 Solution

With primes denoting differentiation in $\tau$, the weighted scaling givesIndeed $\dot{x}={\epsilon }^2{u}^{\prime }$ and $\dot{y}={\epsilon }^3{v}^{\prime }$, so the damping terms carry one extra power of $\epsilon$. ConsequentlyAt $\epsilon =0$ this proves conservation of $H$ directly.

For $\alpha =-a_0^2<0$, the potential is a double well. Its minima are at $u=\pm a_0$, where $H=-a_0^4/4$; these are centre equilibria. The origin is a saddle equilibrium and has $H=0$. Energies $-a_0^4/4<H<0$ give two separate closed contours, one in each well. Energies $H>0$ give a single closed contour surrounding both wells. The intervening figure-eight consists of two homoclinic loops:Their outer turning points are $u=\pm \sqrt{2}{a}_0$. The positive loop, for example, is parametrized by $u=\sqrt{2}{a}_0\mathrm{sech}(a_0\tau )$, $v=u^{\prime }$.