Math 6410 • ODE

\begin{enumerate}

\item
Suppose the function \(f(t,x)\) is periodic in \(t\), \(f(t+1,x)=f(t,x)\). Define the Poincare
map for the differential equation \(\dot{x}=f(t,x)\).



\item
Consider the differential equation

\[
\dot{x}=x(1-x)-h(t)x,
\]


where \(h(t)\ge0\) is periodic, \(h(t+1)=h(t)\). Under what conditions on \(h(t)\) does this
equation have a strictly positive periodic solution?


\item
Suppose \(h(t)=a\sin^2(\pi x)\). What condition on \(a\) guarantees the existence of a positive
periodic solution?


\end{enumerate}