Orinary Differential Equations: Homework 2 M. Gameiro, J.-P. Lessar, J.D. Mireles James, K. Mischaikow January 30, 2017
2 0.1 Eercises Eercise 0.1.1. Let (X, ) be a metric space. function (in the metric space topology). Prove that the metric is a continuous Eercise 0.1.2. Use the contraction mapping theorem to prove that the function f() = cos() has a positive real fie point. Eercise 0.1.3. Suppose that f : R R is a continuously ifferentiable function (i.e. that f is continuous on R). Suppose that there is an 0 R so that f( 0 ) = 0, an that f ( 0 ) < 1. Prove that f is a contraction mapping in some neighborhoo of 0. Eercise 0.1.4. Suppose that f : R R is a continuously ifferentiable mapping (i.e. that the function f is continuous on R). Assume that there is an 0 R an positive constants Y 0 an Z 0 so that f( 0 ) 0 Y 0, an f ( 0 ) Z 0 < 1. Prove that there eists an ɛ > 0 so that for all [ 0 ɛ, 0 + ɛ]. Prove that if f () 1 Z 0, 2 2Y 0 1 Z 0 < ɛ, then there eists a unique fie point for f in [ 0 ɛ, 0 + ɛ]. Eercise 0.1.5. Use the contraction mapping theorem to prove that the function ( ) 2 + y 2 + y f(, y) = 25 + 3 50 3 2 y 2 y 30 + y4 100 has a fie point near = 1, y = 1. Write a MatLab program which computes the fie point to 8 ecimal places. Eercise 0.1.6. In Chapter 1 you worke with the following ifferential equations: t (t) = k(t) k K (t)2, (Logistic Equation)
0.1. EXERCISES 3 ( ) ( = y cy ω sin() ( ) ( ( µ 1 = 3 3 y ) ) 1 µ w z z = = σ(y ) (ρ z) y y βz w (w 2 +y 2 ) 3/2 z y (w 2 +y 2 ) 3/2 ) (The penulum) (van er Pol oscillator) (Lorenz Equation) (The one boy problem ) In each case, etermine the largest open set U R n on which the vector fiel is locally Lipschitz. Repeat this problem for some of the vector fiels from the scavenger hunt. Eercise 0.1.7. Give an eample of a metric space X an a contraction mapping T : X X so that T oes not have any fie points. Eercise 0.1.8. Consier the metric space X = { R : 1 < }, with the metric inherite from R, an let T : X X be the map Prove that T () = + 1. T () T (y) < y for all, y X with y, an the T has no fie points. Eercise 0.1.9. Consier the ifferential equation f(t) = 3f(t), t with the initial conition (0) = 1. Compute the Piccar iteration for this ifferential equation starting with the initial function 0 (t) = 1. (Compute at least 5 iterates of the operator).
4 Eercise 0.1.10. Consier the ifferential equation ( ) ( ) ( (t) 0 1 (t) = (t) 1 0 y(t) with the initial conition (0) = 1, y(0) = 1. Compute the Piccar iteration for this ifferential equation starting with the initial function ( ) ( ) 0 (t) 1 = y 0 (t) 1 (Compute at least 5 iterates of the operator). Eercise 0.1.11. Consier the ifferential equation f(t) = 2f(t) + f(t)2 t with the initial conition (0) = 1/2. Compute the Piccar iteration for this ifferential equation starting with the initial function 0 (t) = 1/2. (Compute at least 3 iterates of the operator). Eercise 0.1.12 (Eistence an uniqueness for linear ifferential equations). Consier J R an open interval. Suppose that g : J R n an a i,j : J R, 1 i, j n, are continuous functions. Set A(t) = a 1,1 (t)... a 1,n (t)..... a n,1 (t)... a n,n (t) Prove that for any t 0 J the initial value problem has a unique solution : J R n. ). ẋ(t) = A(t) + g(t), (t 0 ) = 0, (1) Hint: Define an integrating factor C(t) by writing own a solution of the equation C (t) = C(t)A(t), that is write own an eplicit formula for such a C(t). Show irectly that C(t) is well efine, continuous, an ifferentiable on J.
0.1. EXERCISES 5 C(t 0 ) = I. C(t) is invertible for every t J. Now multiply both sies of Equation (1) by C(t), apply the prouct rule on the left, integrate both sies from t 0 to t with t J, an solve for (t). This provies an eplicit formula for the solution (t). Show that the formula gives a well efine, continuous, ifferentiable function for each t J. Eercise 0.1.13 (Differentiability of the flow with respect to initial conitions). Let ϕ: R R n R n be the flow generate by ẋ = f(), where f : R n R n is C 1 an boune on R n. Proposition?? guarantees that ϕ(t, ) is continuous with respect to initial conitions. The following approach shows that ϕ is actually ifferentiable with respect to. Observe that ifferentiability of ϕ with respect to is equivalent to the statement that for any t R, 0 R n, there eists a unique n n matri Dϕ(t, 0 ) satisfying ϕ(t, 0 + h) ϕ(t, 0 ) Dϕ(t, 0 )h lim = 0. (2) h 0 h Let 0 R n an γ(t) ϕ(t, 0 ) enote the orbit segment through 0. matri Dϕ(t, 0 ) to be the solution of the first variation equation, Define the t Dϕ(t, 0) = Df(γ(t))Dϕ(t, 0 ), Dϕ(0, 0 ) = I. Use the results of problem 0.1.12 to establish that the matri solving this equation eists assuming only that φ(t, 0 ) γ(t) eists an is continuous. Now show that Dϕ(t, 0 ) satisfies (2).