Invariants and reversibility in polynomial systems of ODEs

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 : ζ : ζ 2 : … : ζ n − 1 resonant singularity at the origin, where n is prime and ζ 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.