= Saddle-node bifurcation on an invariant circle
{title2=$T\sim\pi/\sqrt\eta$}
= SNIC bifurcation
{c}
{synonym}
When a saddle-node lies on a returning invariant circle, removing the equilibria creates passage around a periodic orbit through a slow bottleneck. For $\dot x=x^2+\eta$ with $\eta>0$, crossing a fixed interval $[-h,h]$ takes $2\arctan(h/\sqrt\eta)/\sqrt\eta\sim\pi/\sqrt\eta$. If the remaining orbit takes bounded time, this is also its leading period. Strong transverse contraction makes the newborn periodic orbit attracting. This global saddle-node mechanism differs from a local fold occurring away from an already existing cycle.
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