Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homoclinic points of 2-D and 4-D maps via the Parametrization Method   An interesting problem in solid state physics is to compute discrete breather
solutions in $\mathcal{N}$ 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 $2\mathcal{N}$--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 Henon maps ($\mathcal{N}$=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 $\mathcal{N}=2$, we
use the same approach to 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. In dependence of the coupling the
homoclinic intersection is determined, 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.

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The categories of this paper are: funct-an
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph