MATH 557 Homework Set #10
Fall 2015
38. Use Lyapunov theorems to discuss the stability of the zero solution of
y 00 + y 0 + y 3 = 0
Suggestions: Rewrite the equation as a first order system x0 = f (x) and look for a
n
Lyapunov function in the form V (x) = αxm
1 + βx2 for some α, β > 0 and positive even
integers m, n. You should be able to show that 0 is globally asymptotically stable.
39. Let ψ(t) be a bounded solution of x0 = f (x) for t ≥ t0 . Show that ω(ψ) is compact.
Suggestions: Let C + (ξ) = {φ(t, 0, ξ) : t ≥ 0} be the semiorbit starting at ξ, and show
\
ω(ψ) =
C + (ψ(τ ))
τ ≥0
40. Let x0 = f (x) be a gradient system with f (0) = 0. Show that all of the eigenvalues
of ∂f
(0) are real. What does this say about the behavior of solutions of the system near
∂x
the origin?
41. Show that the linear constant coefficient system x0 = Ax (A real) is of gradient type
if and only if A is symmetric. (Suggestion: consider V (x) = xT Ax.)
42. In a domain U ⊂ R3 show that if x0 = f (x) is a gradient system then ∇ × f = 0
in U . The converse is almost, but not quite exactly correct. What is a correct converse
statement? (You may just quote from a textbook here, if you like.) Note that these
results can be applied for U ⊂ R2 also, just by specializing to vector fields with a zero
third component.
43. Let k1 , k2 be constants and consider the system
x01 = k1 x1 − x1 (x21 + x22 ) − x1 x22
x02 = k2 x2 − x2 (x21 + x22 ) − x2 x21
a) Show that the disk B(0, R) is a positive invariant set for sufficiently large R.
b) Is this a gradient system?
c) Find all stable equilibrium points.