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

Anastassiou, Stavros; Bountis, Tassos; B¨acker, Arnd

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

dc.contributor.author	Anastassiou, Stavros
dc.contributor.author	Bountis, Tassos
dc.contributor.author	B¨acker, Arnd
dc.date.accessioned	2018-08-23T09:29:32Z
dc.date.available	2018-08-23T09:29:32Z
dc.date.issued	2017-09
dc.description.abstract	An 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.citation	Stavros Anastassiou, Tassos Bountis, Arnd B¨acker. 2018. Homoclinic Points of 2-D and 4-D Maps via the Parametrization Method. arXiv	en_US
dc.identifier.uri	http://nur.nu.edu.kz/handle/123456789/3396
dc.language.iso	en	en_US
dc.publisher	arXiv	en_US
dc.rights	Attribution-NonCommercial-ShareAlike 3.0 United States	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc-sa/3.0/us/	*
dc.subject	invariant manifolds	en_US
dc.subject	polynomial H´enon maps	en_US
dc.subject	parametrization method	en_US
dc.subject	discrete breathers	en_US
dc.title	Homoclinic Points of 2-D and 4-D Maps via the Parametrization Method	en_US
dc.type	Article	en_US
workflow.import.source	science