The Saddle-Node Separatrix-Loop Bifurcation

We study vector fieldx $\dot x = f(x)$, $x \in \mathbb{R}^2 $, having at some point an equilibrium of saddle-node type with a separatrix loop. Such vector fields fill a codimension two submanifold $\sum $, of an appropriate Banach space. We give analytic conditions that determine whether a two-parameter perturbation of $\dot x = f(x)$ is transverse to $\sum $ The new condition is a version of Melnikov’s integral around the separatrix loop. If it is nonzero, then as one perturbs away from $\dot x = f(x)$ in the direction in which an equilibrium of saddle-node type persists, the separatrix loop breaks in a nondegenerate manner. This integral is shown to be nonzero for the two-parameter pendulum equation $\beta \ddot \phi + \dot \phi + \sin \phi = \rho $ at its organizing center.