Saddle-node bifurcation on an invariant circle ($T\sim \pi /\sqrt{\eta }$)

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.

A periodic orbit entering the local flow $\dot{x}=x^2+k^2$ at a fixed negative $\nu$ and leaving at positive $h$ has passage time $T_{\mathrm{loc}}=[\arctan (h/k)-\arctan (\nu /k)]/k\sim \pi /k$. The same coefficient holds if $-\nu /k\to \infty$. It is not uniform in the joint limit $\nu ,k\to 0$: if $\nu /k\to -C$, the leading coefficient is $\pi /2+\arctan C$. For entry at zero it is $\pi /2$, while entry at a fixed positive $\nu$ gives bounded passage time. These distinctions matter near a saddle-node separatrix-loop bifurcation.