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

TL;DR

This paper studies limit cycles in a class of $ Z_{2n}$-equivariant planar differential equations, generalizing previous symmetric cases, and provides criteria for their uniqueness and hyperbolicity by reducing the problem to an Abel equation.

Contribution

It extends analysis of symmetric systems to $ Z_{2n}$-equivariant equations, offering new criteria for limit cycle existence, uniqueness, and stability.

Findings

Criteria for limit cycle uniqueness and hyperbolicity

Reduction of the problem to an Abel equation

Application to systems with multiple 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.