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Theory of Ordinary Differential Equations

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Theory of Ordinary Differential Equations theoryofodes July 4, 2007 13:20 Page i ¨ © Theory of Ordinary Differential Equations ¨ ¨ © © ¨ © theoryofodes July 4, 2007 13:20 Page ii ¨ © ¨ ¨ © © ¨ © theoryofodes July 4, 2007 13:20 Page i ¨ © Theory of Ordinary Differential Equations Christopher P. Grant Brigham Young University ¨ ¨ © © ¨ © theoryofodes July 4, 2007 13:20 Page ii ¨ © ¨ ¨ © © ¨ © theoryofodes July 4, 2007 13:20 Page i ¨ © Contents Contents i 1 Fundamental Theory 1 1.1 ODEsandDynamicalSystems . 1 1.2 ExistenceofSolutions . ... ... ... ... .. ... ... 6 1.3 UniquenessofSolutions . 10 1.4 Picard-LindelöfTheorem. 14 ¨ ¨ © 1.5 IntervalsofExistence. 16 © 1.6 DependenceonParameters . 20 2 Linear Systems 27 2.1 ConstantCoefficientLinearEquations . 27 2.2 UnderstandingtheMatrixExponential . 30 2.3 GeneralizedEigenspaceDecomposition . 33 2.4 OperatorsonGeneralizedEigenspaces . 37 2.5 RealCanonicalForm .. ... ... ... ... .. ... ... 41 2.6 SolvingLinearSystems. 43 2.7 QualitativeBehaviorofLinearSystems . 50 2.8 ExponentialDecay. ... ... ... ... ... .. ... ... 54 2.9 NonautonomousLinearSystems . 56 2.10NearlyAutonomousLinearSystems. 61 2.11PeriodicLinearSystems . 65 3 Topological Dynamics 71 3.1 InvariantSetsandLimitSets . 71 3.2 RegularandSingularPoints. 75 i ¨ © theoryofodes July 4, 2007 13:20 Page ii ¨ © Contents 3.3 DefinitionsofStability . 80 3.4 PrincipleofLinearizedStability . 85 3.5 Lyapunov’sDirectMethod. 90 3.6 LaSalle’sInvariancePrinciple . 94 4 Conjugacies 101 4.1 Hartman-GrobmanTheorem: Part1 . 101 4.2 Hartman-GrobmanTheorem: Part2 . 103 4.3 Hartman-GrobmanTheorem: Part3 . 105 4.4 Hartman-GrobmanTheorem: Part4 . 109 4.5 Hartman-GrobmanTheorem: Part5 . 111 4.6 ConstructingConjugacies . 115 4.7 SmoothConjugacies . 119 5 Invariant Manifolds 125 5.1 StableManifoldTheorem:Part1 . 125 5.2 StableManifoldTheorem:Part2 . 129 5.3 StableManifoldTheorem:Part3 . 132 ¨ 5.4 StableManifoldTheorem:Part4 . 134 ¨ © 5.5 StableManifoldTheorem:Part5 . 138 © 5.6 StableManifoldTheorem:Part6 . 142 5.7 CenterManifolds .. ... ... ... ... ... ... ... .. 145 5.8 ComputingandUsingCenterManifolds . 148 6 Periodic Orbits 153 6.1 Poincaré-BendixsonTheorem . 153 6.2 Lienard’sEquation . 157 6.3 Lienard’sTheorem . 161 ii ¨ © theoryofodes July 4, 2007 13:20 Page 1 ¨ © 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 ¨ ∈ ¨ © n 1 copies © + F : dom(F) R E E Rj. (1.1) ⊆ × ×···× → z }| { 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) · · · = If R is an interval, then x : E is said to be a solution of (1.2) on I ⊆ I → I if x has derivatives up to order n at every t , and those derivatives ∈ I satisfy (1.2). Often, we will use notation that suppresses the depen- dence 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 (ℓ) on initial conditions which prescribe x (t0) for some fixed t0 R ∈ (called the initial time) and some choices of ℓ 0, 1,...,n . Some 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 ¨ © theoryofodes July 4, 2007 13:20 Page 2 ¨ © 1. Fundamental Theory First-order Equations Every ODE can be transformed into an equivalent first-order equation. In particular, given x : E, suppose we define I → y1 : x = y2 : x˙ = y3 : x¨ = . (n 1) yn : x − , = n and let y : E be defined by y (y1,...,yn). For i 1, 2,...,n 1, I → = = − define n n Gi : R E E E × × → by G1(t,u,p) : p1 u2 = − ¨ G2(t,u,p) : p2 u3 ¨ © = − © G3(t,u,p) : p3 u4 = − . Gn 1(t,u,p) : pn 1 un, − = − − n n j and, given F as in (1.1), define Gn : dom(Gn) R E E R by ⊆ × × → Gn(t,u,p) : F(t,u1,...,un, pn), = where n n dom(Gn) (t,u,p) R E E (t,u1,...,un, pn) dom(F) . = ∈ × × ∈ n n n 1 j Letting G : dom(Gn) R E E E R be defined by ⊆ × × → − × G1 G2 G3 G : , = . . . G n 2 ¨ © theoryofodes July 4, 2007 13:20 Page 3 ¨ © ODEs and Dynamical Systems we see that x satisfies (1.2) if and only if y satisfies G(t, y(t), y(t))˙ = 0. Equations Resolved with Respect to the Derivative Consider the first-order initial-value problem (or IVP) F(t,x, x)˙ 0 = x(t0) x0 (1.3) = x(t˙ 0) p0, = n n n where F : dom(F) R R R R , and x0, p0 are given ele- n ⊆ × × → ments of R satisfying F(t0, x0, p0) 0. The Implicit Function The- = orem says that typically the solutions (t,x,p) of the (algebraic) equa- tion F(t,x,p) 0 near (t0, x0, p0) form an (n 1)-dimensional surface = + that can be parametrized by (t, x). In other words, locally the equation F(t,x,p) 0 is equivalent to an equation of the form p f (t, x) for = n n = some f : dom(f ) R R R with (t0, x0) in the interior of dom(f ). ⊆ × → Using this f , (1.3) is locally equivalent to the IVP ¨ ¨ © x˙ f (t, x) © = x(t0) x0. = Autonomous Equations Let f : dom(f ) R Rn Rn. The ODE ⊆ × → x˙ f (t, x) (1.4) = is autonomous if f doesn’t really depend on t, i.e., if dom(f ) R = × for some Rn and there is a function g : Rn such that f (t, u) ⊆ → Ω= g(u) for every t R and every u . Ω ∈ ∈ Ω Every nonautonomous ODE is actually equivalent to an autonomous Ω n n 1 ODE. To see why this is so, given x : R R , define y : R R + by → →n n y(t) (t, x1(t), . , xn(t)), and given f : dom(f ) R R R , = ⊆ × → define a new function f˜ : dom(f˜ ) Rn 1 Rn 1 by ⊆ + → + 1 f1(p1, (p2,...,pn 1)) f˜ (p) . + , = . . f (p , (p ,...,p )) n 1 2 n 1 + 3 ¨ © theoryofodes July 4, 2007 13:20 Page 4 ¨ © 1. Fundamental Theory T where f (f1,...,fn) and = ˜ n 1 dom(f ) p R + (p1, (p2,...,pn 1)) dom(f ) . = ∈ + ∈ Then x satisfies (1.4) if and only if y satisfies y˙ f˜ (y). = Because of the discussion above, we will focus our study on first- order autonomous ODEs that are resolved with respect to the deriva- tive. 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 abstractness and specificity is one that you will encounter (and have probably already encountered) in other areas of mathematics. Sometimes, when transforming the equa- tion 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 the point x0 we can guarantee that the autonomous IVP ¨ © ∈ Ω © Ω x˙ f (x) = (1.5) x(0) x0 = has a solution defined on some interval containing 0 in its interior, I and this solution will be unique (up to restriction or extension). Fur- thermore, it is possible to “splice” together solutions of (1.5) in a nat- ural way, and, in fact, get solutions to IVPs with different initial times. These considerations lead us to study a structure known as a dynami- cal system. Given Rn, a continuous dynamical system (or a flow) on is a ⊆ function ϕ : R satisfying: Ω × → Ω 1. ϕ(0, x) xΩfor everyΩ x ; = ∈ 2. ϕ(s, ϕ(t, x)) ϕ(s t,x)Ωfor every x and every s,t R; = + ∈ ∈ 3. ϕ is continuous. Ω If f and are sufficiently “nice” we will be able to define a function ϕ : R by letting ϕ( , x0) be the unique solution of (1.5), and × → · 4 Ω Ω Ω ¨ © theoryofodes July 4, 2007 13:20 Page 5 ¨ © ODEs and Dynamical Systems this definition will make ϕ a dynamical system. Conversely, any con- tinuous dynamical system ϕ(t, x) that is differentiable with respect to t is generated by an IVP. Exercise 1 Suppose that: Rn; • ⊆ Ωϕ : R is a continuous dynamical system; • × → ∂ϕ(t,x) ΩexistsΩ for every t R and every x ; • ∂t ∈ ∈ x0 is given; Ω • ∈ y : RΩ is defined by y(t) : ϕ(t, x0); • → = ∂ϕ(s,p) f : RΩn is defined by f (p) : . • → = ∂s s 0 = Show thatΩy solves the IVP ¨ ¨ © y˙ f (y) © = y(0) x0. = In these notes we will also discuss discrete dynamical systems. Giv- en Rn, a discrete dynamical system on is a function ϕ : Z ⊆ × → satisfying: Ω Ω Ω Ω 1. ϕ(0, x) x for every x ; = ∈ 2. ϕ(ℓ, ϕ(m, x)) ϕ(ℓ m,x)Ω for every x and every ℓ,m Z; = + ∈ ∈ 3. ϕ is continuous. Ω There is a one-to-one correspondence between discrete dynamical sys- tems ϕ and homeomorphisms (continuous functions with continuous inverses) F : , this correspondence being given by ϕ(1, ) F. If → · = we relax the requirement of invertibility and take a (possibly noninvert- Ω Ω ible) continuous function F : and define ϕ : 0, 1,... → { }× → 5 Ω Ω Ω Ω ¨ © theoryofodes July 4, 2007 13:20 Page 6 ¨ © 1. Fundamental Theory by n copies ϕ(n, x) F(F( (F(x)) )), = · · · · · · z }| { then ϕ will almost meet the requirements to be a dynamical system, the only exception being that property 2, known as the group property may fail because ϕ(n, x) is not even defined for n < 0.
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