 Nearly Hamiltonian system

A nearly Hamiltonian system is a weak perturbation of a Hamiltonian system. Along its paths, $\dot{H}=ϵ\nabla H\cdot R$. Averaging this drift over an unperturbed periodic orbit is the energy balance method; a simple zero of the averaged drift selects a persistent nearby periodic orbit for sufficiently small $ϵ$. A multiple zero requires higher-order analysis.