Math 6410 • ODE

Consider the scalar differential equation
\begin{equation}
  \ddot{u}-\epsilon u^3+u=0,\qquad 0<\epsilon\ll1.
  \tag{3}
\end{equation}
\begin{enumerate}
\item[(a)] Suppose that (3) has a \(T\)-periodic solution \(u(t;\epsilon)\). Let
\(\mathcal{E}=\frac12(u^2+\dot{u}^2)\). Using the fact that
\(\mathcal{E}(T)=\mathcal{E}(0)\), find the amplitude
\(u(0;\epsilon)=a(\epsilon)\) with \(\dot{u}(0;\epsilon)=0\) and period \(T(\epsilon)\)
for an approximate periodic solution.
\item[(b)] Determine the stability of the periodic solution.
\end{enumerate}