= Solution
The rescaling resolves the small, nearby nonzero saddles while retaining the weak non-Hamiltonian terms. It allows the global <stable manifold>/<unstable manifold> connection to be studied as a <perturbation> of the explicit <heteroclinic cycle>, rather than searched for only in the original variables.
Differentiate the <first integral> using the first-order <perturbation> from part (c):
$$
H'=\varepsilon(\mu-v^2/2)\bigl(2u^2+sv^2-v^4/2\bigr)+O(\varepsilon^2).
$$
On the positive unperturbed connecting branch, $u=(2s-v^2)/(2\sqrt2)$, so
$$
2u^2+sv^2-v^4/2=(s-v^2/2)(s+v^2/2),\qquad
d\tau=\frac{dv}{\sqrt2(s-v^2/2)}.
$$
Integrating the <energy> change from one <saddle equilibrium> to the other therefore gives
$$
\boxed{\Delta H=\frac{\varepsilon}{\sqrt2}M(\mu,s)+O(\varepsilon^2),\quad
M=\int_{-\sqrt{2s}}^{\sqrt{2s}}(\mu-v^2/2)(s+v^2/2)\,dv.}
$$
This is the <heteroclinic Melnikov integral for a quartic Hamiltonian>. It measures the first-order <energy> mismatch, equivalently the transverse splitting between the outgoing <unstable manifold> and the incoming <stable manifold>. A nonzero value means that the <heteroclinic orbit> has broken. A simple zero supplies the leading connection condition; smooth <perturbation> then shifts the curve by higher-order terms. The central inversion <symmetry> makes the two connections have the same condition, so both can be restored on one parameter curve.
For fixed $s>0$, $\partial M/\partial\mu=\int_{-\sqrt{2s}}^{\sqrt{2s}}(s+v^2/2)dv>0$. The coefficient is affine in $\mu$, has a unique simple zero, and changes sign across it. No explicit evaluation is necessary. <Energy> drift on the nearby closed <Hamiltonian> levels can similarly determine a <periodic orbit> and its stability; the cycle born at the supercritical <Hopf bifurcation> grows towards the <separatrix> and can terminate at this <global bifurcation> through a <heteroclinic orbit>. Translating the resulting balance curve back with $\mu_{\rm old}=\varepsilon^2\mu$, $\sigma-1=\varepsilon^2s$ places it near the original double-zero point.
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