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

Research output: Contribution to journalResearch articleContributedpeer-review


  • Stavros Anastassiou - , University of Patras (Author)
  • Tassos Bountis - , University of Patras, Nazarbayev University (Author)
  • Arnd Backer - , Chair of Computational Physics, Max-Planck-Institute for the Physics of Complex Systems (Author)


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 nondissipative 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.


Original languageEnglish
Pages (from-to)3799-3820
Number of pages22
Issue number10
Publication statusPublished - 11 Sept 2017



  • discrete breathers, invariant manifolds, parametrization method, polynomial Henon maps