Document Type

Article

Publication Date

2023

DOI

10.1103/PhysRevResearch.5.043241

Publication Title

Physical Review Research

Volume

5

Issue

4

Pages

043241 (1-21)

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 ultradiscrete Painlevé 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.

Rights

© 2023 American Physical Society.

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Original Publication Citation

Zolkin, T., Kharkov, Y., & Nagaitsev, S. (2023). Machine-assisted discovery of integrable symplectic mappings. Physical Review Research, 5(4), 1-21, Article 043241. https://doi.org/10.1103/PhysRevResearch.5.043241

ORCID

0000-0001-6088-4854 (Nagaitsev)

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