= Solution
For the unperturbed <Hamiltonian> system, choose a conserved energy $H(x,y)$. Along the perturbed system,
$$
\dot H=\varepsilon\bigl(H_xp_1+H_yp_2\bigr).
$$
The <energy balance method> integrates this expression around each unperturbed periodic orbit. A zero of the resulting $O(\varepsilon)$ energy change is a necessary leading-order condition for a nearby periodic orbit. A simple zero at which the drift changes from positive inside to negative outside predicts a stable orbit; the opposite change predicts an unstable one.
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