Math 6410 • ODE

Let \(A\) be a real \(d\times d\) matrix. Consider the initial value problem

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

(3)

\[
x(0)=x_0
\]


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

\item
[9] Define the Poincare Map in this context and give an explicit formula for the Poincare
Map in terms of \(A\), \(h\) and \(x_0\).


\item
[8] Under what conditions on \(A\) and \(h\) is there is a unique \(T\)-periodic solution of (3)?


\item
[8] Compute the derivative of the Poincare map at the periodic solution. What stability
property do you conclude for the periodic solution?

\end{enumerate}