Kjx − yj for all x; y 2 U: n (c) A function f : U −! R is said to be locally Lipschitz on U if for every x0 2 U, there exists an "-neighbourhood N"(x0) ⊂ U and K0 > 0 such that jf(x) − f(y)j ≤ K0jx − yj for all x; y 2 N"(x0): Theorem 1.3.2.
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