Invariants and reversibility in polynomial systems of ODEs

TL;DR

This paper investigates the link between invariants of group actions and time-reversibility in polynomial differential systems, introducing algorithms and algebraic tools to analyze resonant singularities and their invariants.

Contribution

It presents a new algorithm for generating the Sibirsky ideal and explores the algebraic properties of invariants related to reversibility in polynomial systems with resonant singularities.

Findings

Connection between invariants and time-reversibility in 2D systems.

Extension of reversibility concepts to higher-dimensional systems with prime resonance.

Use of binomial ideals to analyze invariants and symmetries.

Abstract

This paper explores a relationship between invariants of certain group actions and the time-reversibility of two-dimensional polynomial differential systems exhibiting a $1:-1$ resonant singularity at the origin. We focus on the connection of time-reversibility with the Sibirsky subvariety of the center (integrability) variety, which encompasses systems possessing a local analytic first integral near the origin. An algorithm for generating the Sibirsky ideal for these systems is proposed and the algebraic properties of the ideal are examined. Furthermore, using a generalization of the concept of time-reversibility we study $n$-dimensional systems with a $1:\zeta:\zeta^2:\dots:\zeta^{n-1}$ resonant singularity at the origin, where $n$ is prime and $\zeta$ is a primitive $n$-th root of unity. We study the invariants of a Lie group action on the parameter space of the system, leveraging the theory of binomial ideals as a fundamental tool for the analysis. Our study reveals intriguing connections between generalized reversibility, invariants, and binomial ideals, shedding light on their complex interrelations.