 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.