Integrable mappings and the notion of anticonfinement

TL;DR

This paper explores anticonfined singularities in discrete systems and demonstrates how their behavior can indicate whether a system is integrable or not, aiding in the analysis of such mappings.

Contribution

It introduces the concept of anticonfinement in singularity analysis and links its behavior to the integrability of discrete mappings, providing a new diagnostic tool.

Findings

Anticonfined singularities are related to the integrability of discrete systems.

Behavior of anticonfined singularities can predict non-integrability.

Concrete examples illustrate the connection between singularity behavior and system integrability.

Abstract

We examine the notion of anticonfinement and the role it has to play in the singularity analysis of discrete systems. A singularity is said to be anticonfined if singular values continue to arise indefinitely for the forward and backward iterations of a mapping, with only a finite number of iterates taking regular values in between. We show through several concrete examples that the behaviour of some anticonfined singularities is strongly related to the integrability properties of the discrete mappings in which they arise, and we explain how to use this information to decide on the integrability or non-integrability of the mapping.