= Solution
For a <nearly Hamiltonian system>, the unperturbed <Hamiltonian> is conserved and its level curves typically contain a family of <periodic orbits>. A weak perturbation changes the energy slowly. The <energy balance method> integrates $\dot H$ over one unperturbed period to obtain the leading energy change $\Delta H(H)$. A zero of this change selects a possible persistent <periodic orbit>. A simple zero persists for sufficiently small perturbation by a <Poincaré return map> and the <implicit function theorem>. A negative slope of the averaged energy drift gives attraction; a positive slope gives repulsion. A multiple zero requires higher-order analysis, and a zero-free leading drift rules out periodic motion in the corresponding perturbative energy range.
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