Homoclinic points of 2-D and 4-D maps via the Parametrization Method

TL;DR

This paper applies the parametrization method to analytically and numerically compute homoclinic points in 2-D and 4-D maps, revealing their existence, tangencies, and dependence on dissipation, with applications to discrete breathers in coupled particle chains.

Contribution

The paper extends the parametrization method to high-dimensional maps, providing a detailed analysis of homoclinic intersections and their behavior under dissipation in coupled systems.

Findings

Homoclinic intersections exist in 2-D and 4-D maps under certain conditions.

Dissipation can lead to homoclinic tangencies and the disappearance of intersections.

Results have implications for the existence of discrete breathers in coupled particle chains.

Abstract

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.