Inverse-square-root period law near a saddle-node bottleneck ($T_{\mathrm{loc}}=\frac{\arctan (h/k)-\arctan (\nu /k)}{k}$)

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.