Validated Numerics for Continuation and Bifurcation of Connecting Orbits of Maps

Abstract

The present work develops validated numerical methods for analyzing continuous branches of connecting orbits—as well as their bifurcations—in one parameter families of discrete time dynamical systems. We use the method of projected boundaries to reduce the connecting orbit problem to a finite dimensional zero finding problem for a high dimensional map. We refer to this map as the connecting orbit operator. We study one parameter branches of zeros for the connecting orbit operator using existing methods of computer assisted proof know as the radii-polynomial approach and validated continuation. The local stable/unstable manifolds of the fixed points are analyzed using validated numerical methods based on the parameterization method. The validated branches of zeros found using our argument correspond to continuous families of transverse connecting orbits for the original dynamical system. Validation of a saddle node bifurcations for the connecting orbit operator correspond to a proof of existence for a tangency. We illustrate the implementation of our method for the Henon map.