Math 519–About the 2nd midterm

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Sample questions about the proofs

1. State the uniqueness theorem for ordinary differential equations of the form $\dot x = f(t, x)$.
2. State the definition for the Poincaré map of a differential equation $\dot x = f(t, x)$, where $f$ satisfies $f(t+T, x) = f(t, x)$ for all $t,x$, and where the uniqueness theorem applies to $\dot x = f(t, x)$.
3. Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$. Assume that $\bar x(t)$ is a given solution with $\bar x(0)=a$.
   1. What is $\phi(a)$?
   2. Explain which linear differential equation you have to solve to compute $\phi'(a)$. Provide details.
4. Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$. State and prove the theorem on monotonicity of the Poincaré map
5. Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$. Suppose $f(t+T, x) = f(t, x)$ for all $t,x$, and suppose that $\bar x(t)$ is a periodic solution with period $3T$. Prove that $\bar x(t)$ is actually periodic with period $T$.
6. Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$ with $f(t+T, x) = f(t, x)$ for all $t,x$. Suppose that we are given a bounded solution $\bar x(t)$, i.e. a solution for which there is a number $M$ such that $|\bar x(t)|\le M$ for all $t\ge 0$.
   1. Show that $\lim_{n\to\infty} \bar x(nT)$ exists
   2. Prove that the differential equation $\dot x = f(t, x)$ has a periodic solution.
7. Show that $x(nT) = \phi\Bigl(\dots \phi\bigl(x(0)\bigr)\dots\Bigr)$ for a solution $x(t)$ of a differential equation $\dot x=f(t, x)$ with $f(t+T,x) = f(t, x)$ (i.e. prove the iteration theorem.)
8. State and prove the Monotone Iterates Theorem.
9. State the definition of asymptotic stability for a periodic solution of a differential equation.
10. State (but do not prove) the criterion involving the Poincaré map for stability or instability of a periodic solution.