Homoclinic balance for the weakly perturbed double-well oscillator

At energy zero, the positive homoclinic orbit of the unperturbed double-well oscillator has $y=\pm x\sqrt{1-x^2/2}$ and turning points zero and $\sqrt{2}$. The first-order drift integral from the energy balance for the weakly perturbed double-well oscillator vanishes whenBoth integrals follow by substituting $u=x^2/2$. Their ratio gives $\alpha =5/4$. The parameter derivative of the balance is $-I_2\ne 0$, so the zero is simple.

This is a leading connecting-orbit condition, rather than an exact formula for the perturbed loop at finite $\epsilon$. Under the usual nondegenerate return-map hypotheses, the Melnikov energy-balance method locates a nearby homoclinic parameter curve with $\alpha =5/4+O(\epsilon )$.