On Some Reversible Cubic Systems

Barbara Arcet,Valery G Romanovski

doi:10.3390/math9121446

Abstract

We study three systems from the classification of cubic reversible systems given by Żoła̧dek in 1994. Using affine transformations and elimination algorithms from these three families the six components of the center variety are derived and limit-cycle bifurcations in neighborhoods of the components are investigated. The invariance of the systems with respect to the generalized involutions introduced by Bastos, Buzzi and Torregrosa in 2021 is discussed. Computations are performed using the computer algebra systems Mathematica and Singular.

Highlights

One of the long-standing problems in the theory of polynomial differential equations is the Poincaré center problem, which involves finding for which values of parameters α pq, β pq a given polynomial differential system of the form n
For the obtained components we discuss the existence of orbital φ-reversibility and study bifurcations of small limit-cycles from the center at the origin
We study limit cycles that bifurcate from the origin under small perturbation of the centers of families given in Theorem 2

Summary

Introduction

One of the long-standing problems in the theory of polynomial differential equations is the Poincaré center problem, which involves finding for which values of parameters α pq , β pq a given polynomial differential system of the form n. In studying the center problem for a given polynomial family of the form (1), one usually computes a few first focus quantities v1 (α, β), . The reversible families of [29] contain systems with a center and systems that do not have a center, so the relation of the classification to the center variety is not discussed in [29] This relation was investigated in [30,31], where the authors were looking for an affine transformation. For the obtained components we discuss the existence of orbital φ-reversibility and study bifurcations of small limit-cycles from the center at the origin.

Preliminaries

Some Center Conditions

Bifurcations of Limit Cycles

Conclusions