 Saddle-node bifurcation on an invariant circle

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.