= Energy balance for the weakly perturbed double-well oscillator
{title2=$\dot x=y,\quad\dot y=x-x^3+\varepsilon(1-\alpha x^2)y$}
For
$$
\dot x=y,\qquad \dot y=x-x^3+\varepsilon(1-\alpha x^2)y,
$$
the unperturbed <Hamiltonian function> $H=y^2/2-x^2/2+x^4/4$ satisfies the exact drift identity $\dot H=\varepsilon(1-\alpha x^2)y^2$. The perturbation adds energy where $\alpha x^2<1$ and removes it where $\alpha x^2>1$, for positive $\varepsilon$.
On a regular unperturbed <periodic orbit> of energy $h$, let $x_1<x_2$ be its turning points and write $y=\pm\sqrt{2h+x^2-x^4/2}$. Since $dt=dx/y$, both halves contribute the same integral:
$$
\Delta H=2\varepsilon\int_{x_1}^{x_2}(1-\alpha x^2)\sqrt{2h+x^2-x^4/2}\,dx+O(\varepsilon^2).
$$
A simple zero of this energy increment selects a nearby perturbed periodic orbit by the <energy balance method>. The approximation is local to regular closed levels; at $h=0$, the <homoclinic balance for the weakly perturbed double-well oscillator> uses a convergent drift integral despite the infinite traversal time.
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