** Authors: **

Jean-Philippe Lessard

Jason D. Mireles James

Christian Reinhardt

** Abstract:** 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.

Journal of Dynamics and Differential Equations, Vol 26, Issue 2, pp. 267-313 (2014).

The final The final version is available here.