Energy balance for the weakly perturbed double-well oscillator ($\dot{x}=y,\dot{y}=x-x^3+\epsilon (1-\alpha x^2)y$)

Forthe perturbation gives $\dot{H}=\epsilon (1-\alpha x^2)y^2$. Along an unperturbed orbit $H=H_0$ with turning points $x_1,x_2$,

For $u^{\prime }=v$, $v^{\prime }=u-u^3+\epsilon (\beta -u^2)v$, let $H=v^2/2-u^2/2+u^4/4$. On an outer periodic orbit with $H>0$, leading energy balance method requires $\beta =B(H)$, whereHere $u_{\max }^2=1+\sqrt{1+4H}$. The homoclinic orbit limit is $B(0)=4/5$. Differentiating the denominator gives a period integral diverging at $H=0$, whereas the derivative of the numerator stays finite; hence $B$ decreases immediately above zero. At large $H$, rescaling $u=H^{1/4}s$ gives $B(H)\propto \sqrt{H}$. Its first nondegenerate minimum selects a saddle-node bifurcation of periodic orbits; numerical quadrature gives ${\beta }_{\mathrm{fold}}\simeq 0.75226$. The cycle on the decreasing portion is unstable and the one on the increasing portion is stable, since the net energy drift has sign $\beta -B(H)$. Restoring an unperturbed coefficient $a^2$ in the linear restoring term multiplies the threshold by $a^2$.

On either unperturbed homoclinic loop, $H_0=0$ and the positive-side turning points are zero and $\sqrt{2}$. The first-order persistence condition iswhich gives $\alpha =5/4$.

For the periodic energy levels $-1/4<H_0<0$ enclosing one well, the energy-balance value of $\alpha$ ranges from one at the center to $5/4$ at the homoclinic loop. Thus for small positive $\epsilon$, one expects one symmetric pair of single-well periodic orbits when $1<\alpha <5/4$.