Coalgebra symmetry for discrete systems

TL;DR

This paper introduces coalgebra symmetry for discrete systems, demonstrating that radially symmetric systems are quasi-integrable and exploring their integrability properties, including generalizations of discrete Painlevé and McMillan maps.

Contribution

It defines coalgebra symmetry for discrete systems and proves quasi-integrability for radially symmetric systems, extending integrability analysis to multi-degree systems.

Findings

Radially symmetric systems in standard form are quasi-integrable.

Variational discrete quasi-radially symmetric systems are Poincaré--Lyapunov--Nekhoroshev maps.

Generalizations of discrete Painlevé I and McMillan maps are analyzed.

Abstract

In this paper we introduce the notion of coalgebra symmetry for discrete systems. With this concept we prove that all discrete radially symmetric systems in standard form are quasi-integrable and that all variational discrete quasi-radially symmetric systems in standard form are Poincar\'e--Lyapunov--Nekhoroshev maps of order $N-2$, where $N$ are the degrees of freedom of the system. We also discuss the integrability properties of several vector systems which are generalisations of well-known one degree of freedom discrete integrable systems, including two $N$ degrees of freedom autonomous discrete Painlev\'e I equations and an $N$ degrees of freedom McMillan map.