= Averaged area criterion for perturbed Hamiltonian cycles
{title2=$\widehat\mu=2\int_{\mathcal D_h}r^2\,dA/|\mathcal D_h|$}
For a planar <Hamiltonian system> with vector field $(H_y/2,-H_x/2)$ perturbed by $\varepsilon(\widehat\mu-r^2)(x,y)$, the <divergence theorem> expresses the leading drift of the <first integral> around a closed <periodic orbit> as
$$
\oint H_t\,dt=4\varepsilon\left[\widehat\mu|\mathcal D_h|-2\int_{\mathcal D_h}r^2\,dA\right].
$$
A vanishing drift is a necessary first-order selection condition for a persisting <periodic orbit>. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting <limit cycle>. In the <Hamiltonian limit of three-to-one forcing>, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\widehat\mu=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.
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