Invariants and reversibility in polynomial systems of ODEs

Abstract

We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with $1:-1$ resonant singularity at the origin. The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety of systems admitting a local analytic first integral near the origin. We propose a new algorithm to obtain a generating set for the Sibirsky ideal of such polynomial systems and investigate some algebraic properties of this ideal. Then, we discuss a generalization of the concept of time-reversibility in the three-dimensional case considering the systems with $1:\zeta:\zeta^2$ resonant singularity at the origin (where $\zeta$ is a primitive cubic root of unity) and study a connection of such reversibility with the invariants of some group actions in the space of parameters of the system