WINTER 2011 - PH.D. PRELIMINARY EXAMINATION
ORDINARY DIFFERENTIAL EQUAITONS
(1) Consider the following differential equation in Rn
x0=Ax +B(t)x+g(x, t).
Assume that
(a) Ais a constant n×nmatrix with only eigenvalues with
negative real parts,
(b) B(t) is the n×nmatrix, continuously dependent on tsuch
that ||B(t)|| → 0 as t→ ∞,
(c) g(x, t) is C2and there are constants a > 0 and k > 0 such
that ||g(x, t)|| ≤ k||x||2for all t≥0 and ||x|| < a.
Prove that there are constants C > 1, δ > 0, λ > 0 such that
||x(t, t0, x0)|| ≤ C||x0||e−λ(t−t0), t ≥t0,
whenever ||x0|| ≤ δ/C, where x(t, t0, x0) is the solution of above
equation with x(t0, t0, x0) = x0.
(2) Prove that the following differential equations in R2
x0=x−y−x3, y0=x+y−y3
has a unique globally attracting limit cycle on the punctured
plane R2− {0}.
(3) Consider a smooth differential equation on the plane
x0=g(x, y), y0=h(x, y )
and let f(x, y)=(g(x, y), h(x, y)). If the divergence of fgiven
by
div f(x, y) = ∂
∂x g(x, y) + ∂
∂y h(x, y )
is not identically zero and of fixed sign in a simply connected
region Ω. Prove that the system has no periodic orbits in Ω.
(4) Suppose that Vis a smooth function defined on an open neigh-
borhood Uof an equilibrium point ¯xof the differential equation
x0=f(x), where fis a smooth function from Rninto itself, such
that V(¯x) = 0 and ˙
V(x) = (grad V(x), f (x)) >0 on U− {¯x}.
Assume that for each neighborhood Nof ¯x, there is a point
x∗∈Nsuch that V(x∗)>0. Prove ¯xis unstable.
1
