The Saddle-Node Separatrix-Loop Bifurcation

We study vector fieldx x = f(x), x ∈ R², having at some point an equilibrium of saddle-node type with a separatrix loop. Such vector fields fill a codimension two submanifold ∑, of an appropriate Banach space. We give analytic conditions that determine whether a two-parameter perturbation of x = f(x) is transverse to ∑ The new condition is a version of Melnikov’s integral around the separatrix loop. If it is nonzero, then as one perturbs away from 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 β φ + φ + sin φ = ρ at its organizing center.