Homoclinic balance for the weakly perturbed double-well oscillator
= Homoclinic balance for the weakly perturbed double-well oscillator
On either unperturbed homoclinic loop, $H_0=0$ and the positive-side turning points are zero and $\sqrt2$. The first-order persistence condition is
$$
\int_0^{\sqrt2}(1-\alpha x^2)x\sqrt{1-x^2/2}dx=0,
$$
which gives $\alpha=5/4$.