Math 6410 • ODE

Consider the differential equation

\[
\ddot{x}+\eta(x)\dot{x}+U'(x)=0,\quad \eta(x)>0.
\]

Suppose that \(U(x) \to \infty\) as \(x \to \pm \infty\). Find a Lyapunov function for this equation and use
this to show that
\begin{enumerate}

\item
there are no periodic solutions.


\item
minima of \(U(x)\) are asymptotically stable steady solutions.


\item
Under what conditions on \(U(x)\) is a minimum of \(U(x)\) exponentially stable?

\end{enumerate}