= Saddle-node separatrix-loop bifurcation
= Saddle-node loop bifurcation
{synonym}
A saddle-node separatrix-loop point occurs when a saddle-node equilibrium simultaneously has a global returning separatrix at its strong stable manifold. In the local form $\dot x=x^2-\mu$, $\dot y=-\lambda y$, a global return to $y=h$ at $x=\nu$ makes the codimension-two point $\mu=\nu=0$. For $\mu>0$ a saddle loop occurs on $\nu=\sqrt\mu$. The saddle-node boundary $\mu=0$ has an invariant-circle segment on the $\nu<0$ side, while for $\nu>0$ a nearby periodic orbit avoids the bottleneck and survives the local equilibrium fold. The return geometry, rather than the local saddle-node normal form alone, determines the global orbit.
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