= Outer-cycle fold in a weakly perturbed double-well oscillator
For $u'=v$, $v'=u-u^3+\varepsilon(\beta-u^2)v$, let $H=v^2/2-u^2/2+u^4/4$. On an outer <periodic orbit> with $H>0$, leading <energy balance method> requires $\beta=B(H)$, where
$$
B(H)=\frac{\int_0^{u_{\max}(H)}u^2\sqrt{2H+u^2-u^4/2}\,du}{\int_0^{u_{\max}(H)}\sqrt{2H+u^2-u^4/2}\,du}.
$$
Here $u_{\max}^2=1+\sqrt{1+4H}$. 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 $\beta_{\rm fold}\simeq0.75226$. 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 $a^2$.
Back to article page
