# On the representations and $\mathbb{Z}_2$-equivariant normal form for solenoidal Hopf-zero singularities

## TL;DR

This paper explores the structure and representations of a Lie algebra related to Hopf-zero singularities, focusing on $ ext{Z}_2$-equivariant normal forms and illustrating results with a modified Chua's oscillator.

## Contribution

It introduces new representations of the solenoidal Lie algebra for Hopf-zero singularities and analyzes the $ ext{Z}_2$-equivariant normal form, including non-potential properties.

## Key findings

- List of Lie algebra representations including Hamiltonian, Euler's form, and Clebsch representations.
- Analysis of non-potential vector fields with Hopf-zero singularity.
- Normal form with non-zero cubic terms under $ ext{Z}_2$ symmetry, illustrated with Chua's oscillator.

## Abstract

In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and $\mathbb{Z}_2$-equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group $\mathbb{Z}_2$ . The theoretical results of normal form theory are illustrated with the modified Chua's oscillator.