MATH 557 Homework Set #7
Fall 2015
28. Let φ(t) be a real continuous function, with period T , and consider Hill’s equation
x00 + φ(t)x = 0 t ∈ R
Let z1 , z2 be the solutions with initial conditions
z1 (0) = z20 (0) = 1
z2 (0) = z10 (0) = 0
a) Show that the Floquet multipliers (computed with t0 = 0) are the roots of the
quadratic
r2 − Dr + 1
where D = z1 (T ) + z20 (T ). (Hint: make a fundamental matrix from z1 , z2 . Note that D
is the trace of this matrix.)
b) If |D| < 2 show that all solutions of x00 + φ(t)x = 0 are bounded on R, whereas if
|D| > 2 there exists at least one unbounded solution.
29. (Problem 4, page 177 in text) If f : Rn → Rn is locally Lipschitz, and
F (x) =
f (x)
1 + |f (x)|
verify that is also locally Lipschitz.
30. Let A be a 2 × 2 constant matrix with a real eigenvalue λ of algebraic multiplicity
2 and geometric multiplicity 1. Show that the system x0 = Ax has a fundamental set of
the form
x1 (t) = eλt v
x2 (t) = eλt (tv + w)
for some vectors v, w. Clarify exactly how v, w may be determined from A.
31. Classify the critical point at the origin, and sketch a phase portrait for x0 = Ax in
each of the following cases.
1 3
A=
3 1
0 −1
A=
2 0
1 −3
A=
1 1
−1 3
A=
0 −1