= Inverse-square-root period law near a saddle-node bottleneck
{title2=$T_{\rm loc}=\frac{\arctan(h/k)-\arctan(\nu/k)}k$}
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 $T_{\rm loc}=[\arctan(h/k)-\arctan(\nu/k)]/k\sim\pi/k$. The same coefficient holds if $-\nu/k\to\infty$. It is not uniform in the joint limit $\nu,k\to0$: 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>.
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