Math 6410 • ODE

Let \(A\) be a \(d\times d\) matrix with all eigenvalues having negative
real part and consider the system

\begin{equation}
\tag{1}
\dot{x}=Ax+h(t)
\end{equation}

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

\item
Define the Poincar\'e map in this context and give an explicit formula
in terms of \(h\) and \(A\).


\item
Show that there is a unique \(T\)-periodic solution \(x(t)\) of equation (1).
Hint: there is again an explicit formula in terms of \(A\) and \(h\)


\item
What stability property does this periodic orbit have? (You do need
to justify your answer with explicit calculation/proof)


\item
How could the requirements stipulated on the eigenvalues of \(A\) be
changed and still maintain the existence and uniqueness result for
the periodic orbit from (b).

\end{enumerate}