= 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 $\sqrt2$. The first-order drift integral from the <energy balance for the weakly perturbed double-well oscillator> vanishes when
$$
I_0-\alpha I_2=0,\qquad
I_0=\int_0^{\sqrt2}x\sqrt{1-x^2/2}\,dx=\frac23,\qquad
I_2=\int_0^{\sqrt2}x^3\sqrt{1-x^2/2}\,dx=\frac8{15}.
$$
Both integrals follow by substituting $u=x^2/2$. Their ratio gives $\alpha=5/4$. The parameter derivative of the balance is $-I_2\ne0$, so the zero is simple.
This is a leading connecting-orbit condition, rather than an exact formula for the perturbed loop at finite $\varepsilon$. Under the usual nondegenerate return-map hypotheses, the <Melnikov energy-balance method> locates a nearby homoclinic parameter curve with $\alpha=5/4+O(\varepsilon)$.
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