= Heteroclinic Melnikov integral for a quartic Hamiltonian
{title2=$M(\mu,s)=\int_{-\sqrt{2s}}^{\sqrt{2s}}(\mu-v^2/2)(s+v^2/2)dv$}
Perturb the <Hamiltonian blow-up of a reflection-symmetric double-zero point> by $u'= -sv+v^3/2+\varepsilon(\mu-v^2/2)u+O(\varepsilon^2)$ and $v'=2u+\varepsilon(\mu-v^2/2)v+O(\varepsilon^2)$. Then $H'=\varepsilon(\mu-v^2/2)(2u^2+sv^2-v^4/2)+O(\varepsilon^2)$. On the positive heteroclinic branch $dt=dv/[\sqrt2(s-v^2/2)]$, so its first energy change is $\varepsilon M/\sqrt2$, where $M=\int_{-\sqrt{2s}}^{\sqrt{2s}}(\mu-v^2/2)(s+v^2/2)dv$. A simple zero gives cancellation of the leading stable/unstable manifold splitting and continues to a nearby heteroclinic bifurcation. Since $\partial M/\partial\mu>0$, there is a unique first-order balance for fixed $s>0$.
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