FALL 2002
APPLIED MATHEMATICS
PH.D. PRELIMINARY EXAMINATION
Option: Ordinary Differential Equations
Instructions: Give solutions to exactly 5 of the following 8 problems. If you give more
than 5 solutions, your grade will be determined by the first five that appear.
Some Notation
R– the set of real numbers
˙x– the derivative of xwith respect to t
φt– the flow generated by ˙x=f(x), x= (x1, ..., xn)∈Rn
P2– the set of bi-infinite sequences of elements of {0,1}
σ– the shift map on P2
1. (a) State the definition of a section for φt.
(b) State the definition of a Poincar´e section.
(c) State the definition of a Poincar´e map.
(d) Given a section Sfor φ, and a p∈Sand smallest positive T > 0 for which φT(p) = p,
prove that there is an open subset Pof Scontaining pon which the Poincar´e map is smooth.
2. (a) State the definitions of a Lyapunov function and a strict Lyapunov function for
˙x=f(x) at x0.
(b) State Lyapunov’s Stability Theorem.
(c) Using Lyapunov’s Stability Theorem, determine the stability of the rest points of
˙x1=x2−(x1)3
˙x2=−x1−(x2)3
3. Suppose that φis a complete flow.
(a) State the definition of the omega limit set, ω(p), of the orbit t→φt(p) through p.
(b) Prove that if the forward orbit of φthrough p, which is the set {φt(p) : t≥0}, has
compact closure, then ω(p) is nonempty, compact, and connected.
(c) Give an example to show that the compact closure hypothesis is necessary.
4. (a) State the definition of a fundamental matrix solution of ˙x=A(t)x.
(b) Assuming that t→Φ(t) is a fundamental matrix solution of ˙x=A(t)xdefined on an
interval J0of t0, derive the Variation of Constants Formula for the solution of the initial
value problem
˙x=A(t)x+g(x, t), x(t0) = x0.
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