Math 6410 • ODE

Consider the ODE system

\[
\begin{cases}
\dot{x}=p,\\
\dot{p}=-V'(x),
\end{cases}
\qquad
\text{where } V(x)=\frac14(1-x^2)^2.
\]

\begin{enumerate}

\item
Define Lyapunov stability. Are the stationary points at \((\pm 1,0)\) Lyapunov
stable? Justify.


\item
Find all fixed points, classify their stability (with justification), and
make an accurate drawing of the phase plane including any heteroclinic /
homoclinic orbits.


\item
Define the notion of asymptotically stable stationary point. Are any
of the stationary points asymptotically stable?

\end{enumerate}