 Solution

The linear $\cos x$ mode has frequency squared $k^2-1$, so $k_c(0)=1$. To resolve the small frequency near threshold, writeThe leading equationand the initial data give

At $O({\epsilon }^2)$, the equation for ${\theta }_2$ has a forcing whose $\cos x$ component isThe Fredholm solvability condition removes this secular term and yields the slow amplitude equationIt has the conserved energyThe potential has maxima atA periodic orbit launched from $A(0)=1$, $A^{\prime }(0)=0$ exists only when that turning point lies inside the two maxima, namelyAt equality the orbit is the separatrix; below it, the assumed real periodic oscillation is lost. Therefore