Singularity confinement and chaos in two-dimensional discrete systems

TL;DR

This paper introduces a novel two-dimensional lattice equation that exhibits both integrable singularity confinement and chaotic behavior, expanding understanding of quasi-integrability in discrete systems.

Contribution

It presents the first known example of a quasi-integrable two-dimensional lattice equation with confined singularities and positive algebraic entropy.

Findings

The equation satisfies singularity confinement criterion.

It exhibits exponential growth in degrees indicating chaos.

Reductions produce a hierarchy of difference equations with similar properties.

Abstract

We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a criterion of integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its iterates exhibit exponential growth. By systematic reduction to one-dimensional systems, it gives a hierarchy of ordinary difference equations with confined singularities, but with positive algebraic entropy including a generalized form of the Hietarinta-Viallet mapping. We believe that this is the first example of such quasi-integrable equations defined over a two-dimensional lattice.