Homoclinic Points of 2-D and 4-D Maps via the Parametrization Method
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Date
2017-09
Authors
Anastassiou, Stavros
Bountis, Tassos
B¨acker, Arnd
Journal Title
Journal ISSN
Volume Title
Publisher
arXiv
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.
Description
Keywords
invariant manifolds, polynomial H´enon maps, parametrization method, discrete breathers
Citation
Stavros Anastassiou, Tassos Bountis, Arnd B¨acker. 2018. Homoclinic Points of 2-D and 4-D Maps via the Parametrization Method. arXiv