QUALIFYING EXAM –ODE, Fall 2007
Name:
Choose six problems
1. A solution x(t), 0 ≤t < ∞, is called recurrent if x(tn)→x(0) for some sequence
tn→ ∞. Prove that a gradient dynamical system has no nonconstant recurrent solution.
2. Sketch the phase portraits of
(i) the pendulum without friction
u00 + sinu= 0,
(ii) the pendulum with friction
u00 −u0+ sinu= 0,
and point out the stable and unstable equilibrium points.
3. Let Abe a n×nmatrix and h: IR →IR be a bounded continuous function. Assume
that the real parts of all eigenvalues of Aare negative. Prove that x0=Ax +h(t) has a
unique bounded solution x(t) defined on (−∞,∞), which is given by
x(t) = Zt
−∞
eA(t−s)h(s)ds
4. Prove the following instability theorem: Let Vbe a C1real-valued function defined on
a neighborhood Uof an equilibrium point ¯xof x0=f(x). Suppose V(¯x) = 0 and ˙
V(x)>0
in U−¯x. If there is a convergent sequence xn→¯xsuch that V(xn)>0, then ¯xis unstable.
5. Let fk: IRn→IRn,k= 1,· · · ,satisfy
|fk(x)−fk(y)| ≤ K|x−y|,for x, y ∈IRn
and
lim
k→∞
fk(x) = f(x),uniformly.
Let xk(t) be the solution of ˙x=fk(x), x(0) = x0and x(t) be the solution of ˙x=
f(x), x(0) = x0. Prove
lim
k→∞
xk(t) = x(t),uniformly.
6. Assume that F: IR →IR is C1and odd, F(0) = 0, that F(x)→ ∞ as x→ ∞ and that
Fsatisfies F(x)<0 on 0 <x<aand F(x)>0 on x>a, where ais a positive constant.
Show that
y00 +F(y0) + y= 0
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