Machine-assisted discovery of integrable symplectic mappings

TL;DR

This paper introduces an automated method to discover integrable symplectic maps in the plane, identifying over 100 new families by analyzing polygon invariants and trajectory shapes, thus advancing understanding of discrete integrable systems.

Contribution

The paper presents a novel algorithm that automatically finds integrable symplectic maps, rediscovering known cases and discovering numerous new integrable families with polygon invariants.

Findings

Over 100 new integrable families identified

Method successfully rediscovered known integrable mappings

Analysis of polygon invariants distinguishes integrable from chaotic dynamics

Abstract

We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the algorithm is based on the knowledge that the evolution of an integrable system in the phase space is restricted to a lower-dimensional submanifold. Limiting ourselves to polygon invariants of motion, we analyze the shape of individual trajectories thus successfully distinguishing integrable motion from chaotic cases. For example, our method rediscovers some of the famous McMillan-Suris integrable mappings and discrete Painlev\'e equations. In total, over 100 new integrable families are presented and analyzed; some of them are isolated in the space of parameters, and some of them are families with one parameter (or the ratio of parameters) being continuous or discrete. At the end of the paper, we suggest how newly discovered maps are related to a general 2D symplectic map via an introduction of discrete perturbation theory and propose a method on how to construct smooth near-integrable dynamical systems based on mappings with polygon invariants.