#  On the representations and ${{\mathbb{Z}_{2}}}$-equivariant normal form for
solenoidal Hopf-zero singularities

Fahimeh Mokhtari Corresponding author. Email<EMAIL_ADDRESS>
Department of Mathematics, Faculty of Sciences
Vrije Universiteit, De Boelelaan 1081a,
1081 HV Amsterdam, The Netherlands

###### 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.

Key words. Hopf-pitchfork singularity; Conservative and solenoidal vector
field; Clebsch representation; Euler’s form; Vector potential; Normal form.

2010 Mathematics Subject Classification. 34C20; 34A34.

Dedicated to Professor Jan. A Sanders on the occasion of his 70th birthday

##  1 Introduction

This investigation is a continuation of [16] in which the maximal solenoidal
conservative Lie algebra of classical normal form of Hopf-zero singularities
(in the sense of [7]) was introduced. This Lie algebra was denoted by
$\mathscr{L}.$ The simplest normal form, simplest parametric normal form, and
radius of convergence corresponding to the second level normal form of this
type of singularities were explored there. This present paper has two
purposes: first, we intend to list some of the representations that the vector
fields from $\mathscr{L}$ admit. Solenoidal property of vector fields from
$\mathscr{L}$ allows us to express these vector fields by local Euler’s
potentials, vector potential, and local Clebsch potentials, see [24, 10]. The
second purpose is to explore the ${{\mathbb{Z}_{2}}}$-equivariant unique
normal form of the solenoidal conservative family associated to Hopf-pitchfork
singularities. We denote the Lie algebra of such a family by
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}.$ Now, we elaborate in general terms our
results in the following.

In the framework of classical mechanics, the Poisson structure has a key role
in the description of Hamiltonian dynamics (see preface of [13]). In the
literature of normal form theory, this structure is employed to facilitate the
normal form study of singularities, an exhaustive treatment of this subject
can be found in [8, 9]. The first issue we address here involves constructing
the Poisson algebra $\mathcal{P}$ for a Lie subalgebra of $\mathscr{L}$, in
order to give a representation for $\mathscr{L}$. Having established the
Poisson structure we will be able to present the vector fields from this Lie
algebra by a set of ordinary differential equations, the two first of them are
in the canonical form of a one-degree of freedom Hamiltonian system and the
third one depends upon the two first variables. Noting that, for a given
three-dimensional solenoidal vector field which possesses one-parameter
symmetry group with solenoidal and infinitesimal generators, one would be able
to present the system in the normal form as given in [24]. Hence, beside the
Poisson structure following [16, Remark 2.4] and [24] that form could be
derived.

Solenoidal dynamical systems are important in a huge variety of applications.
These vector fields, also known as incompressible vector fields, occur in many
settings: velocity field of an incompressible fluid, magnetic field, etc., see
for instance [29, 11]. Exploiting tools from geometric dynamics, solenoidal
vector fields may be represented in some interesting and physically
significant representations such as Euler’s form and vector potential (see
[10] and references therein). In the following, we briefly review these
representations. If $v$ be a solenoidal vector field then there exists a
vector potential $\bf A$ such that $v=\nabla\times\bf A.$ From the point of
view of electromagnetism, $\bf A$ is called magnetic vector potential [33].
Euler’s form is another representation that solenoidal vector fields admit
(see [37, page 22] and [38, page 48]). Based on this form, one can find two
independent invariant functions $\alpha$ and $\beta$ from which the vector
field $v$ is derivable via $v=\nabla\alpha\times\nabla\beta.$ Applications of
these representations can be found in various books and papers, for instance,
see [1, 10, 30, 33]. In [10] the application of Euler’s form in the partial
differential equation has been discussed. For the significance of vector
potential in quantum theory, we refer the reader to [1].

In [19] the authors express a Lie algebra of completely integrable solenoidal
triple-zero singularities via Euler’s form and vector potential. As in their
studies, due to the solenoidal property of $\mathscr{L},$ here we shall
present any vector field in this Lie algebra using vector potentials and
Euler’s form. Further, the non-potential property of this family with Hopf-
zero singularity is examined. Thus, these vector fields cannot be derived by
the gradient of a scalar valued function, instead, these vector fields are
expressible in terms of the lamellar and complex lamellar vector fields. More
precisely, any vector field $v\in\mathscr{L}$ could be expressed by
$v=f_{1}\nabla f_{2}+\nabla f_{3}$ where $f_{1},f_{2},$ and $f_{3}$ are scalar
valued functions. This representation is named Monge representation or Clebsch
representation (see [37, page 27] and [38, Section 2.4]). For the application
of this representation in thermodynamics, we refer to [38, Subsection 9.11].

Progress towards deriving these representations has significant practical
implications for magnetic fields since these vector fields are solenoidal. For
instance, in [29, 11] these representations have been employed for studying
magnetic reconnection at three–dimensional null points. In [23] the authors
used the magnetic fields to interpret the solar flares. Hence, the results of
the present paper should provide the researchers in this area with
sufficiently powerful tools to analyze the motion of magnetic fields. As a
matter of fact, each of these representation gives an interpretation about the
structure of magnetic fields. Thereby, from these researches, we believe that
these studies would be remarkably useful in application. In the remainder of
introduction, we focus on the second aim of the paper.

In order to apply the general methods of bifurcation theory to singularities,
it is necessary to apply normal form theory. Roughly speaking, normal form
theory is to simplify the nonlinear part of vector fields with permissible
transformations, see [27, 31] and [32, chapters 9-13]. In this paper, we are
also interested in treating the simples normal form classification of
solenoidal family associated to the classical normal form of Hopf-pitchfork
singularities. The problem of the normal form of singular dynamical systems
has been studied by many authors. Before explaining our results in detail, we
explain some of this previous work in the following.

Baider and Sanders [8, 9] studied the unique normal form of Bogdanov-Takens
and Hamiltonian Bogdanov-Takens singularities. The paper [18] studied the
infinite level normal form of the Lie algebra of quasi-Eulerian Hopf-zero
vector fields. For the conservative-nonconservative decomposition of the
classical normal form of Hopf-zero dynamical systems and complete
classifications of the simplest normal form of this singularity, the reader is
referred to [17]. In [2] the ${{\mathbb{Z}_{2}}}$-equivariant normal form for
Hopf-zero vector fields are computed under the assumption that the cubic terms
be non-zero, for results on the bifurcation of Hopf-pitchfork singularities,
see [5, 4, 3]. More studies regarding the normal form of dynamical system
could be found in [15, 14, 19, 26].

In the studies mentioned above and in the literature, the researchers did not
investigate the problem of classifying the unique normal form for
${{\mathbb{Z}_{2}}}$-equivariant solenoidal conservative Hopf-zero vector
fields. As announced at the beginning of this section, our aim is to treat the
unique normal form of the following system

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm d}t}&=2a^{0}_{1}x\rho^{2}+\sum
a^{l}_{k}(k-{2l}+1){x}^{{2l}+1}{\rho}^{2(k-{2l})},\\\ \frac{{\rm
d}{\rho}}{{\rm d}t}&=-\frac{a^{0}_{1}\rho^{3}}{2}-\sum
a^{l}_{k}\frac{({2l}+1)}{2}{x}^{2l}{\rho}^{2(k-{2l})+1},\\\ \frac{{\rm
d}{\theta}}{{\rm d}t}&=1+\sum
b^{l}_{k}{x}^{{2l}}\rho^{2(k-{2l})},\end{aligned}\right.$ (1.1)

where constants $a^{l}_{k},b^{l}_{k}$ are real numbers, $0\leqslant
2l\leqslant k,0\leqslant k$ $b^{0}_{0}=a^{0}_{0}=0,$ and $a^{0}_{1}\neq 0.$
The associated first integral of this system is

$\displaystyle s(x,\rho)$ $\displaystyle:=$ $\displaystyle
a^{0}_{1}x\rho^{4}+\sum a^{l}_{k}x^{2l+1}{\rho}^{2({k-2l+1})},$

with $0\leqslant 2l\leqslant k.$ In this work, the unique normal form study of
the above system proceeds in a manner parallel to the study of the solenoidal
Hopf-zero vector fields without symmetry [16]. We use naturally their
algebraic structures to construct the Lie algebra
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}.$ We wish to stress that, despite the fact
that ${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}$ is the Lie subalgebra of
$\mathscr{L}$, the unique normal form of (1.1) cannot be obtained from the
unique normal form of volume preserving Hopf-zero vector fields given there.
The leading term that plays a dominant role in our normal form study is cubic,
whereas there the quadratic term was the leading term. Hence, the
normalization problem that is performed here differs from that was studied in
[16]. Finally, we shall present the unique normal form of (1.1) in four
different representations, see Theorem (4.4).

###  1.1 Outline of the paper

This paper has the following organization. In Section 2, first, we recall some
notations and definitions that needed throughout the paper. Then, we provide
the associated Poisson algebra $\mathcal{P}$ for one Lie subalgebra of
$\mathscr{L}.$ Following the Poisson algebra, a form for solenoidal Hopf-zero
vector fields using Hamilton’s equations as introduced in [24] is presented.
It is also in this section that the representations such as Euler’s form,
vector potential, and Clebsch representation for Lie algebra $\mathscr{L}$ are
given.

In Section 3, we introduce and formulate the Lie algebra
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}.$ We also recall the general framework for
computing the normal form for ${\Gamma}$-equivariant singularities required to
study the unique normal form of the class of singularities under
consideration.

In Section 4, we study the unique normal form of (1.1). One symmetry of unique
normal form is detected. We also present the four alternative representations
of the unique normal from rely on the results on Section 2.

The final section is dedicated to make symbolic computations of our normal
form study. Some sufficient conditions on the coefficients of any Hopf-
pitchfork system, under which the lower order truncation of the classical
normal form of the original system takes the form (1.1) are given. Moreover,
the modified Chua’s oscillator serves to demonstrate our main results in the
normal form. All of the computations are performed using Maple [12].

##  2  Representations of Lie algebra $\mathscr{L}$

###  2.1 Preliminaries

In this section, we present the main results regarding the ways that the
vector fields from $\mathscr{L}$ would be expressible. The Poisson algebra
associated to the sub-family of $\mathscr{L}$ and tools from geometric
dynamics are adapted to obtain these forms. Before going to the main results,
we give a review of some definitions and facts from the Lie algebra
$\mathscr{L}$ which are fundamental to what follows. Furthermore, after we
establish some notation, we recall some general information required to study
the representations of vector fields in $\mathscr{L}.$

The classical normal form of solenoidal conservative Hopf-zero vector field in
cylindrical coordinates is given by

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm
d}t}&=\sum(k-l+1)a^{l}_{k}x^{l+1}{\rho^{2}}^{(k-l)},\\\ \frac{{\rm
d}{\rho}}{{\rm d}t}&=-\sum\frac{(l+1)}{2}a^{l}_{k}x^{l}{\rho}^{2(k-l)+1},\\\
\frac{{\rm d}{\theta}}{{\rm d}t}&=\sum
b^{m}_{n}x^{m}{\rho}^{2(n-m)},\end{aligned}\right.$ (2.1)

where $-1\leqslant l\leqslant k,0\leqslant k,$ $0\leqslant m\leqslant n,$
$b^{-1}_{0}=a^{0}_{0}=0,$ and $a^{l}_{k},b^{m}_{n}\in\mathbb{R},$ see [16,
Equation 1.1]. This class of vector fields was derived by
$\mathfrak{sl}_{2}$-decomposition of the classical normal form of Hopf-zero
bifurcation [17]. َAnd indeed, the above system could be formulated using the
Lie algebraic structure as follows.

We recall[16, 17] that the maximal Lie algebra of solenoidal Hopf-zero
classical normal form is given by
${\mathscr{L}}=\mathscr{F}\oplus\mathscr{T},$ where

$\displaystyle\mathscr{F}={\rm span}\left\\{\sum
a^{l}_{k}F^{l}_{k}\,|\,0\leqslant
k,-1\leqslant{l}\leqslant{k},a^{l}_{k}\in\mathbb{R}\right\\},\quad\mathscr{T}={\rm
span}\left\\{\sum
b^{l}_{k}\Theta^{l}_{k}\,|\,0\leqslant{l}\leqslant{k},b^{l}_{k}\in\mathbb{R}\right\\},$

and

$\displaystyle F^{l}_{k}$ $\displaystyle=$ $\displaystyle
x^{l}{(y^{2}+z^{2})}^{k-l}\left((k-l+1)x\frac{\partial}{\partial
x}-\frac{(l+1)}{2}y\frac{\partial}{\partial
y}-\frac{(l+1)}{2}z\frac{\partial}{\partial
z}\right),\quad-1\leqslant{l}\leqslant{k},\;\;\;$ (2.2)
$\displaystyle\Theta^{l}_{k}$ $\displaystyle=$ $\displaystyle
x^{l}(y^{2}+z^{2})^{k-l}\left(z\frac{\partial}{\partial
y}-y\frac{\partial}{\partial
z}\right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\,\,\;\;0\leqslant{l}\leqslant{k}.\quad\;\;\;$
(2.3)

Furthermore, the algebra of the first integral for $\mathscr{F}$ is as
$\left\langle
x^{l+1}(y^{2}+z^{2})^{k-l+1}\right\rangle_{-1\leqslant{l}\leqslant{k}}^{1\leqslant
l+k}$ and for $\mathscr{T}$ is as $\left\langle x,(y^{2}+z^{2})\right\rangle$.
Expressed in terms of the cylindrical polar coordinates $(x,\rho,\theta),$ the
preceding vector fields will be given by the expressions

$\displaystyle F^{l}_{k}$ $\displaystyle=$ $\displaystyle
x^{l}\rho^{2(k-l)}\left((k-l+1)x\frac{\partial}{\partial
x}-\frac{(l+1)}{2}\rho\frac{\partial}{\partial\rho}\right),$ (2.4)
$\displaystyle\Theta^{l}_{k}$ $\displaystyle=$
$\displaystyle{x}^{{l}}\rho^{2(k-{l})}\frac{\partial}{\partial\theta}.$ (2.5)

Using the structures given above, system (2.1) can be recast to

$\displaystyle
v=\sum_{k=0}^{\infty}\sum_{l=-1}^{k}a^{l}_{k}F^{l}_{k}+\sum_{n=0}^{\infty}\sum_{m=0}^{n}b^{m}_{n}\Theta^{m}_{n}.$

As mentioned before, the simplest normal form and simplest parametric normal
form of foregoing system with the assumption $a^{-1}_{0}\neq 0$ were explored
using $\mathfrak{sl}_{2}$-style in [16]. Now, we fix some notation.

######  Notation 2.1.

The following notation is used throughout the paper.

  * •

Define $v=(v_{1},v_{2},v_{3})\in\mathbb{R}^{3}$ by
$v=v_{1}\cdot\mathbf{{e}_{x}}+v_{2}\cdot\mathbf{{e}_{y}}+v_{3}\cdot\mathbf{{e}_{z}}.$
We denote $F^{l}_{k}=dx(F^{l}_{k})\frac{\partial}{\partial
x}+dy(F^{l}_{k})\frac{\partial}{\partial
y}+dz(F^{l}_{k})\frac{\partial}{\partial z}.$

  * •

The symbol $\nabla$ indicates the gradient operator of the vector field.

  * •

The Pochhammer $k$-symbol notation for any $a,b\in\mathbb{R}$ and
$k\in\mathbb{N},$ is given by

$\displaystyle\left(a\right)^{k}_{b}:=\prod_{j=0}^{k-1}(a+jb).$

######  Definition 2.2.

Consider the dynamical system ${\dot{x}}=v,$ with $x\in\mathbb{R}^{3}.$

  * •

The vector field $v$ is said to be potential, or locally potential if
$\nabla\times v=0$ for all $x\in\mathbb{R}^{3}.$ The system is named non-
potential, otherwise, see [35, page 1].

  * •

A vector field $v$ which derives from the gradient of a function is called
lamellar (also known as gradient, or globally potential) vector field. The
function is called the potential function. It follows that a vector field is
lamellar if and only if the system is potential, see [37, page 23] and [35,
page 1].

  * •

The three-dimensional vector field $v$ which has the representation of the
form $v=f\nabla g,$ in which $f$ and $g$ are functions, is named complex
lamellar. The vector field $v$ is complex lamellar if and only if
$v\cdot(\nabla\times v)=0,$ that is, this type of vector field is orthogonal
to its curl, see [37, page 23].

###  2.2 Poisson structure

As mentioned, the algebra of the first integral for $\mathscr{F}$ is spanned
by $\left\langle
x^{l+1}(y^{2}+z^{2})^{k-l+1}\right\rangle_{-1\leqslant{l}\leqslant{k}}^{1\leqslant
l+k}.$ In what follows, we extend this algebra to the Poisson structure for
$\mathscr{F}$ by equipping it with an appropriate Poisson bracket. In
accordance with this Poisson algebra, one representation of any vector fields
in $\mathscr{F}$ by the Hamiltonian equation is given.

Performing the change of variable $r=\rho^{2}$ into the vector field given by
(2.4) we obtain

$\displaystyle F^{l}_{k}$ $\displaystyle=$ $\displaystyle
x^{l}r^{k-l}\left((k-l+1)x\frac{\partial}{\partial
x}-(l+1)r\frac{\partial}{\partial r}\right).$ (2.6)

Now, define

$\displaystyle\mathcal{P}:=\left\\{\sum c^{l}_{k}f^{l}_{k}\mid-1\leqslant
l\leqslant k\right\\},$ (2.7)

where $f^{l}_{k}:=x^{l+1}r^{k-l+1}$ is the first integral of $F^{l}_{k}.$ Now,
we have the following result.

######  Theorem 2.3.

  * Consider the vector field $F^{l}_{k}$ given by (2.6) and the algebra $\mathcal{P}$ given by (2.7). The following statements hold.

  * 1.

$({\mathcal{P}},\\{\cdot,\cdot\\})$ is Poisson algebra where the Poisson
bracket is given by

$\\{f,g\\}:={\frac{\partial}{\partial r}}f\frac{\partial}{\partial
x}g-{\frac{\partial}{\partial r}}g\frac{\partial}{\partial
x}f,\qquad{\hbox{for all}}\,\,f,g\in\mathcal{P}.$ (2.8)

  * 2.

Hamilton’s equations of $F^{l}_{k}$ are

$\displaystyle F^{l}_{k}=\\{f^{l}_{k},x\\}\frac{\partial}{\partial
x}+\\{f^{l}_{k},r\\}\frac{\partial}{\partial r}.$ (2.9)

  * 3.

$({\mathcal{P}},\\{\cdot,\cdot\\})$ and $(\mathscr{F},[\cdot,\cdot])$ are
isomorphic Lie algebras where Lie isomorphism is defined by
$\varphi:({\mathcal{P}},\\{\cdot,\cdot\\})\rightarrow(\mathscr{F},[\cdot,\cdot]),$
with $\varphi({f}^{l}_{k})=F^{l}_{k}.$

###### Proof.

One can readily check that $({\mathcal{P}},\\{\cdot,\cdot\\})$ is a Poisson
algebra (see [25, Section 5] and [13, Chapter one ]). A direct calculation
using the Hamiltonian function given by the first item and Poisson bracket
given by (2.8) establish (2.9) (see also preface of [13]). The last claim
follows from the straightforward calculation of structure constants given by
(2.8) and [16, Lemma 2.5]. ∎

######  Corollary 2.4.

The class of Hopf-zero singular system given by (2.1) may be represented
through appropriate coordinates transformations to the form

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm d}t}&=\frac{\partial
H(x,r)}{\partial r}=\\{H(x,r),x\\},\\\ \frac{{\rm d}{r}}{{\rm
d}t}&=-\frac{\partial H(x,r)}{\partial x}=\\{H(x,r),r\\},\\\ \frac{{\rm
d}{\theta}}{{\rm d}t}&=G(x,r),\end{aligned}\right.$ (2.10)

where $H(x,r):=\sum a^{l}_{k}x^{l+1}{r}^{{k-l+1}}$ is a constant of the motion
and $G(x,r):=1+\sum b^{m}_{n}{x}^{{m}}r^{n-m}.$

###### Proof.

The corollary can be verified in two ways: $(1)$ recast the dynamical system
(1.1) by employing the coordinate change $r=\rho^{2}.$ Then, taking Hamilton’s
equations corresponding to $F^{l}_{k}$ given by Equation (2.9) into account,
immediately verify the statement of the corollary. Then $(2)$ follows from
[16, Remark 2.4] and the procedure given in the proof of [24, Theorem 2.2]
with the slight modification. In fact, replace $J$ by $2J,$ into that proof.
First by employing the change of variables
$y=\rho\cos(\theta),z=\rho\sin(\theta),$ and $x=x,$ the differential equation
(2.1) goes over into the system

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm d}t}&=\frac{\partial
K(x,\rho)}{2J\partial\rho},\\\ \frac{{\rm d}{\rho}}{{\rm d}t}&=-\frac{\partial
K(x,\rho)}{2J\partial x},\\\ \frac{{\rm d}{\theta}}{{\rm
d}t}&=G(x,\rho),\end{aligned}\right.$

where $J\\!=\rho$ is the Jacobian of the cylindrical transformation,
$K(x,\rho):=\sum a^{l}_{k}x^{l+1}{\rho}^{2({k-l+1})},$ and $G(x,\rho):=1+\sum
b^{m}_{n}{x}^{{m}}\rho^{2(n-m)}.$ To proceed further, carrying out the
transformation $r=\int 2J=\rho^{2}$ into the former system. This turns the
above system into the form given by (2.10), see also [24, Example 3]. The
corollary follows. ∎

###  2.3 Euler’s form for $\mathscr{L}$

Any solenoidal vector field may be represented by Euler’s form as

$\displaystyle v=h(g_{1},g_{2})\nabla g_{1}\times\nabla g_{2},$ (2.11)

where $h,g_{1},$ and $g_{2}$ are scalar valued functions. We remark that
$g_{1}$ and $g_{2}$ may not be defined everywhere on the domain of $v$. The
functions $g_{1}$ and $g_{2}$ are first integrals or vector sheets of vector
field $v$ and named Euler’s potentials. The geometrical meaning of Euler’s
form is that the vector lines of solenoidal vector fields are the intersection
of level surfaces of Euler’s potentials, see [37, page 22], Euler’s Theorem
[38, page 48], and [23].

In the following result, we express the vector fields in $\mathscr{L}$ by
local Euler’s potentials.

######  Theorem 2.5.

The Euler’s form of ${F}^{l}_{k}$ and $\Theta^{l}_{k}$ are given as follows

$\displaystyle{F}^{l}_{k}$ $\displaystyle=$
$\displaystyle\nabla\left(\frac{1}{2}\arctan({\frac{z}{y}})\right)\times\nabla\left(x^{l+1}(y^{2}+z^{2})^{k-l+1}\right),\qquad\hbox{for}\quad
y\neq 0,$ (2.12) $\displaystyle{{\Theta}^{l}_{k}}$ $\displaystyle=$
$\displaystyle\nabla\left(x^{l+1}\right)\times\nabla\left({\frac{-\left({y}^{2}+{z}^{2}\right)^{k-l+1}}{2\left(l+1\right)\left(k-l+1\right)}}\right).$
(2.13)

###### Proof.

We prove this theorem only for ${F}^{l}_{k}.$ The proof for $\Theta^{l}_{k}$
can be done analogously. Due to the solenoidal property of vector fields in
$\mathscr{L}$ and Euler’s Theorem [38, page 48], $F^{l}_{k}$ may be written as
(2.11). Setting the first integrals of $F^{l}_{k}$ as local Euler’s
potentials, in fact, $g_{1}:=x^{l+1}(y^{2}+z^{2})^{k-l+1}$ and
$g_{2}:=\frac{1}{2}\arctan({\frac{z}{y}}).$ Now, we show that $h(f_{1},f_{2})$
given by (2.11) equals one. By straightforward calculation, one has

$\displaystyle\nabla\left(\frac{1}{2}\arctan({\frac{z}{y}})\right)$
$\displaystyle=$
$\displaystyle{\frac{-z}{2({y}^{2}+{z}^{2})}}\cdot\mathbf{{e}_{y}}+{\frac{y}{2({y}^{2}+{z}^{2})}}\cdot\mathbf{{e}_{z}},$
$\displaystyle\nabla\left(x^{l+1}(y^{2}+z^{2})^{k-l+1}\right)$
$\displaystyle=$ $\displaystyle
x^{l}(y^{2}+z^{2})^{k-l+1}\Big{(}(l+1)(y^{2}+z^{2})\cdot\mathbf{{e}_{x}}+2(k-l+1)xy\cdot\mathbf{{e}_{y}}$
$\displaystyle+2(k-l+1)xz\cdot\mathbf{{e}_{z}}\Big{)}.$

By taking the cross product of the above vector fields and considering (2.2)
one finds

$\displaystyle\nabla\left(\frac{1}{2}\arctan({\frac{z}{y}})\right)\times\nabla\left(x^{l+1}(y^{2}+z^{2})^{k-l+1}\right)=F^{l}_{k}.$

This implies that $h(f_{1},f_{2})=1$ and concludes our assertion. ∎

###  2.4 Vector potential for $\mathscr{L}$

Another way that solenoidal vector field could be constructed, is by mean of
the vector potential. These vector fields are expressible from another vector
field by taking its curl. To find more information about the physical
significant of vector potential, see [22, 28, 36].

######  Theorem 2.6.

The vector fields ${F}^{l}_{k}$ and $\Theta^{l}_{k}$ could be expressed in the
following form.

$\displaystyle{F}^{l}_{k}$ $\displaystyle=$
$\displaystyle\nabla\times\left(\frac{1}{2}x^{l+1}(y^{2}+z^{2})^{k-l}({{-z}}\cdot\mathbf{{e}_{y}}+{{y}}\cdot\mathbf{{e}_{z}})\right),$
$\displaystyle\Theta^{l}_{k}$ $\displaystyle=$
$\displaystyle\nabla\times\left({\frac{-{x}^{l}\left({y}^{2}+{z}^{2}\right)^{k-l+1}}{2(k-l+1)}}\cdot\mathbf{{e}_{x}}\right).$

###### Proof.

The proof follows from [23, Equation 5] and Euler’s form given by Theorem 2.5.
∎

The vector potential is not unique since the curl of a gradient is zero. In
the following theorem we use an alternative approach to construct a vector
potential associated to the solenoidal vector fields $F^{l}_{k}$ and
$\Theta^{l}_{k}.$

######  Theorem 2.7.

The vector fields $F^{l}_{k}$ and $\Theta^{l}_{k}$ can be produced in the
following form as

$\displaystyle F^{l}_{k}$ $\displaystyle=$ $\displaystyle\nabla\times{\bf
A}^{l}_{k},$ (2.14) $\displaystyle\Theta^{l}_{k}$ $\displaystyle=$
$\displaystyle\nabla\times{\bf B}^{l}_{k},$ (2.15)

where the vector potentials ${\bf A}^{l}_{k}$ and ${\bf B}^{l}_{k}$ are
defined as follows

$\displaystyle{\bf A}^{l}_{k}$ $\displaystyle:=$
$\displaystyle\sum_{j=0}^{k-l}{\frac{{y}^{2j}{z}^{2(k-l-j)+1}{x}^{l}{k-l\choose
j}}{2(k-l-j)+1}}\left(\frac{l+1}{2}y\cdot\mathbf{{e}_{x}}+\left(k-l+1\right){x}\cdot\mathbf{{e}_{y}}\right),$
$\displaystyle{\bf B}^{l}_{k}$ $\displaystyle:=$
$\displaystyle\frac{{x}^{l}}{{2(k-l+1)}}\left({{({y}^{2}+{z}^{2})^{k-l+1}}}-{{{y}^{2(k-l+1)}}}\right)\cdot\mathbf{{e}_{x}}-{\frac{{x}^{l+1}{y}^{2k-2l+1}}{l+1}}\cdot\mathbf{{e}_{y}}.$

###### Proof.

To prove the theorem, we follow the approach given in [34]. By solving the
equality $F^{l}_{k}=\nabla\times{{\bf A}^{l}_{k}},$ we obtain the following
equalities

$\displaystyle\frac{\partial}{\partial z}{{\bf
A}^{l}_{k}}\cdot\mathbf{{e}_{x}}=dy(F^{l}_{k}),\qquad-\frac{\partial}{\partial
z}{{\bf
A}^{l}_{k}}\cdot\mathbf{{e}_{y}}=dx(F^{l}_{k}),\qquad\frac{\partial}{\partial
x}{{\bf A}^{l}_{k}}\cdot\mathbf{{e}_{y}}-\frac{\partial}{\partial y}{{\bf
A}^{l}_{k}}\cdot\mathbf{{e}_{x}}=dz(F^{l}_{k}).$ (2.16)

The first identity of (2.16) implies that

$\displaystyle{{\bf A}^{l}_{k}}\cdot\mathbf{{e}_{x}}$ $\displaystyle=$
$\displaystyle-\frac{l+1}{2}x^{l}\sum_{j=0}^{k-l}\frac{{k-l\choose
j}{y}^{2j+1}{z}^{2(k-l-j)+1}}{2(k-l-j)+1}+p_{1}(x,y),$ (2.17)

analogously the second identity of (2.16) implies that

$\displaystyle{{\bf
A}^{l}_{k}}\cdot\mathbf{{e}_{y}}=-(k-l+1)x^{l+1}\sum_{j=0}^{k-l}\frac{{k-l\choose
j}{y}^{2j}{z}^{2(k-l-j)+1}}{2(k-l-j)+1}+p_{2}(x,y),$ (2.18)

where $p_{1}(x,y)$ and $p_{2}(x,y)$ are functions. Now by substituting (2.17)
and (2.18) into the last identity of (2.16) one can find,
$\frac{\partial}{\partial z}p_{1}(x,y)-\frac{\partial}{\partial
x}p_{2}(x,y)=0.$ Thus, without loss of generality we may assume that
$p_{1}(x,y)=p_{2}(x,y)=0,$ the equality (2.14) concludes. By repeating this
procedure for $\varTheta^{l}_{k}$ we can verify equality (2.15). ∎

###  2.5 Clebsch representation for $\mathscr{L}$

In this part, we shall study the following problems: namely, we shall study
the non-potential property of Hopf-zero singularities from $\mathscr{L},$ and
we shall present an alternative representation that is the Clebsch
representation of any vector fields from $\mathscr{L}.$ Based on this
representation for a given vector field $v\in\mathbb{R}^{3},$ for any point
$x\in\mathbb{R}^{3}$ in which $\nabla\times v\neq 0,$ one can find three
scalar valued functions $f_{1},f_{2},$ and $f_{3}$ such that

$\displaystyle v=f_{1}\nabla f_{2}+\nabla f_{3}.$ (2.19)

These scalar valued functions are called Clebsch potentials of $v.$ Noting
that $f_{1},f_{2}$ and $f_{2}$ may not be defined everywhere on the domain of
$v.$ Moreover, this representation shows that any vector fields may be
presented by summation of the lamellar field and complex lamellar field, see
Definition 2.2. For those interested in knowing that how this representation
is constructed, we include the following discussion from the proof of
Clebsch’s Theorem of [38, Section 2.4].

Since $\nabla\times v$ is a solenoidal vector field then, there exist local
Euler’s potentials $f_{1}$ and $f_{2}$ such that $\nabla\times v=\nabla
f_{1}\times\nabla f_{2}.$ Making use of the vector calculus identity we have
$\nabla\times(v-f_{1}\nabla f_{2})=0.$ Hence, $v-f_{1}\nabla f_{2}$ is
irrotational vector field, then there exists potential function $f_{3}$ such
that $v-f_{1}\nabla f_{2}=\nabla f_{3},$ see Definition 2.2. Then, the
relation (2.19) follows.

For the Euler’s form (and consequently the representation given by vector
potential) to be possible, the vector field $v$ has to be solenoidal. We
remark that deriving the vector field $v$ from Clebsch potentials
$f_{1},f_{2},$ and $f_{3}$ through the relation (2.19), is not related to the
solenoidal property of vector field $v.$ In fact, any vector field may be
represented in the Clebsch representation, see [38, Section 2.4].

######  Theorem 2.8.

Any $v\in\mathscr{L}$ with Hopf-zero singularity is the non-potential vector
field.

###### Proof.

Recall from [16], the grading function for generators of $\mathscr{L}$ given
by $\delta({\rm F}^{l}_{k})=\delta({\Theta}^{l}_{k})=k.$ Suppose that the non-
zero Hopf-zero vector field $v=\sum_{j=0}^{\infty}v_{j}\in\mathscr{L}$ where
$\delta(v_{j})=j$ is given. Assume that the claim of the theorem does not
hold. Similar to argument given in [19], since the vector fields with
different grades do not have any monomial in common, it implies that
$\nabla\times v=0$ if and only if $\nabla\times v_{j}=0,$ for all
$j\in\mathbb{N}_{0}.$ Thus without loss of generality, it suffices to show
that $\nabla\times v_{j}=0.$ Taking into account the defined grading function,
the vector field $v_{j}$ may be represented by

$v_{j}:=\sum_{i=-1}^{\lfloor\frac{j}{2}\rfloor}a_{i}{F}^{i}_{j}+\sum_{i=0}^{\lfloor\frac{j}{2}\rfloor}b_{i}\Theta^{i}_{j},$

where $a_{i}$ and $b_{i}$ are real constants for all $i.$ Applying the curl
operator to $v_{j}$ yields

$\displaystyle\nabla\times v_{j}$ $\displaystyle=$
$\displaystyle\sum_{i=-1}^{\lfloor\frac{j}{2}\rfloor}(0,0,a_{i}\frac{i({i}+1)}{2}{x}^{{i}-1}{\rho}^{2(j-{i})+1}+2a_{i}(j-i)(j-i+1){x}^{{i}+1}{\rho}^{2(j-{i})-1})$
$\displaystyle-\sum_{i=0}^{\lfloor\frac{j}{2}\rfloor}\left(ib_{i}{x}^{{i}-1}{\rho}^{2(j-{i})},0,2(j-i+1)b_{i}{x}^{{i}}{\rho}^{2(j-{i})-1}\right).$

Hence, the only way that the foregoing relation vanishes is either $v_{j}=0$
or $v_{0}=x\frac{\partial}{\partial
x}-\frac{1}{2}\rho\frac{\partial}{\partial\rho}.$ This implies that either
$v=0$ or the vector field is not Hopf-zero singularity, which are in
contradiction to our assumption. Hence, the claim is proved. ∎

In the terminology of Definition 2.2, this result shows that the solenoidal
family of Hopf-zero singularities are not lamellar vector fields. In this
sense, these vector fields can not be expressed by the gradient of the scalar
valued function. In what follows, the Clebsch representation of $\mathscr{L}$
is given.

######  Theorem 2.9.

  * The following hold.

  * •

For given $F^{l}_{k}\in\mathscr{F}$ there exist Clebsch potentials
$f_{1},f_{2}$, and $f_{3}$ such that

$\displaystyle{F}^{l}_{k}=f_{1}\nabla f_{2}+\nabla f_{3},$ (2.20)

where for $l\neq 0,$

$\displaystyle f_{1}$ $\displaystyle:=$ $\displaystyle x^{l},$ $\displaystyle
f_{2}$ $\displaystyle:=$
$\displaystyle-\frac{\left(l+1\right)}{4({k-l+1})}{{\left({y}^{2}+{z}^{2}\right)^{k-l+1}}}-\frac{\left(k-l+1\right)}{l}{{{x}^{2}\left({y}^{2}+{z}^{2}\right)^{k-l}}},$
(2.21) $\displaystyle f_{3}$ $\displaystyle:=$
$\displaystyle\frac{1}{l}{{\left({y}^{2}+{z}^{2}\right)^{k-l}\left(k-l+1\right){x}^{l+2}}}.$

Otherwise

$\displaystyle f_{1}:=-x,\quad f_{2}:=(k+1)x(y^{2}+z^{2})^{k},\quad
f_{3}:=-(k+1)x^{2}(y^{2}+z^{2})^{k}+\frac{1}{4(k+1)}(y^{2}+z^{2})^{k+1}.$

  * •

For each vector field ${\Theta}^{l}_{k}\in\mathscr{T}$ there exist local
Clebsch potential $g_{1}$ and global Clebsch potentials $g_{2},g_{3}$ as
follows

$\displaystyle g_{1}:={\frac{z}{y}},\,\,\,\qquad
g_{2}:={y}^{2}{x}^{l}\left({y}^{2}+{z}^{2}\right)^{k-l},\qquad
g_{3}:=-yz{x}^{l}\left({y}^{2}+{z}^{2}\right)^{k-l},\qquad\hbox{for}\quad
y\neq 0,$

such that

${\Theta}^{l}_{k}=g_{1}\nabla g_{2}+\nabla g_{3}.$

###### Proof.

Following the discussions at the beginning of this part, first we need to find
the Euler’s form for $\nabla\times F^{l}_{k}.$ Writing $\nabla\times{\rm
F}^{l}_{k},$ given by the preceding theorem using Cartesian coordinates gives

$\displaystyle\nabla\times F^{l}_{k}$ $\displaystyle=$
$\displaystyle\left({y}^{2}+{z}^{2}\right)^{k-l-1}{x}^{l-1}\left(\frac{l(l+1)}{2}\left({y}^{2}+{z}^{2}\right)+2\left(k-l\right)\left(k-l+1\right){x}^{2}\right)(z\cdot\mathbf{{e}_{y}}-y\cdot\mathbf{{e}_{z}}).$

Suppose that $l\neq 0,$ then by straightforward calculation one can verify
that $\nabla\times F^{l}_{k}=\nabla f_{1}\times\nabla f_{2},$ where
$f_{1},f_{2}$ are given by Equation (• ‣ 2.9). In order that (2.20) to be true
the following equalities must hold

$\displaystyle dx(F^{l}_{k})-f_{1}\nabla f_{2}\cdot\mathbf{{e}_{x}}=\nabla
f_{3}\cdot\mathbf{{e}_{x}},$ (2.22) $\displaystyle dy(F^{l}_{k})-f_{1}\nabla
f_{2}\cdot\mathbf{{e}_{y}}=\nabla f_{3}\cdot\mathbf{{e}_{y}}.$ (2.23)

Equation (2.22) follows after some computations that

$\displaystyle
f_{3}={x}^{l}\left({y}^{2}+{z}^{2}\right)^{k-l}\left({\frac{\left(k-l+1\right){x}^{2}}{l+1}}-{\frac{{y}^{2}+{z}^{2}}{4(k-l+1)}}\right)+h(y,z).$

Substituting $f_{3}$ in (2.23) results in $h(y,z)=0.$ In this way, we obtain
the expression (2.20). If $l=0$ one can check that

$\displaystyle F^{0}_{k}$ $\displaystyle=$
$\displaystyle(y^{2}+z^{2})^{k}\left((k+1)x\cdot\mathbf{{e}_{x}}-\frac{1}{2}y\cdot\mathbf{{e}_{y}}-\frac{1}{2}z\cdot\mathbf{{e}_{z}}\right)$
$\displaystyle=$
$\displaystyle-x\nabla\left((k+1)x(y^{2}+z^{2})^{k}\right)+\nabla\left(-(k+1)x^{2}(y^{2}+z^{2})^{k}+\frac{1}{4(k+1)}(y^{2}+z^{2})^{k+1}\right).$

The proof for $\Theta^{l}_{k}$ is analogous to the proof of ${F}^{l}_{k}.$ ∎

##  3 Solenoidal conservative ${{\mathbb{Z}_{2}}}$-equivariant Lie algebra

In this section, we shall introduce the solenoidal conservative Lie algebra
associated to the classical normal form of Hopf-pitchfork singularities (1.1).
We also recall the theory of unique normal form at the end of this part.

To start, we recall the following definition from [21, Chapter XII ].

######  Definition 3.1.

Let $\dot{x}=v$ with $x\in\mathbb{R}^{n},$ be an autonomous dynamical system
where $v,$ is smooth. Let ${\Gamma},$ be a compact Lie group in ${\rm Gl}(n)$.
This system is called ${\Gamma}$-equivariant if $v(\gamma x)=\gamma v$ for all
$\gamma\in{\Gamma}$ and $x\in\mathbb{R}^{n}.$

Define

$\displaystyle{\rm H}^{{l}}_{k}$ $\displaystyle:=$
$\displaystyle(k-{2l}+1){x}^{{2l}+1}{\rho}^{2(k-{2l})}\frac{\partial}{\partial
x}-\frac{({2l}+1)}{2}{x}^{2l}{\rho}^{2(k-{2l})+1}\frac{\partial}{\partial\rho},$
(3.1) $\displaystyle{\varTheta}^{l}_{k}$ $\displaystyle:=$
$\displaystyle{x}^{{2l}}\rho^{2(k-{2l})}\frac{\partial}{\partial\theta},$
(3.2)

where $0\leqslant 2l\leqslant k.$ These vector fields are invariant under the
linear map

$\displaystyle\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}:\qquad(x,y,z)\mapsto(-x,-y,-z).$
(3.3)

However, the foregoing vector fields are ${{\mathbb{Z}_{2}}}$-equivariant
version of those that are given by equations (2.4) and (2.5), in order to
avoid any confusion, we would like to change the notations. Thereby, instead
of $F$-terms and $\Theta$-terms we shall write ${\rm H}$-terms and
$\varTheta$-terms.

Denote

${\mathscr{F}}^{\mathbb{Z}_{2}}:=\Big{\langle}\sum a^{l}_{k}{\rm
H}^{l}_{k}\;|\;a^{l}_{k}\in\mathbb{R},{0\leqslant 2l\leqslant k,1\leqslant
k}\Big{\rangle}$ (3.4)

and

${\mathscr{T}}^{{{\mathbb{Z}_{2}}}}:=\Big{\langle}\sum
b^{l}_{k}{\varTheta}^{l}_{k}\;|\;b^{l}_{k}\in\mathbb{R},{0\leqslant
2l\leqslant k,0\leqslant k}\Big{\rangle}.$ (3.5)

Then we define the maximal Lie algebra of solenoidal conservative
${{\mathbb{Z}_{2}}}$-equivariant classical normal form of Hopf-zero vector
fields by
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}:={\mathscr{F}}^{{{\mathbb{Z}_{2}}}}\oplus{\mathscr{T}}^{{{\mathbb{Z}_{2}}}}.$
Since ${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}$ is a Lie subalgebra of
$\mathscr{L},$ then ${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}$ inherits the
geometrical properties such as conservation, incompressibility, and
rotationality, from $\mathscr{L}.$ See also [16, Theorem 2.4].

Our next task is to present the structure constants for
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}.$

######  Lemma 3.2.

The following relations always hold.

$\displaystyle{[{\rm H}^{{l}}_{k},{\rm H}^{{m}}_{n}]}$ $\displaystyle=$
$\displaystyle\left((2m+1)(k+2)-(2l+1)(n+2)\right){\rm H}^{l+m}_{k+n},$
$\displaystyle{[{\rm H}^{{l}}_{k},{\varTheta}^{{m}}_{n}]}$ $\displaystyle=$
$\displaystyle\left(2m(k+2)-n(2l+1)\right){\varTheta}^{l+m}_{k+n},$
$\displaystyle{[{\varTheta}^{{l}}_{k},{\varTheta}^{{m}}_{n}]}$
$\displaystyle=$ $\displaystyle 0.$

###### Proof.

The proof follows from [16, Lemma 2.5]. ∎

The following specific cases of preceding result will be useful in the rest of
the paper.

$\displaystyle{[{\rm H}^{0}_{1},{\rm H}^{m}_{n}]}$ $\displaystyle=$
$\displaystyle\left(6m-n+1\right){\rm H}^{m}_{{n+1}},$ (3.6)
$\displaystyle{[{\rm H}^{0}_{1},{\varTheta}^{m}_{n}]}$ $\displaystyle=$
$\displaystyle\left(6m-n\right){\varTheta}^{m}_{{n+1}}.$ (3.7)

The remainder of this section is devoted to revisiting a concise but detailed
outline of the theory and method concerning finding the unique normal form for
${\Gamma}$-equivariant singularities.

  * •

Denote ${\rm ad}(u){v}$ for ${\rm ad}(u){v}=uv-vu,$ where $u$ and $v$ are two
arbitrary vector fields.

  * •

Denote ${\mathscr{L}}^{\Gamma}$ for the space of all ${\Gamma}$-equivariant
vector fields.

  * •

Denote ${{{\mathscr{L}}_{k}}}^{\Gamma}$ for the space of
${\Gamma}$-equivariant graded Lie algebra with grade $k.$
${\Gamma}$-equivariant graded Lie algebra means that ${\rm
ad}(v_{i})v_{j}\in{{{\mathscr{L}}_{i+j}}}^{\Gamma},$ for arbitrary
$v_{i}\in{{{\mathscr{L}}_{i}}}^{\Gamma}$ and
$v_{j}\in{{{\mathscr{L}}_{j}}}^{\Gamma}.$

Consider the differential equation $v_{s}=\sum^{\infty}_{i=0}v_{i},$ where
$v_{s}\in{\mathscr{L}}^{\Gamma}.$ First, we define a linear map to find the
first level normal form as follows

$\displaystyle
d^{k,1}_{s}:{{{\mathscr{L}}_{k}}}^{\Gamma}\rightarrow{{{\mathscr{L}}_{k}}}^{\Gamma},$
$\displaystyle d^{k,1}_{s}(Y_{k}):={\rm ad}(Y_{n}){v_{0}}.$

The first level normal form is given by
$v^{(1)}=\sum^{\infty}_{i=0}v^{(1)}_{i},$ where
$v^{(1)}_{i}\in\mathcal{C}^{k,1}$ for all $i$ and $\mathcal{C}^{k,1}$ is the
complement space to ${\rm Im}\,(d^{k,1}_{s}).$ Note that in order to preserve
the symmetric structure of $v_{s}$, the transformations should be taken from
${{{\mathscr{L}}_{k}}}^{\Gamma}.$ Proceeding inductively, we define

$\displaystyle d^{k,n}_{s}:{{\mathscr{L}}_{k}}^{\Gamma}\times\ker
d^{k,n-1}_{s}\rightarrow{{\mathscr{L}}_{k}}^{\Gamma},$ $\displaystyle
d^{k,n}_{s}(Y^{k}_{s},Y^{k-1}_{s},\ldots,Y^{k-n+1}_{s}):=\sum^{n-1}_{i=0}{\rm
ad}(Y^{k-i}_{s})v_{i},\quad\hbox{ for any }n\leqslant k.$

Then, there exists the complement subspace $\mathcal{C}^{k,n}$ such that ${\rm
Im}\,(d^{k,n}_{s})\oplus\mathcal{C}^{k,n}={{\mathscr{L}}_{k}}^{\Gamma},$ where
$\mathcal{C}^{k,n}$ follows the normal form style. Then the $n$-level normal
form of $v_{s}$ is given by $w_{s}=\sum^{\infty}_{i=0}w_{i},$ where
$w_{i}\in\mathcal{C}^{k,n}$ for all $i.$ For the more detailed treatment of
this theory, we refer to [20, 31, 27] and [32, sections 9-13].

##  4 Normal form

In this section, we examine the unique normal form of dynamical system (1.1).
We closely follow the approach in [17, 8, 16]. In terms of $\rm H$-terms and
$\varTheta$-terms as introduced in (3.1) and (3.2), this system leads to the
following expression

$w^{(1)}:=\varTheta^{0}_{0}+b^{0}_{1}{\varTheta}^{0}_{1}+a^{0}_{1}{\rm
H}^{0}_{1}+\sum a^{l}_{k}{\rm H}^{l}_{k}+\sum
b^{l}_{k}{\varTheta}^{l}_{k},\quad\hbox{for}\,\,0\leqslant 2l\leqslant
k,1\leqslant k,$ (4.1)

where $a^{0}_{1}\neq 0,$ $a^{l}_{k},$ and $b^{l}_{k}$ are real constants which
could be computed explicitly in terms of the coefficients of the original
system using our Maple program. We shall depart from the foregoing system.

######  Lemma 4.1.

Given the dynamical system defined by (4.1), there exists a sequence of
${{\mathbb{Z}_{2}}}$-equivariant transformations that send the system to the
following second level normal form

$\displaystyle w^{(2)}$ $\displaystyle:=$
$\displaystyle{\varTheta}^{0}_{0}+b^{(2)}_{0}{\varTheta}^{0}_{1}+a^{(2)}_{1}{\rm
H}^{0}_{1}+\sum_{i=1}^{\infty}a^{(2)}_{i}{\rm
H}^{i}_{2i}+\sum_{i=1}^{\infty}b^{(2)}_{i}{\varTheta}^{i}_{2i},$ (4.2)

in which all of the coefficients are real constants.

###### Proof.

Let $\delta$ be grading function defined by $\delta({\rm
H}^{m}_{n}):=n,\delta({\varTheta}^{m}_{n}):=n+1.$ Using the structure
constants given by (3.6) and (3.7) one can deduce that $\mathcal{C}^{k,2}={\rm
span}\\{{\varTheta}^{0}_{1},{\rm
H}^{m}_{2m},{\varTheta}^{m}_{2m}\,|\,m\geqslant 1\\}$ for any $k\geqslant 1$
and the lemma follows. ∎

Similar to [16], one can remove ${\varTheta}^{0}_{0}$ using the linear change
of variables, for further details, see [27, Lemma 5.3.6] and [17]. In what
follows, we intend to simplify the second level normal form; $w^{(2)}.$ First,
nonetheless, we need some notational conventions.

Suppose that there exists a non-zero ${\rm H}^{l}_{2l}$ in $w^{(2)}$ for some
$l$ and there exists a non-zero ${\varTheta}^{k}_{2k}$ in $w^{(2)}$ for some
$k.$ Then, define

$\displaystyle\mathtt{r}:=\min\\{l\,|\,a^{(2)}_{l}\neq 0,l\geqslant
1\\},\qquad\quad\mathtt{s}:=\min\\{k\,|\,b^{(2)}_{k}\neq 0,k\geqslant 1\\}.$

Define the following grading function

$\displaystyle\delta({\rm
H}^{m}_{n}):=\mathtt{r}(n-2m)+m,\,\,\,\,\,\delta({\varTheta}^{m}_{n}):=\mathtt{r}(n-2m)+\mathtt{r}+m+1.$

Let us denote the leading order term of (4.2) by $\mathbb{H}_{\mathtt{r}}.$
According to the above grading function, we have that

$\displaystyle\mathbb{H}_{\mathtt{r}}:={\rm
H}^{0}_{1}+a^{(2)}_{\mathtt{r}}{\rm H}^{\mathtt{r}}_{2\mathtt{r}}.$ (4.3)

This vector field plays a prominent role in the sequel. By performing re-
scaling $x$ in (4.2) as

$\displaystyle
x\rightarrow\left|{\frac{a_{\mathtt{r}}}{a^{(2)}_{\mathtt{r}}}}\right|^{\frac{1}{2{\mathtt{r}}}}x,$

then $\mathbb{H}_{{\mathtt{r}}}$ could be replaced by

$\displaystyle\mathbb{H}_{\mathtt{r}}:={\rm H}^{0}_{1}+a_{\mathtt{r}}{\rm
H}^{\mathtt{r}}_{2\mathtt{r}},$ (4.4)

which implies that $a_{r}$ could be taken an arbitrary real constant.
Henceforth, for simplicity, we set $a_{\mathtt{r}}=1.$

For further reduction of (4.2), the following lemma is quite useful.

######  Lemma 4.2.

For each ${\rm H}^{m}_{n}\in{\mathscr{F}}^{{{\mathbb{Z}_{2}}}}$ and
${\varTheta}^{m}_{n}\in{\mathscr{T}}^{{{\mathbb{Z}_{2}}}}$ there exist
transformations $\mathfrak{H}^{m}_{n}$ and $\mathfrak{R}^{m}_{n}$, such that
the following hold.

$\displaystyle[\mathfrak{H}^{m}_{n},\mathbb{H}_{{\mathtt{r}}}]+{\rm
H}^{m}_{n}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}(4m{\mathtt{r}}-2n{\mathtt{r}}+4m-n+1)^{n-2m}_{4{\mathtt{r}}+1}}{(6m-n+2)^{n-2m}_{4{\mathtt{r}}+1}}{\rm
H}^{n{\mathtt{r}}+m-2m{\mathtt{r}}}_{2(n{\mathtt{r}}+m-2m{\mathtt{r}})},$
(4.5)
$\displaystyle{[\mathfrak{R}^{m}_{n},\mathbb{H}_{{\mathtt{r}}}]+{\varTheta}^{m}_{n}}$
$\displaystyle=$
$\displaystyle\frac{(-1)^{n}(4m{\mathtt{r}}-2n{\mathtt{r}}+4m-n+2{\mathtt{r}}+1)^{n-2m-1}_{4{\mathtt{r}}+1}}{2(6m-n+1)^{n-2m-1}_{4{\mathtt{r}}+1}}{\varTheta}^{n{\mathtt{r}}+m-2m{\mathtt{r}}}_{2(n{\mathtt{r}}+m-2m{\mathtt{r}})}.$
(4.6)

###### Proof.

Define

$\displaystyle\mathfrak{H}^{m}_{n}$ $\displaystyle:=$
$\displaystyle\sum_{j=0}^{n-2m-1}\frac{(-1)^{j+1}(4m{\mathtt{r}}-2n{\mathtt{r}}+4m-n+1)^{j}_{4{\mathtt{r}}+1}}{(6m-n+2)^{j+1}_{4{\mathtt{r}}+1}}{\rm
H}^{j{\mathtt{r}}+m}_{2j{\mathtt{r}}-j+n-1},$
$\displaystyle\mathfrak{R}^{m}_{n}$ $\displaystyle:=$
$\displaystyle\sum_{j=0}^{n-2m-1}\frac{(-1)^{j+1}(4m{\mathtt{r}}-2n{\mathtt{r}}+4m-n+2{\mathtt{r}}+1)^{j}_{4{\mathtt{r}}+1}}{2(6m-n+1)^{j+1}_{4{\mathtt{r}}+1}}{\varTheta}^{j{\mathtt{r}}+m}_{2j{\mathtt{r}}-j+n-1}.$

Then, substitute the previous transformations into the left hand side of (4.5)
and (4.6), respectively. The right hand side could be obtained readily. ∎

The following theorem is the main result of this section.

######  Theorem 4.3.

The unique normal form of system (1.1) under assumption $a^{0}_{1}\neq 0$ is
given by

$\displaystyle w^{(\infty)}$ $\displaystyle:=$
$\displaystyle{\varTheta}^{0}_{0}+{\rm H}^{0}_{1}+{\rm
H}^{{\mathtt{r}}}_{2{\mathtt{r}}}+b_{0}{\varTheta}^{0}_{1}+\sum_{i={\mathtt{r}}+1}^{\infty}a_{i}{\rm
H}^{i}_{2i}+\sum_{i={\mathtt{s}}}^{\infty}b_{i}{\varTheta}^{i}_{2i},$ (4.7)

or equivalently in the cylinder coordinates $w^{(\infty)}$ takes the form

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm
d}t}&=2x\rho^{2}+{x}^{{2\mathtt{r}}+1}+\sum_{i={\mathtt{r}}+1}^{\infty}a_{i}{x}^{{2i}+1},\\\
\frac{{\rm d}{\rho}}{{\rm
d}t}&=-\frac{1}{2}\rho^{3}-\frac{({2\mathtt{r}}+1)}{2}{x}^{2\mathtt{r}}{\rho}-\sum_{i={\mathtt{r}}+1}^{\infty}\frac{({2i}+1)}{2}a_{i}{x}^{2i}{\rho},\\\
\frac{{\rm d}{\theta}}{{\rm
d}t}&=1+b_{0}\rho^{2}+b_{\mathtt{s}}{x}^{{2\mathtt{s}}}+\sum_{i={\mathtt{s}}+1}^{\infty}b_{i}{x}^{{2i}},\end{aligned}\right.$

where $b_{i}=0$ for all $i\equiv_{4{\mathtt{r}}+1}(\mathtt{s}+\mathtt{r})$ and
$\mathtt{s}\not\equiv_{4{\mathtt{r}}+1}0.$ The corresponding first integral of
the unique normal form is given by

$\displaystyle s^{(\infty)}(x,\rho)$ $\displaystyle:=$ $\displaystyle
x\rho^{4}+{\rho}^{2}x^{2\mathtt{r}+1}+\sum_{i=\mathtt{r}+1}^{\infty}a_{i}{\rho}^{2}x^{2i+1}.$
(4.8)

###### Proof.

Due to special structure constants given by equations (3.6) and (3.7) we
conclude that the generators of $\ker({\rm ad}_{{\rm H}^{0}_{1}})$ are
$\\{{\rm H}^{k}_{6k+1},{\varTheta}^{k}_{6k}\\}$ for all $k\in\mathbb{N}.$
Applying the method given at the end of Section 3 and using Lemma 4.2, we have
that

$\displaystyle{[{\rm H}^{k}_{6k+1},{\rm
H}^{{\mathtt{r}}}_{2{\mathtt{r}}}]+[\mathfrak{H}^{{\mathtt{r}}+k}_{2{\mathtt{r}}+6k+1},\mathbb{H}_{r}]}$
$\displaystyle=$
$\displaystyle\frac{(4{\mathtt{r}}+1)(-2k)^{4k+2}_{1}}{(4k+1)!}{\rm
H}^{4{\mathtt{r}}k+k+2{\mathtt{r}}}_{8{\mathtt{r}}k+2k+4{\mathtt{r}}}=0,$
$\displaystyle{[{\rm
H}^{k}_{6k+1},{\varTheta}^{{\mathtt{s}}}_{2{\mathtt{s}}}]+[\mathfrak{R}^{{\mathtt{s}}+k}_{6k+2{\mathtt{s}}+1},\mathbb{H}_{r}]}$
$\displaystyle=$
$\displaystyle{\frac{-2{\mathtt{s}}\left(2k+1\right)(2{\mathtt{s}}-2k-8{\mathtt{r}}k)_{4{\mathtt{r}}+1}^{4k}}{(4{\mathtt{s}})^{4k}_{4{\mathtt{r}}+1}}}{\varTheta}^{4{\mathtt{r}}k+k+{\mathtt{s}}+{\mathtt{r}}}_{8{\mathtt{r}}k+2k+2{\mathtt{s}}+2{\mathtt{r}}}.$

Thus, the above relations imply that ${\rm H}^{k}_{6k+1}$ can not eliminate
any ${\rm H}$-terms and ${\varTheta}^{m}_{2m}\in{\rm
Im}\,(d^{m,{\mathtt{s}}+1}_{s})$ for any
$m\equiv_{4{\mathtt{r}}+1}({\mathtt{r}}+{\mathtt{s}}),$ where
$\mathtt{s}\not\equiv_{4{\mathtt{r}}+1}0.$ Furthermore, one has

$\displaystyle{[\varTheta^{k}_{6k},{\rm
H}^{{\mathtt{r}}}_{2{\mathtt{r}}}]}+{[{\mathfrak{R}}^{k+{\mathtt{s}}}_{6k+2{\mathtt{s}}},\mathbb{H}_{r}]}$
$\displaystyle=$
$\displaystyle{\frac{-k\left(4{\mathtt{r}}+1\right)(1-2k)^{4k-1}_{1}}{({4k-1})!}}{\varTheta}^{4{\mathtt{r}}k+k+2{\mathtt{r}}}_{8{\mathtt{r}}k+2k+4{\mathtt{r}}}=0,$

which turns out that $\varTheta^{k}_{6k}$ generates a symmetry for the unique
normal form of (1.1). This completes the proof. ∎

We close this section by giving four representations of (4.7) based on the
given discussions in Section 2.

######  Theorem 4.4.

The unique normal form of (1.1) can be expressed in the following
representations.

  * 1.

The unique normal form can be presented in the following form

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm d}t}&=\frac{\partial
H(x,r)}{\partial r},\\\ \frac{{\rm d}{r}}{{\rm d}t}&=-\frac{\partial
H(x,r)}{\partial x},\\\ \frac{{\rm d}{\theta}}{{\rm
d}t}&=1+b_{0}\rho^{2}+b_{\mathtt{s}}{x}^{{2\mathtt{s}}}+\sum_{i={\mathtt{s}}+1}^{\infty}b_{i}{x}^{{2i}},\end{aligned}\right.$

in which
$H(x,r):=xr^{2}+a_{\mathtt{r}}rx^{2\mathtt{r}+1}+\sum_{i=\mathtt{r}+1}^{\infty}a_{i}x^{2i+1}r$
and $r=\rho^{2}.$

  * 2.

Euler’s form for $y\neq 0$

$\displaystyle w^{(\infty)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nabla\left(x(y^{2}+z^{2})^{2}\right)\times\nabla\left(\arctan({\frac{z}{y}})\right)-\frac{1}{2}\nabla\left({{{y}^{2}+{z}^{2}}}\right)\times\nabla
x+\frac{b_{0}}{4}\nabla\left({({y}^{2}+{z}^{2})^{2}}\right)\times\nabla x$
$\displaystyle+\sum_{i={\mathtt{r}}}^{\infty}\frac{a_{i}}{2}\nabla\left((y^{2}+z^{2})x^{2i+1}\right)\times\nabla\left(\arctan({\frac{z}{y}})\right)-\sum_{i={\mathtt{s}}}^{\infty}\frac{b_{i}}{2(2i+1)}\nabla\left({{({y}^{2}+{z}^{2})}}\right)\times\nabla\left(x^{2i+1}\right).$

  * 3.

Vector potential

$w^{(\infty)}={\nabla}\times{\bf A},$

where

$\displaystyle{\bf A}$ $\displaystyle=$
$\displaystyle-{\frac{1}{2}x({y}^{2}+{z}^{2})}(z\cdot\mathbf{{e}_{y}}-y\cdot\mathbf{{e}_{z}})-{{\frac{1}{2}({y}^{2}+{z}^{2})}}\cdot\mathbf{{e}_{x}}-\frac{b_{0}({y}^{2}+{z}^{2})^{2}}{4}\cdot\mathbf{{e}_{x}}$
$\displaystyle-\sum_{i={\mathtt{r}}}^{\infty}{\frac{a_{i}}{2}z}{x^{2i+1}\left(\mathbf{{e}_{y}}+{{y}}\cdot\mathbf{{e}_{z}}\right)}-\sum_{i={\mathtt{s}}}^{\infty}{{\frac{b_{i}}{2}({y}^{2}+{z}^{2}){x}^{2i}}}\cdot\mathbf{{e}_{x}}.$

  * 4.

Monge representation or Clebsch representation for $y\neq 0$

$\displaystyle w^{(\infty)}$ $\displaystyle=$
$\displaystyle-\left(\nabla\left(x^{2}-\frac{1}{4}(y^{2}+z^{2})\right)+x\nabla
x\right)-\left(\nabla\left(zy({y}^{2}+{z}^{2})\right)-{\frac{z}{y}}\nabla\left({y}^{2}\left({y}^{2}+{z}^{2}\right)\right)\right)$
$\displaystyle-\left(\nabla\left(zy\right)-{\frac{z}{y}}\nabla\left({y}^{2}\right)\right)+\sum_{i={\mathtt{r}}}^{\infty}a_{i}\left(\frac{1}{2i}\nabla\left({{x}^{2i+2}}\right)-x^{2i}\nabla\left(\frac{1}{4}\left(2\,i+1\right)\left({y}^{2}+{z}^{2}\right)+\frac{1}{2i}{{{x}^{2}}}\right)\right)$
$\displaystyle-\sum_{i={\mathtt{s}}}^{\infty}b_{i}\left(\nabla\left(zy{x}^{2i}\right)-{\frac{z}{y}}\nabla\left({y}^{2}{x}^{2i}\right)\right).$

For all representations are given above the constants $b_{i}$ for all
$i\in\mathbb{N}_{0},$ satisfy the conditions that are given in the Theorem
4.3.

###### Proof.

Follow Corollary 2.4, theorems 2.5, 2.6, and 2.9, respectively. See [19] for
the relevant result. ∎

##  5 Practical formulas

In this section, we would like to give some fruitful formulas which are
fundamentally significant for applications. First, some necessary relations
between the coefficients of given Hopf-pitchfork differential system are
provided. These relations guarantee that the classical normal form of the
given system up to third order belongs to
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}.$ Then, several formulas for the unique
normal form’s coefficients of the system
$w\in{\mathscr{L}}^{{{\mathbb{Z}_{2}}}}$ are given. Note that all of the
results in this part valid up to third order truncation. It is quite possible
to derive these computations for any finite order using our Maple program.
Finally, we illustrate our results with the modified Chua’s circuit.

Consider the Hopf-pitchfork differential system governed by

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm d}t}&=\sum
a_{i,j,k}x^{i}y^{j}z^{k},\\\ \frac{{\rm d}{y}}{{\rm d}t}&=z+\sum
b_{i,j,k}x^{i}y^{j}z^{k},\\\ \frac{{\rm d}{z}}{{\rm d}t}&=-y+\sum
c_{i,j,k}x^{i}y^{j}z^{k},\end{aligned}\right.$ (5.1)

where $i+j+k=3$ and the monomials $x^{i}y^{j}z^{k}$ are assumed to be odd
functions. If the following relations between the coefficients of cubic terms
of system (5.1) hold, then the classical normal form of this system up to
third order belongs to ${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}.$

$\displaystyle a_{1,0,2}$ $\displaystyle=$
$\displaystyle-2c_{{0,2,1}}-6c_{{0,0,3}}-6b_{{0,3,0}}-2b_{{0,1,2}}-2a_{{1,2,0}},$
(5.2) $\displaystyle c_{2,0,1}$ $\displaystyle=$ $\displaystyle-
b_{{2,1,0}}-3a_{{3,0,0}}.$ (5.3)

Following Theorem 4.3, the unique normal form of (5.1) under the above
conditions is as

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm
d}t}&=a_{0}2x\rho^{2}+a_{1}{x}^{3},\\\ \frac{{\rm d}{\rho}}{{\rm
d}t}&=-\frac{a_{0}}{2}\rho^{3}-a_{1}\frac{3}{2}{x}^{2}{\rho},\\\ \frac{{\rm
d}{\theta}}{{\rm d}t}&=1+b_{0}\rho^{2}+b_{1}{x}^{4},\end{aligned}\right.$

where the coefficients of previous system are given explicitly by

$\displaystyle a_{0}$ $\displaystyle:=$
$\displaystyle\frac{-1}{4}(c_{{0,2,1}}+3c_{{0,0,3}}+3b_{{0,3,0}}+b_{{0,1,2}}),$
$\displaystyle b_{0}$ $\displaystyle:=$
$\displaystyle\frac{1}{8}(-3c_{{0,3,0}}-c_{{0,1,2}}+b_{{0,2,1}}+3b_{{0,0,3}}),$
$\displaystyle a_{1}$ $\displaystyle:=$ $\displaystyle a_{{3,0,0}},\qquad
b_{1}:=-\frac{1}{2}(c_{{2,1,0}}+b_{{2,0,1}}).$

The above relations, Equation (5.2), and (5.3) are obtained using our Maple
program and employing beneficial formulas regarding the coefficients of the
classical normal form of Hopf-zero singularities derived in [2].

We finish with an example.

######  Example 5.1.

Consider modified Chua’s oscillator defined by the set of ordinary
differential equations

$\left\\{\begin{aligned} \frac{{\rm d}{x}}{{\rm d}t}&=-\gamma x-\beta y,\\\
\frac{{\rm d}{y}}{{\rm d}t}&=z-y+x+{\mu_{1}}{z}^{2}y,\\\ \frac{{\rm
d}{z}}{{\rm
d}t}&=\alpha\left(-cz+y-a{z}^{3}\right)+{\mu_{2}}{z}^{2}y.\end{aligned}\right.$
(5.4)

This system possesses the ${{\mathbb{Z}_{2}}}$-symmetry given in (3.3). If one
puts $(\mu_{1},\mu_{2})=(0,0)$ in (5.4), then one gets the Chua’s oscillator.
The linearization of this system at origin has eigenvalues
$0,\pm{i}\omega_{0}$ when

$\displaystyle(c,\beta)=\Big{(}-\frac{\gamma+1}{\alpha},-{\frac{\gamma\left(\alpha+\gamma+1\right)}{\gamma+1}}\Big{)},\quad\frac{\left(\gamma+1\right)^{3}+\alpha\left(2\gamma+1\right)}{\gamma+1}>0,$
(5.5)

where
$\omega_{0}^{2}:={-{\frac{\left(\gamma+1\right)^{3}+\alpha\left(2\gamma+1\right)}{\gamma+1}}},$
$a\neq 0,$ $\alpha\neq 0,$ and $\gamma\approx 0,$ see [6, 2]. The following
linear transformations and rescaling the time

$\displaystyle x$ $\displaystyle\rightarrow$
$\displaystyle\frac{\left(\gamma+1\right)}{\left(\alpha+\gamma+1\right)\gamma}\Big{(}{{\omega_{{0}}\left(\gamma+1\right)}}x+{{\Big{(}\alpha+(\gamma+1)^{2}\big{)}}}y-{\frac{\gamma\alpha}{\gamma+1}}z\Big{)},$
$\displaystyle y$ $\displaystyle\rightarrow$
$\displaystyle\frac{\left(\gamma+1\right)}{\left(\alpha+\gamma+1\right)}\left({\frac{\omega_{{0}}}{\gamma}}x+y+z\right),$
$\displaystyle z$ $\displaystyle\rightarrow$ $\displaystyle y+z,$
$\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle\omega_{0}t,$

bring the linear part of (5.4) to the Jordan canonical form. If in addition to
the conditions already cited (5.5), we require

$\displaystyle\mu_{1}:={\frac{3\omega_{0}{\alpha}^{2}a\left(2\alpha+\left(\gamma+1\right)^{2}\right)}{\left(2\alpha-\left(\gamma+1\right)^{2}\right)\left(\alpha-2\left(\gamma+1\right)^{2}\right)}},\qquad\mu_{2}:={\frac{3\omega_{0}{\alpha}^{2}a\left(\gamma+1\right)\left(3\alpha-\left(\gamma+1\right)^{2}\right)}{\left(2\alpha-\left(\gamma+1\right)^{2}\right)\left(\alpha-2\left(\gamma+1\right)^{2}\right)}},$

where $\alpha\neq 2(\gamma+1)^{2},$ then the classical normal form of modified
Chua’s oscillator up to third order belongs to
${\mathscr{L}}^{{{\mathbb{Z}_{2}}}}$. The previous relations are derived with
the aid of equations (5.2) and (5.3). Following Theorem 4.3, the unique normal
form of (5.4) is given by

$\displaystyle C^{(\infty)}$ $\displaystyle:=$
$\displaystyle\frac{{\alpha}^{3}a(\alpha+(\gamma+1)^{2})}{\omega_{{0}}(\alpha+\gamma+1)\left(2\alpha-(\gamma+1)^{2}\right)}\Big{(}{\frac{3(\gamma+1)}{2\gamma{}}}{\rm
H}^{0}_{1}+{\frac{2{\alpha}\gamma}{(\gamma+1)(\alpha+\gamma+1)}{\rm
H}^{1}_{2}\Big{)}}$
$\displaystyle+\frac{3{\alpha}^{3}a\left((4\gamma+5)(\gamma+1)^{2}+\alpha(3\gamma+5)\right)}{2(\alpha+\gamma+1)\left(2\alpha-(\gamma+1)^{2}\right)\left(\alpha-2(\gamma+1)^{2}\right)}\Big{(}{\frac{(\gamma+1)^{2}}{4{\gamma}}}\varTheta^{0}_{1}+{\frac{{\alpha}}{(\alpha+\gamma+1)}}\varTheta^{1}_{2}\Big{)}.$

We refer the reader to [16] for relevant results.

##  6 Acknowledgement

I would like to thank my thesis advisor, Professor Majid Gazor, for the
encouragement to write this paper and Professor Jan A. Sanders for invaluable
remarks and comments.

## References

  * [1] Yakir Aharonov and David Bohm.  Significance of electromagnetic potentials in the quantum theory.  Physical Review, 115(3):485, 1959.
  * [2] Antonio Algaba, Emilio Freire, and Estanislao Gamero.  Hypernormal form for the Hopf-zero bifurcation.  International Journal of Bifurcation and Chaos, 8(10):1857–1887, 1998.
  * [3] Antonio Algaba, Emilio Freire, Estanislao Gamero, and Alejandro J Rodríguez-Luis.  On a codimension-three unfolding of the interaction of degenerate Hopf and pitchfork bifurcations.  International Journal of Bifurcation and Chaos, 9(07):1333–1362, 1999.
  * [4] Antonio Algaba, Emilio Freire, Estanislao Gamero, and Alejandro J Rodríguez-Luis.  A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation.  Nonlinearity, 12:1177–1206, 1999.
  * [5] Antonio Algaba, Emilio Freire, Estanislao Gamero, and Alejandro J Rodríguez-Luis.  A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol–Duffing oscillator.  Nonlinear Dynamics, 22(3):249–269, 2000.
  * [6] Antonio Algaba, Manuel Merino, Emilio Freire, Estanislao Gamero, and Alejandro J Rodríguez-Luis.  Some results on Chua’s equation near a triple-zero linear degeneracy.  International Journal of Bifurcation and Chaos, 13(03):583–608, 2003.
  * [7] Vladimir I. Arnold.  On matrices depending on parameters.  Russian Mathematical Surveys, 26(2):29–43, 1971.
  * [8] Alberto Baider and Jan A Sanders.  Unique normal forms: The nilpotent Hamiltonian case.  Journal of Differential Equations, 92(2):282–304, 1991.
  * [9] Alberto Baider and Jan A Sanders.  Further reduction of the Takens-Bogdanov normal form.  Journal of Differential Equations, 99(2):205–244, 1992.
  * [10] Cristian Barbarosie.  Representation of divergence-free vector fields.  Quarterly of applied mathematics, 69(2):309–316, 2011.
  * [11] Daniel S Brown and Eric R Priest.  Topological bifurcations in three–dimensional magnetic fields.  In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 455, pages 3931–3951. The Royal Society, 1999.
  * [12] Bruce W Char, Keith O Geddes, Gaston H Gonnet, Benton L Leong, Michael B Monagan, and Stephen Watt.  Maple V library reference manual.  Springer Science & Business Media, 2013.
  * [13] Rui Loja Fernandes and Ioan Marcut.  Lectures on Poisson geometry, Springer 2014.
  * [14] Estanislao Gamero, Emilio Freire, Alejandro J Rodríguez-Luis, Enrique Ponce, and Antonio Algaba.  Hypernormal form calculation for triple-zero degeneracies.  Bulletin of the Belgian Mathematical Society Simon Stevin, 6(3):357–368, 1999.
  * [15] Majid Gazor and Mojtaba Moazeni.  Parametric normal forms for Bogdanov–Takens singularity; the generalized saddle-node case.  Discrete and Continuous Dynamical Systems, 35:205–224, 2015.
  * [16] Majid Gazor and Fahimeh Mokhtari.  Volume-preserving normal forms of Hopf-zero singularity.  Nonlinearity, 26(10):2809–2832, 2013.
  * [17] Majid Gazor and Fahimeh Mokhtari.  Normal forms of Hopf-zero singularity.  Nonlinearity, 28(2):311, 2015.
  * [18] Majid Gazor, Fahimeh Mokhtari, and Jan A Sanders.  Normal forms for Hopf-zero singularities with nonconservative nonlinear part.  Journal of Differential Equations, 254(3):1571–1581, 2013.
  * [19] Majid Gazor, Fahimeh Mokhtari, and Jan A Sanders.  Vector potential normal form classification for completely integrable solenoidal nilpotent singularities.  https://arxiv.org/abs/1711.09126, 2017.
  * [20] Majid Gazor and Pei Yu.  Spectral sequences and parametric normal forms.  Journal of Differential Equations, 252(2):1003–1031, 2012.
  * [21] Martin Golubitsky, Ian Stewart, and David G Schaeffer.  Singularities and groups in bifurcation theory, volume 1–2.  Springer Science & Business Media, 1985 and 1988.
  * [22] Emil Jan Konopinski.  What the electromagnetic vector potential describes.  American Journal of Physics, 46(5):499–502, 1978.
  * [23] Dana W Longcope.  Topological methods for the analysis of solar magnetic fields.  Living Reviews in Solar Physics, 2(1):1–72, 2005.
  * [24] Igor Mezić and Stephen Wiggins.  On the integrability and perturbation of three-dimensional fluid flows with symmetry.  Journal of Nonlinear Science, 4(1):157–194, 1994.
  * [25] Fahimeh Mokhtari.  The simplest normal forms of Hopf-zero singularity, master thesis (in Persian).  Isfahan University of Technology, Isfahan, Iran, September 2011\.
  * [26] James Murdock.  On the structure of nilpotent normal form modules.  Journal of Differential Equations, 180(1):198–237, 2002.
  * [27] James Murdock.  Normal forms and unfoldings for local dynamical systems.  Springer Science & Business Media, 2006.
  * [28] Charudatta Phatak, Amanda K Petford-Long, and Marc De Graef.  Three-dimensional study of the vector potential of magnetic structures.  Physical review letters, 104(25):253901, 2010.
  * [29] Eric Priest, David Pontin Lonie, and Viacheslav Titov.  Bifurcations of magnetic topology by the creation or annihilation of null points.  Journal of plasma physics, 56(3):507–530, 1996.
  * [30] Ernest C Ray.  A topological study of the motion of particles trapped in a magnetic field.  Annals of Physics, 42(3):508–512, 1967.
  * [31] Jan A Sanders.  Normal form theory and spectral sequences.  Journal of Differential Equations, 192(2):536–552, 2003.
  * [32] Jan A Sanders, Ferdinand Verhulst, and James A Murdock.  Averaging methods in nonlinear dynamical systems, volume 59.  Springer, 2007.
  * [33] Mark D Semon and John R Taylor.  Thoughts on the magnetic vector potential.  American Journal of Physics, 64(11):1361–1369, 1996.
  * [34] Michael Sullivan.  Finding Vector Potentials.  http://galileo.math.siu.edu/mikesullivan/Courses/251/S12/vpot.pdf/, 2012\.
  * [35] Vasily Tarasov.  Quantum Mechanics of Non-Hamiltonian and Dissipative Systems, volume 7.  Elsevier, 2008.
  * [36] Akira Tonomura.  Applications of electron holography.  Reviews of modern physics, 59(3):639, 1987.
  * [37] Clifford Truesdell.  The kinematics of vorticity, volume 954.  Indiana University Press Bloomington, 1954.
  * [38] Constantin Udriste.  Geometric Dynamics.  Mathematics and Its Applications. Springer Netherlands, 2012.