Limit cycles for a class of quintic $\mathbb{Z}_6-$equivariant systems without infinite critical points

Abstract

We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the parameters, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points. The method used is the reduction of the problem to an Abel equation.