 Solution

Substitution of the weighted scaling and $\tau =\epsilon t$ givesThe derivative of the candidate Hamiltonian isThus $H$ is a conserved quantity at $\epsilon =0$. For $\alpha >0$ put $a=\sqrt{\alpha }$. The conservative equilibrium point $(-a,0)$ is a center with $H=-2a^3/3$, and $(a,0)$ is a saddle equilibrium with saddle-loop energyThe closed component of a level between these energies surrounds the center. At $H=H_s$,is the homoclinic orbit, with left turning point $u=-2a$. The upper branch is traversed to the right and the lower branch to the left. Levels above the barrier have escaping trajectories; the saddle level also has unbounded branches to the right. For $\alpha =0$ the two critical points coalesce; for $\alpha <0$ the potential is monotone and there is no center or saddle loop.