Theory of Ordinary Differential Equations Theory of Ordinary Differential Equations CHRISTOPHER P. GRANT Brigham Young University Contents Contents i 1 Fundamental Theory 1 1.1 ODEsandDynamicalSystems . 1 1.2 Existence of Solutions . 6 1.3 Uniqueness of Solutions . 9 1.4 Picard-Lindel¨of Theorem . 13 1.5 Intervals of Existence . 15 1.6 Dependence on Parameters . 18 2 Linear Systems 25 2.1 Constant Coefficient Linear Equations . 25 2.2 Understanding the Matrix Exponential . 27 2.3 Generalized Eigenspace Decomposition . 31 2.4 Operators on Generalized Eigenspaces . 34 2.5 Real Canonical Form . 37 2.6 Solving Linear Systems . 39 2.7 Qualitative Behavior of Linear Systems . 46 2.8 Exponential Decay . 50 2.9 Nonautonomous Linear Systems . 52 2.10 Nearly Autonomous Linear Systems . 56 2.11 Periodic Linear Systems . 59 3 Topological Dynamics 65 3.1 Invariant Sets and Limit Sets . 65 3.2 Regular and Singular Points . 69 3.3 Definitions of Stability . 72 3.4 Principle of Linearized Stability . 77 3.5 Lyapunov’s Direct Method . 82 i CONTENTS 3.6 LaSalle’s Invariance Principle . 85 4 Conjugacies 91 4.1 Hartman-Grobman Theorem: Part 1 . 91 4.2 Hartman-Grobman Theorem: Part 2 . 92 4.3 Hartman-Grobman Theorem: Part 3 . 95 4.4 Hartman-Grobman Theorem: Part 4 . 98 4.5 Hartman-Grobman Theorem: Part 5 . 101 4.6 Constructing Conjugacies . 104 4.7 Smooth Conjugacies . 107 5 Invariant Manifolds 113 5.1 Stable Manifold Theorem: Part 1 . 113 5.2 Stable Manifold Theorem: Part 2 . 116 5.3 Stable Manifold Theorem: Part 3 . 119 5.4 Stable Manifold Theorem: Part 4 . 122 5.5 Stable Manifold Theorem: Part 5 . 125 5.6 Stable Manifold Theorem: Part 6 . 129 5.7 CenterManifolds ..... ...... ..... ...... ... 132 5.8 Computing and Using Center Manifolds . 134 6 Periodic Orbits 139 6.1 Poincar´e-Bendixson Theorem . 139 6.2 Lienard’s Equation . 143 6.3 Lienard’s Theorem . 147 ii 1 Fundamental Theory 1.1 ODEs and Dynamical Systems Ordinary Differential Equations An ordinary differential equation (or ODE) is an equation involving derivatives of an unknown quantity with respect to a single variable. More precisely, suppose j;n N, E is a Euclidean space, and 2 n C 1 copies F dom.F/ R E E Rj : (1.1) W ! ‚ …„ ƒ Then an nth order ordinary differential equation is an equation of the form F.t;x.t/; x.t/; x.t/;x.3/.t/; ;x.n/.t// 0: (1.2) P R D If I R is an interval, then x I E is said to be a solution of (1.2) on I if W ! x has derivatives up to order n at every t I, and those derivatives satisfy (1.2). 2 Often, we will use notation that suppresses the dependence of x on t. Also, there will often be side conditions given that narrow down the set of solutions. In these .`/ notes, we will concentrate on initial conditions which prescribe x .t0/ for some fixed t0 R (called the initial time) and some choices of ` 0;1;:::;n . Some 2 2f g ODE texts examine two-point boundary-value problems, in which the value of a function and its derivatives at two different points are required to satisfy given algebraic equations, but we won’t focus on them in this one. 1 1. FUNDAMENTAL THEORY First-order Equations Every ODE can be transformed into an equivalent first-order equation. In partic- ular, given x I E, suppose we define W ! y1 x WD y2 x WD P y3 x WD R : : .n1/ yn x ; WD n and let y I E be defined by y .y1;:::;yn/. For i 1;2;:::;n 1, W ! D D define n n Gi R E E E W ! by G1.t;u;p/ p1 u2 WD G2.t;u;p/ p2 u3 WD G3.t;u;p/ p3 u4 WD : : Gn1.t;u;p/ pn1 un; WD n n j and, given F as in (1.1), define Gn dom.Gn/ R E E R by W ! Gn.t;u;p/ F.t;u1;:::;un;pn/; WD where n n dom.Gn/ .t;u;p/ R E E .t;u1;:::;un;pn/ dom.F/ : D 2 2 n n n1 j Letting G dom.G˚ n/ R E E Eˇ R be defined by « W ! ˇ G1 G2 0 1 G G3 ; WD B : C B : C B C BG C B nC @ A we see that x satisfies (1.2) if and only if y satisfies G.t;y.t/; y.t// 0. P D 2 ODEs and Dynamical Systems Equations Resolved with Respect to the Derivative Consider the first-order initial-value problem (or IVP) F.t;x; x/ 0 P D x.t0/ x0 (1.3) 8 D <ˆx.t0/ p0; P D n ˆ n n n where F dom.F/ R R : R R , and x0;p0 are given elements of R W ! satisfying F.t0;x0;p0/ 0. The Implicit Function Theorem says that typically D the solutions .t;x;p/ of the (algebraic) equation F.t;x;p/ 0 near .t0;x0;p0/ D form an .n 1/-dimensional surface that can be parametrized by .t;x/. In other C words, locally the equation F.t;x;p/ 0 is equivalent to an equation of the D n n form p f.t;x/ for some f dom.f / R R R with .t0;x0/ in the D W ! interior of dom.f /. Using this f , (1.3) is locally equivalent to the IVP x f.t;x/ P D x.t0/ x0: ( D Autonomous Equations Let f dom.f / R Rn Rn. The ODE W ! x f.t;x/ (1.4) P D is autonomous if f doesn’t really depend on t, i.e., if dom.f / R for some D Rn and there is a function g Rn such that f.t;u/ g.u/ for every t R and every u . W ! D 2 2 Every nonautonomous ODE is actually equivalent to an autonomous ODE. To see why this is so, given x R Rn, define y R RnC1 by y.t/ W ! W n ! n D .t;x1.t/;:::;xn.t//, and given f dom.f / R R R , define a new W ! function f dom.f/ RnC1 RnC1 by Q W Q ! 1 f1.p1;.p2;:::;pnC1// f .p/ 0 : 1 ; Q D : B C Bf .p ;.p ;:::;p //C B n 1 2 nC1 C @ A T where f .f1;:::;fn/ and D nC1 dom.f/ p R .p1;.p2;:::;pnC1// dom.f / : Q D 2 2 ˚ ˇ « Then x satisfies (1.4) if and only ifˇ y satisfies y f .y/. P D Q 3 1. FUNDAMENTAL THEORY Because of the discussion above, we will focus our study on first-order au- tonomous ODEs that are resolved with respect to the derivative. This decision is not completely without loss of generality, because by converting other sorts of ODEs into equivalent ones of this form, we may be neglecting some special structure that might be useful for us to consider. This trade-off between abstract- ness and specificity is one that you will encounter (and have probably already encountered) in other areas of mathematics. Sometimes, when transforming the equation would involve too great a loss of information, we’ll specifically study higher-order and/or nonautonomous equations. Dynamical Systems As we shall see, by placing conditions on the function f Rn Rn and W ! the point x0 we can guarantee that the autonomous IVP 2 x f.x/ P D (1.5) x.0/ x0 ( D has a solution defined on some interval I containing 0 in its interior, and this so- lution will be unique (up to restriction or extension). Furthermore, it is possible to “splice” together solutions of (1.5) in a natural way, and, in fact, get solu- tions to IVPs with different initial times. These considerations lead us to study a structure known as a dynamical system. Given Rn, a continuous dynamical system (or a flow) on is a function ' R satisfying: W ! 1. '.0;x/ x for every x ; D 2 2. '.s;'.t;x// '.s t;x/ for every x and every s; t R; D C 2 2 3. ' is continuous. If f and are sufficiently “nice” we will be able to define a function ' W R by letting '. ;x0/ be the unique solution of (1.5), and this defi- ! nition will make ' a dynamical system. Conversely, any continuous dynamical system '.t;x/ that is differentiable with respect to t is generated by an IVP. Exercise 1 Suppose that: Rn; 4 ODEs and Dynamical Systems ' R is a continuous dynamical system; W ! @'.t;x/ exists for every t R and every x ; @t 2 2 x0 is given; 2 y R is defined by y.t/ '.t;x0/; W ! WD @'.s;p/ f Rn is defined by f.p/ . W ! WD @s ˇsD0 ˇ ˇ Show that y solves the IVP ˇ y f.y/ P D y.0/ x0: ( D In these notes we will also discuss discrete dynamical systems. Given Rn, a discrete dynamical system on is a function ' Z satisfying: W ! 1. '.0;x/ x for every x ; D 2 2. '.`;'.m;x// '.` m;x/ for every x and every `;m Z; D C 2 2 3. ' is continuous. There is a one-to-one correspondence between discrete dynamical systems ' and homeomorphisms (continuous functions with continuous inverses) F , W ! this correspondence being given by '.1; / F .