 Saddle-node separatrix-loop bifurcation

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.