Limit cycles for a class of eleventh $\mathbb{Z}_{12}-$equivariant systems without infinite critical points

TL;DR

This paper studies the dynamics of a specific class of symmetric planar systems, identifying conditions for the existence and uniqueness of stable limit cycles around certain equilibria using Abel equation reduction.

Contribution

It introduces new parameter conditions ensuring unique, hyperbolic limit cycles in $ ext{Z}_{12}$-equivariant systems, expanding understanding of their complex dynamics.

Findings

Conditions for unique limit cycles are derived.

Limit cycles can surround specific equilibria (1, 13, or 25).

Hyperbolicity of limit cycles is established.

Abstract

We analyze the complex dynamics dynamics of a family of $\mathbb{Z}_{12}-$equivariant planar systems, by using their reduction to an Abel equation. We derive conditions in the parameter space that allow uniqueness and hyperbolicity of a limit cycle surrounding either $1,~13$ or $25$ equilibria.