MAY 2011 PH.D. PRELIMINARY EXAMINATION
ORDINARY DIFFERENTIAL EQUATIONS
Instructions: Give solutions to exactly 6 of the following 7 problems.
(1) Prove that the origin is an uniformly asymptotically stable solution for the equation
dx
dt =y,
dy
dt =−x−y3(1 −5 sin t).
(2) Let A(t) be a continuous n×nmatrix with period T. Suppose that
x0=A(t)x
has no periodic solutions of period Tand there is no non-zero constant solutions. Let g(t)∈Rnbe
any continuous vector function with period T. Prove
x0=A(t)x+g(t)
has a periodic (possibly constant) solution of period T.
(3) Use the Poincare-Bendixson Theorem to prove the Brouwer fixed point theorem for a C1mapping.
(4) Let Abe a 2 ×2 matrix with one positive eigenvalue and one negative eigenvalue and g(t) be a
bounded continuous function from (−∞,∞) to R2. Prove that
x0=Ax +g(t)
has an unique bounded solution over (−∞,∞).
(5) Prove that
x0=−3x+1
1 + |x|4sin tcos t
has 2πperiodic solution.
(6) Consider x0=f(x) and its perturb ed equation x0=f(x) + h(t, x) where x∈Rn,fand hare
C2functions, and h(t, x) is T−periodic in t, 0 < is a parameter. Prove that if x0=f(x) has
a hyperbolic equilibrium point p∗, then the perturbed equation x0=f(x) + h(t, x) has a unique
periodic solution p(t, ) such that
p(t, )−p∗=O().
(7) Consider the following differential equation
x0=Ax +f(x)
where ∈x∈R2and f:R2×R2is globally Lipschitz continuous and f(0) = 0. Assume that Ahas a
positive and a negative eigenvalues. Prove that there exists a constant δ > 0 such that if Lip(f)< δ,
then the global unstable manifold Wu(0) is given by the graph of a Lipschitz continuous function.
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