Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems

A rigorous numerical proof for establishing existence of a transversal homoclinic orbit for a saddle fixed point with higher dimensional unstable eigenspaces is presented. As the first component of this method, a shadowing theorem that guarantees the existence of such a homoclinic orbit near a suitable pseudo orbit given the invertibility of a certain Jacobian is proved. The second component consists of a refinement procedure for numerically computing a pseudo homoclinic orbit with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component verifies that the homoclinic orbit is transversal. In [6], they proved the existence of transversal homoclinic orbits near anti-integrable limits and near singularities for the Arneodo-Coullet-Tresser maps. In this paper, the existence of transversal homoclinic orbits were proved far away from anti-integrable limits and singularities for these maps.