Math 6410 • ODE

Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname{Re}\lambda<0\) for all eigenvalues of \(A\). Consider the
system

\[
\dot{x}=Ax+h(t) \tag{3}
\]

where \(h(t)\in\mathbb{R}^d\) is a \(T\)-periodic continuous function.
\begin{enumerate}

\item
Define the Poincare Map in this context and give an explicit formula for the Poincare
Map in terms of \(A\) and \(h\)


\item
Show that there is a unique \(T\)-periodic solution of (3). [Hint: there is again an explicit
equation in terms of \(A\) and \(h\).]


\item
What stability property does the periodic orbit have? Provide a justification of your
answer with explicit calculation/proof.


\item
How could the requirements stipulated on the eigenvalues of \(A\) be generalized to still
maintain the existence and uniqueness results for the periodic orbit from (b)?

\end{enumerate}