Homoclinic Points of 2-D and 4-D Maps via the Parametrization Method

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dc.contributor.authorAnastassiou, Stavros
dc.contributor.authorBountis, Tassos
dc.contributor.authorB¨acker, Arnd
dc.date.accessioned2018-08-23T09:29:32Z
dc.date.available2018-08-23T09:29:32Z
dc.date.issued2017-09
dc.description.abstractAn interesting problem in solid state physics is to compute discrete breather solutions in N coupled 1–dimensional 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 2–dimensional (2–D) family of generalized H´enon 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 4–D map consisting of two coupled 2–D cubic H´enon 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 1–dimensional particle chains with nearest–neighbor interactions and a Klein–Gordon on site potential.en_US
dc.identifier.citationStavros Anastassiou, Tassos Bountis, Arnd B¨acker. 2018. Homoclinic Points of 2-D and 4-D Maps via the Parametrization Method. arXiven_US
dc.identifier.urihttp://nur.nu.edu.kz/handle/123456789/3396
dc.language.isoenen_US
dc.publisherarXiven_US
dc.rightsAttribution-NonCommercial-ShareAlike 3.0 United States*
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/3.0/us/*
dc.subjectinvariant manifoldsen_US
dc.subjectpolynomial H´enon mapsen_US
dc.subjectparametrization methoden_US
dc.subjectdiscrete breathersen_US
dc.titleHomoclinic Points of 2-D and 4-D Maps via the Parametrization Methoden_US
dc.typeArticleen_US
workflow.import.sourcescience

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