A Lipschitz Version of the λ-Lemma and a Characterization of Homoclinic and Heteroclinic Orbits

In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney's Extension Theorem on compact manifolds to obtain a version of the well-known lambda-lemma for Lipschitz functions. The notions of Lipschitz transversality and hyperbolicity are investigated in the finite dimensional framework with a norm between C-1-norm and C-0-norm. As an application, we study homoclinic and heteroclinic orbits obtaining, as a consequence, a stability result for Lipschitz Morse-Smale functions.