Limit cycles for a class of ${\mathbb{Z}}_{2n}-$equivariant systems without infinite equilibria

Abstract

We analyze the dynamics of a class of ${\mathbb{Z}}_{2n}$-equivariant differential equations on the plane, depending on 4 real parameters. This study is the generalisation to ${\mathbb{Z}}_{2n}$ of previous works with ${\mathbb{Z}}_4$ and ${\mathbb{Z}}_6$ symmetry. We reduce the problem of finding limit cycles to an Abel equation, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds either 1, $2n+1$ or $4n+1$ equilibria, the origin being always one of these points.