 Solution

Set $\tau =\epsilon t$ and use the stated weighted spatial and parameter scales. Dividing the two equations by ${\epsilon }^3$ and ${\epsilon }^4$ respectively giveswhere prime now means differentiation with respect to $\tau$. The Hamiltonian function satisfiesThus $H=v^2/2+\alpha u-u^3/3$ is conserved for $\epsilon =0$.

For $\alpha >0$, let $s=\sqrt{\alpha }$. The conservative system has a center at $(-s,0)$ with $H_c=-2{\alpha }^{3/2}/3$ and a saddle at $(s,0)$ withClosed contours in the left potential well have $H_c<H<H_h$. At $H_h$ the left-hand separatrix is a homoclinic orbit, sinceIt leaves and returns to $(s,0)$, with its left turning point at $u=-2s$. The right separatrix branches are unbounded. The energy levels above the saddle value are open, as are the separate right-hand components of lower energy levels. For $\alpha \le 0$ there is no nondegenerate potential well and no such saddle loop.