Shadowing of non-transversal heteroclinic chains in lattices

TL;DR

This paper investigates shadowing phenomena in complex lattice dynamical systems with non-transversal heteroclinic chains, revealing conditions under which finite chains can be shadowed despite non-transversality, with implications for energy transfer models.

Contribution

It introduces a geometric framework for shadowing non-transversal heteroclinic chains in lattice systems, extending previous results limited to transversal cases, and applies covering relations as a key technique.

Findings

Finite chains can be shadowed despite non-transversality.

Block diagonal dynamics enable shadowing in certain cases.

The approach applies to Hamiltonian and non-Hamiltonian systems.

Abstract

We deal with dynamical systems on complex lattices possessing chains of non-transversal heteroclinic connections between several periodic orbits. The systems we consider are inspired by the so-called \emph{toy model systems} (TMS) used to prove the existence of energy transfer from low to high frequencies in the \emph{nonlinear cubic Schr\"odinger equation} (NLS) or generalizations. Using the geometric properties of the complex projective space as a base space, we generate in a natural way collections of such systems containing this type of chains, both in the Hamiltonian and in the non-Hamiltonian setting. On the other hand, we characterize the property of block diagonal dynamics along the heteroclinic connections that allows these chains to be shadowed, a property which in general only holds for transversal heteroclinic connections. Due to the lack of transversality, only finite chains are shadowed, since there is a dropping dimensions mechanism in the evolution of any disk close to them. The main shadowing technical tool used in our work is the notion of covering relations as introduced by one of the authors.