JAN 2013- PH.D. PRELIMINARY EXAMINATION
ORDINARY DIFFERENTIAL EQUATIONS
Instructions: Give solutions to the following problems and show all your work.
(1) Consider the system
x′=x2+y
y′=xy +a
where ais a parameter.
(a) Find all equilibrium points and compute the linearized equation at
each.
(b) Describe the behavior of the linearized system at each equilibrium
point.
(2) Let A(t) be a continuous family of n×nmatrices and let P(t) be the matrix
solution to the initial value problem d
dt P=A(t)P, P (0) = P0. Show that
detP(t) = (detP0)exp (∫t
0
TrA(s)ds).
(3) A solution x(t) of a system of differential equations is called recurrent if
x(tn)→x(0) for some sequence tn→ ∞. Prove that a gradient dynamical
system has no nonconstant recurrent solutions.
(4) Let γbe a closed orbit for a planar system, and let Ube the bounded, open
region inside γ. Show that γis not simultaneously the omega and alpha
limit set of points of U. Use this fact and the Poincar´e-Bendixson theorem
to prove that Ucontains an equilibrium that is not a saddle.
(5) Let Ube an open set of Rncontaining x0. Suppose that f:U→Rnis
C1and f(x0) = 0. Suppose further that there is a C1function V:U→R
satisfying V(x0) = 0 and V(x)>0 if x=x0. Prove
(a) if ˙
V(x) = gradV(x)·f(x)≤0 for all x∈U, then x0is stable.
(b) if ˙
V(x)<0 for all x∈U− {x0}, then x0is asymptotically stable.
(6) Consider the following differential equation
x′=Ax +f(x)
where x∈R2,Ais a 2 ×2 matrix, and 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 stable manifold Ws(0) is given by the graph of a Lipschitz
continuous function.
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