= Heteroclinic Melnikov function for a periodic planar flow
{title2=$M=\int_{-\infty}^{\infty}\nabla\psi_0(q_0(t-t_0))\cdot u_1(q_0(t-t_0),t)\,dt$}
For $\dot q=u_0(q)+\epsilon u_1(q,t)$ with planar <Hamiltonian flow> $u_0=(-\partial_y\psi_0,\partial_x\psi_0)$, let $q_0$ be a <heteroclinic orbit> between two <hyperbolic fixed points>. Its first-order displacement satisfies $\dot q_1=Du_0q_1+u_1$. The covector $w=\nabla\psi_0(q_0)$ satisfies $\dot w=-(Du_0)^Tw$, obtained by differentiating $\nabla\psi_0\cdot u_0=0$. Thus $(w\cdot q_1)'=w\cdot u_1$. Integrating the stable correction from the future and the unstable correction from the past gives the displayed function. At a regular section the first-order normal separation is $\epsilon M/|\nabla\psi_0|$. A simple zero yields a transverse intersection of the perturbed manifolds, by the <implicit function theorem>.
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