Averaged first-integral obstruction to persistence of a periodic orbit

Suppose $V$ is a first integral of an unperturbed planar system and $\dot{V}=\epsilon G+O({\epsilon }^2)$ after perturbation. A periodic orbit converging to an unperturbed orbit $\gamma$ can persist only ifIndeed, the net change of $V$ over every period is zero. A nonzero averaged drift therefore obstructs persistence.

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA 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 $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.