WINTER 2012 - PH.D. PRELIMINARY EXAMINATION
ORDINARY DIFFERENTIAL EQUATIONS AND
DYNAMICAL SYSTEMS
Instructions: Give solutions to exactly 6 of the following 8 prob-
lems. If you give more than 6 solutions, your grade will be determined
by the first six that appear.
(1) Prove that the planar system
˙x=x+y−x3
˙y=−x+y−y3
has a nonconstant periodic solution.
(2) Find all real numbers asuch that the initial-value problem
¨x=x+ae−t
x(0) = 1
˙x(0) = 0
has a solution x(t) that is bounded as t→ ∞.
(3) Prove that for every ε > 0 there exists (x0, y0)∈R2such that
0< x2
0+y2
0< ε2and the solution (x(t), y(t)) of the initial-value
problem
˙x= (x2+y2)(4y−x2)
˙y= (x2+y2)(x+ 7y3)
x(0) = x0
y(0) = y0
satisfies limt→∞(x(t), y(t)) = (0,0).
(4) Consider the initial-value problem
˙x=xsin(x−t)+1
x(τ) = 0.
It is straightforward to check that x(t) := tis the solution if
τ= 0 and that the solution xwill be different for different initial
times τ. Let’s write the solution xas x(t, τ ) to make explicit
its dependence on τ. Compute xτ(t, 0) (where the subscript τ
stands for partial differentiation with respect to τ).
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