 Heteroclinic Melnikov function for a periodic planar flow

For $\dot{q}=u_0(q)+ϵu_1(q,t)$ 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 ${\dot{q}}_1=D{u}_0{q}_1+u_1$. The covector $w=\nabla {\psi }_0(q_0)$ satisfies $\dot{w}=-(D{u}_0{)}^{T}w$, obtained by differentiating $\nabla {\psi }_0\cdot u_0=0$. Thus $(w\cdot q_1{)}^{\prime }=w\cdot u_1$. 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 $ϵM/|\nabla {\psi }_0|$. A simple zero yields a transverse intersection of the perturbed manifolds, by the implicit function theorem.