Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the NewtonKantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.