= Melnikov energy-balance method
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The <Melnikov energy-balance method> measures the leading separation of invariant manifolds after a weak perturbation of a planar <Hamiltonian system>. For
$$
\dot z=J\nabla H(z)+\varepsilon R(z,t),
$$
the exact identity is $\dot H=\varepsilon\nabla H\cdot R$. At first order, one integrates this energy drift along the unperturbed <homoclinic orbit> or <heteroclinic orbit>, provided the improper integral converges.
For an autonomous perturbation depending on a parameter $\mu$, write the leading splitting as $\varepsilon M(\mu)+O(\varepsilon^2)$. A simple zero $M(\mu_0)=0$, $M'(\mu_0)\ne0$ permits a nearby connecting-orbit parameter curve by the <implicit function theorem>. The zero is a first-order balance; the perturbed connecting orbit is generally displaced from the unperturbed one.
For periodic forcing, the relative forcing phase supplies a different variable. A simple zero of $M(t_0)$ can give a transverse intersection of the perturbed stable and unstable manifolds in a return section. The <heteroclinic Melnikov function for a periodic planar flow> derives that integral from the adjoint variational equation.
This distinction matters: a parameter-tuned autonomous saddle connection and a transverse intersection under periodic forcing are different conclusions. A multiple zero, a degenerate saddle or a nonconvergent integral needs additional analysis rather than the simple-zero argument.
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