Homoclinic points of 2D and 4D maps via the parametrization method

An interesting problem in solid state physics is to compute discrete breather solutions in N coupled 1D Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of 2N-dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure for a two-dimensional (2D) family of generalized Henon maps (N = 1), prove the existence of a hyperbolic set in the non-dissipative case and show that it is directly connected to the existence of a homoclinic orbit at the origin. Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond which the homoclinic intersection disappears. Proceeding to N = 2, we use the same approach to accurately determine the homoclinic intersections of the invariant manifolds of a saddle point at the origin of a 4D map consisting of two coupled 2D cubic Henon maps. For small values of the coupling we determine the homoclinic intersection, which ceases to exist once a certain amount of dissipation is present. We discuss an application of our results to the study of discrete breathers in two linearly coupled 1D particle chains with nearest-neighbor interactions and a Klein-Gordon on site potential.