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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 48, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/48\hfil Hopf bifurcations and small amplitude limit cycles]
{Hopf bifurcations and small amplitude limit cycles in
 Rucklidge systems}

\author[F. S. Dias, L. F. Mello \hfil EJDE-2013/48\hfilneg]
{Fabio Scalco Dias, Luis Fernando Mello}  % in alphabetical order

\address{Fabio Scalco Dias \newline
Instituto de Matem\'atica  e Estat\'istica,
Universidade Federal de Itajub\'a\\
Avenida BPS 1303, Pinheirinho, CEP 37.500--903,
Itajub\'a, MG, Brazil}
\email{scalco@unifei.edu.br}

\address{Luis Fernando Mello \newline
Instituto de Matem\'atica  e Estat\'istica,
Universidade Federal de Itajub\'a\\
Avenida BPS 1303, Pinheirinho, CEP 37.500--903,
Itajub\'a, MG, Brazil \newline
Tel: 00--55--35--36291217, Fax: 00--55--35--36291140}
\email{lfmelo@unifei.edu.br}

\thanks{Submitted February 7, 2012. Published February 12, 2013.}
\subjclass[2000]{34A34, 34D20, 34C07}
\keywords{Rucklidge system; limit cycle; stability; Lyapunov coefficient}

\begin{abstract}
 In this article we study Hopf bifurcations and small amplitude limit
 cycles in a family of quadratic systems in the three dimensional
 space called Rucklidge systems.
 Bifurcation analysis at the equilibria of Rucklidge system is pushed
 forward toward the calculation of the second Lyapunov coefficient,
 which makes possible the determination of the Lyapunov and higher
 order structural stability.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}\label{S:1}

In this article we study Hopf bifurcations and small amplitude limit
cycles in the following family of quadratic systems, called
Rucklidge system,
\begin{equation}\label{eq:01}
x' = -ax+by-yz, \quad y' = x, \quad z' = -z+y^2,
\end{equation}
where $(x,y,z) \in \mathbb{R}^3$ are the state variables and 
$(a, b) \in \mathcal{W} = \mathbb{R}^2$ are real parameters. Despite the
simplicity, system \eqref{eq:01} has a rich local dynamical behavior
and was widely analyzed (see \cite{wang} and references therein).

Quadratic systems in $\mathbb{R}^3$ are some of the simplest systems
after linear ones and have been extensively studied in the last five
decades. Examples of such systems are the Lorenz system, the Chen
system, the Liu system, the R\"ossler system, the Rikitake system,
the L\"u system, the Genesio system among several others. See
\cite{DM1} and references therein.

An interesting problem related to quadratic systems defined in
$\mathbb{R}^3$ is the determination of the number of their limit
cycles. In $\mathbb{R}^2$ this number is finite \cite{Eca, Ilia}. For
quadratic systems in $\mathbb{R}^n$, $n \geq 3$ the scenario is very
different. Recently Ferragut, Llibre and Pantazi \cite{FLP} provided
an example of quadratic vector field in $\mathbb{R}^3$ and an
analytical proof that it has infinitely many limit cycles.

It is well known (see \cite{wang} and references therein) that
system \eqref{eq:01} has at most three equilibria $E_0=(0,0,0)$ and
$E_{\pm}=(0,\pm\sqrt{b},b)$, when $b \geq 0$. In order to study the
stability of $E_{\pm}$ it is sufficient only to study the stability
of $E_+$ due to the symmetry $(x, y, z)\to (-x,-y,z)$
presented by system \eqref{eq:01}.

In general, to decide the stability of a non--hyperbolic equilibrium
point of a system in $\mathbb{R}^3$ is very difficult even for
quadratic systems. As far as we know, the stabilities of $E_0$ and
$E_{\pm}$ were analyzed in \cite{wang}. But the studies of Hopf bifurcations
presented in \cite{wang} are incomplete and are not correct.

Consider the subset $\mathcal{U} \subset \mathcal{W}$ of the parameter
plane where $b \neq 0$. Write $\mathcal{U} = \mathcal{U}_1 \cup
\mathcal{U}_2 \cup \mathcal{U}_3 \cup \mathcal{H}_0$, where
\begin{gather*}%\label{H0}
\mathcal{U}_1 = \{ a \in \mathbb{R}, \; b > 0 \} , \quad
\mathcal{U}_2 = \{ a < 0, \; b < 0 \}, \\
\mathcal{U}_3 = \{a > 0, \; b < 0 \}, \quad
\mathcal{H}_0 = \{a = a_c = 0, \; b <0 \}.
\end{gather*}
From the linear analysis of system \eqref{eq:01} at $E_{0}$ the
following statements hold: if $(a, b) \in \mathcal{U}_1 \cup
\mathcal{U}_2$ then $E_0$ is unstable; if $(a, b) \in \mathcal{U}_3$
then $E_0$ is locally asymptotically stable; if
 $(a, b) \in \mathcal{H}_0$ then $E_0$ is a non--hyperbolic equilibrium 
of Hopf type, that is the Jacobian matrix of system \eqref{eq:01} 
at $E_{0}$ has one
negative real eigenvalue and a pair of purely imaginary eigenvalues
\[
\theta_1 = - 1 < 0, \quad \theta_{2,3} = \pm i \sqrt{-b}.
\]

Now consider the subset $\mathcal{W}^{+} \subset \mathcal{W}$ of the
parameter plane where $b > 0$. Write $\mathcal{W^{+}} = \mathcal{W}_1
\cup \mathcal{W}_2 \cup \mathcal{W}_3 \cup \mathcal{H}_+$, where
\begin{gather*}%\label{H+}
\mathcal{W}_1 = \{ a \leq 0, \; b > 0 \} , \quad
\mathcal{W}_2 = \big\{ a > 0, \; b > \frac{a(a+1)}{2} \big\}, \\
\mathcal{W}_3 = \big\{a > 0, \; 0 < b < \frac{a(a+1)}{2} \big\}, \quad
\mathcal{H}_+ = \big\{a > 0, \; b = b_c = \frac{a(a+1)}{2} \big\}.
\end{gather*}
From the linear analysis of system \eqref{eq:01} at $E_{+}$ the
following statements hold: if $(a, b) \in \mathcal{W}_1 \cup
\mathcal{W}_2$ then $E_+$ is unstable; if $(a, b) \in \mathcal{W}_3$
then $E_+$ is locally asymptotically stable; if $(a, b) \in \mathcal{H}_+$ then $E_+$ is a non--hyperbolic equilibrium of Hopf type, that
is the Jacobian matrix of system \eqref{eq:01} at $E_{+}$ has one
negative real eigenvalue and a pair of purely imaginary eigenvalues
\[
\lambda_1 = - (a+1) < 0, \quad \lambda_{2,3} = \pm i \sqrt{a}.
\]

The sets $\mathcal{H}_0$ and $\mathcal{H}_+$ are called the Hopf
curves of the equilibria $E_0$ and $E_{+}$, respectively. From the
Center Manifold Theorem, at a Hopf point a two dimensional center
manifold is well--defined, it is invariant under the flow generated
by \eqref{eq:01} and can be continued with arbitrary high class of
differentiability to nearby parameter values (see \cite[p.
152]{kuznet}). These center manifolds are normally attracting since
$\theta_1 < 0$ and $\lambda_1 < 0$. So it is enough to study the
stability of $E_0$ and $E_{+}$ for the flow restricted to the family
of parameter--dependent continuations of these center manifolds.

It is important to emphasize that the study the stability of $E_0$
and $E_{+}$ for the flow of system \eqref{eq:01} restricted to the
center manifolds is in fact the study of the center--focus problem
in an extended version to systems in $\mathbb{R}^3$. Although this
problem has a solution for quadratic systems in the plane
\cite{bautin} it remains open for quadratic systems in $\mathbb{R}^3$.

The study carried out in the present article may contribute to
understand analytically the stability of the equilibria $E_0$ and
$E_{+}$ of system \eqref{eq:01}. By using the classical projection
method which allows us to calculate the first and the second
Lyapunov coefficients associated to the Hopf points, we study the
stability of $E_0$ and $E_{+}$ as well as the number of small
amplitude limit cycles in system \eqref{eq:01}. More precisely, in
this article we prove the following two theorems.

\begin{theorem}\label{thm:main0}
Consider system \eqref{eq:01} with parameter values in
$\mathcal{H}_{0}$; that is, $a = a_c = 0$ and $b < 0$. Then the
first Lyapunov coefficient associated to $E_0$ is positive, so $E_0$
is an unstable equilibrium point.
\end{theorem}

\begin{theorem}\label{thm:main1}
Consider system \eqref{eq:01} with parameter values in
$\mathcal{H}_+$. Define $a_1=6+\sqrt{37}$. The following
statements hold.
\begin{enumerate}
\item If $0 < a < a_1$ and $b = b_c$ then the first Lyapunov coefficient
associated to $E_+$ is negative, so $E_+$ is locally asymptotically
stable.
\item If $a > a_1$ and $b = b_c$ then the first Lyapunov coefficient
associated to $E_+$ is positive, so $E_+$ is unstable.
\item If $a = a_1$ and $b = b_c$ then the first Lyapunov coefficient
associated to $E_+$ vanishes and the second Lyapunov coefficient is
positive, so $E_+$ is unstable.
\end{enumerate}
\end{theorem}

The proofs of Theorems \ref{thm:main0} and \ref{thm:main1}, and
the study of the small amplitude limit cycles of system
\eqref{eq:01} are presented in Section \ref{S:3}. 
In Section \ref{S:2}, we present a review on the methods of Hopf 
bifurcation analysis. Some
concluding remarks are presented in Section \ref{S:4}.

\section{Review on Hopf bifurcation}\label{S:2}

In this section we present a review of the projection method
described in \cite{kuznet} for the calculation of the first and
second Lyapunov coefficients associated to Hopf bifurcations. This
method was extended to the calculation of the third and fourth
Lyapunov coefficients in \cite{smb1} and \cite{smb2}, respectively.

Consider the differential equation
\begin{equation}\label{diffequat}
x'= f (x, \zeta),
\end{equation}
where $x \in \mathbb{R}^3$ and $\zeta \in \mathbb{R}^2$ are
respectively vectors representing phase variables and control
parameters. Assume that $f$ is of class $C^{\infty}$ in $\mathbb{R}^3
\times \mathbb{R}^2$. Suppose that \eqref{diffequat} has an
equilibrium point $x = x_0$ at $\zeta = \zeta_0$ and, denoting the
variable $x-x_0$ also by $x$, write
\begin{equation} \label{Fhomo}
F(x) = f (x, \zeta_0)
\end{equation}
as
\begin{equation} \label{taylorexp}
\begin{aligned}
F(x) &= Ax + \frac{1}{2}
B(x,x) + \frac{1}{6}  C(x, x,
x) +  \frac{1}{24}  D( x, x,x, x) \\
&\quad +  \frac{1}{120}  E(x, x, x, x, x) + O(|| x ||^6),
\end{aligned}
\end{equation}
where $A = f_{x}(0,\zeta_0)$ and, for $i = 1,2,3$,
\[
B_i (x,y) = \sum_{j,k=1}^3 \frac{\partial ^2 F_i(\xi)}{
\partial \xi_j  \partial \xi_k} \Big|_{\xi=0} x_j \; y_k, \quad
 C_i (x,y,z) = \sum_{j,k,l=1}^3 \frac{\partial
^3 F_i(\xi)}{\partial \xi_j \: \partial \xi_k \: \partial \xi_l}
\Big|_{\xi=0} x_j \; y_k \: z_l,
\]
and so on for $D_i$ and $E_i$.

Suppose that $(x_0, \zeta_0) = (0, \zeta_0)$ is an equilibrium point
of \eqref{diffequat} where the Jacobian matrix $A$ has a pair of
purely imaginary eigenvalues $\lambda_{2,3} = \pm i \omega_0$,
$\omega_0 > 0$, and the other eigenvalue $\lambda_1 \neq 0$. Let
$T^c$ be the generalized eigenspace of $A$ corresponding to
$\lambda_{2,3}$. By this it is meant the largest subspace invariant
by $A$ on which the eigenvalues are $\lambda_{2,3}$. Let $p, q \in
\mathbb C^3$ be vectors such that
\begin{equation}\label{normalization}
A q = i \omega_0  q, \quad A^{\top} p = -i \omega_0  p, \quad
\langle p,q \rangle = \sum_{i=1}^3 \bar{p}_i  q_i  = 1,
\end{equation}
where $A^{\top}$ is the transpose of the matrix $A$. Any vector 
$y \in T^c$ can be represented as $y = w q + \bar w \bar q$, where 
$w = \langle p , y \rangle \in \mathbb C $. The two dimensional center
manifold associated to the eigenvalues 
$\lambda_{2,3} = \pm i \omega_0$ can be parameterized by the variables 
$w$ and $\bar w$ by means of an immersion of the form $x = H (w, \bar w)$,
where $H:\mathbb C^2 \to \mathbb{R}^3$ has a Taylor expansion of the form
\begin{equation}\label{defH}
H(w,{\bar w}) = w q + {\bar w}{\bar q} + \sum_{2 \leq j+k \leq 5}
\frac{1}{j!k!}  h_{jk}w^j{\bar w}^k + O(|w|^6),
\end{equation}
with $h_{jk} \in \mathbb C^3$ and $h_{jk}={\bar h}_{kj}$.
Substituting this expression into \eqref{diffequat} we obtain the
following differential equation
\begin{equation}\label{ku}
H_w w'+ H_{\bar w} {\bar w}'= F (H(w,{\bar w})),
\end{equation}
where $F$ is given by \eqref{Fhomo}. The complex vectors $h_{ij}$
are obtained solving the system of linear equations defined by the
coefficients of \eqref{ku}, taking into account the coefficients of
$F$ (see Remark 3.1 of \cite[p. 27]{smb1}), so that system
\eqref{ku}, on the chart $w$ for a central manifold, writes as
follows, with $G_{jk} \in \mathbb C$,
\[
w'= i \omega_0 w + \frac{1}{2} \; G_{21} w |w|^2 +
\frac{1}{12} \; G_{32} w |w|^4 + O(|w|^6).
\]

The \emph{first Lyapunov coefficient} $l_1$ is defined by
\begin{equation}\label{coef1}
l_1 = \frac{1}{2}  \operatorname{Re} G_{21},
\end{equation}
where $G_{21}= \langle p, \mathcal{H}_{21} \rangle$ and 
$\mathcal{H}_{21} = C(q,q,\bar q) + B(\bar q, h_{20}) + 2 B(q, h_{11})$.


The \emph{second Lyapunov coefficient} is defined by
\begin{equation}\label{coef2}
l_2= \frac{1}{12} \operatorname{Re} G_{32},
\end{equation}
where $G_{32}=\langle p, \mathcal{H}_{32} \rangle$ and
\begin{align*}
\mathcal{H}_{32} 
&= 6 B(h_{11},h_{21})+ B({\bar h}_{20},h_{30}) + 3
B({\bar h}_{21},h_{20})+ 3 B(q,h_{22}) + 2 B(\bar q, h_{31})
  \\ 
&\quad +6 C(q,h_{11},h_{11}) + 3 C(q, {\bar h}_{20}, h_{20})+
3 C(q,q,{\bar h}_{21}) +6 C(q,\bar q, h_{21})
\\
&\quad  + 6 C(\bar q, h_{20}, h_{11}) + C(\bar q, \bar q, h_{30}) +
D(q,q,q,{\bar h}_{20}) + 6 D(q,q,\bar q,h_{11}) 
  \\ 
&\quad + 3 D(q, \bar q,\bar q, h_{20}) + E(q,q,q,\bar q,\bar q) 
 -6 G_{21}h_{21} - 3 {\bar G}_{21} h_{21},
\end{align*}

A \emph{Hopf point of codimension one} is an equilibrium point
$(x_0,\zeta_0)$ such that linear part of the vector field $f$ has
eigenvalues $\lambda_2$ and $\lambda_3 = \overline{\lambda}$ with
$\lambda = \lambda(\zeta) = \gamma(\zeta) + i \eta(\zeta)$,
$\gamma(\zeta_0) = 0$, $\eta(\zeta_0) = \omega_0 > 0$, the other
eigenvalue $\lambda_1 \neq 0$ and the first Lyapunov coefficient,
$l_1(\zeta_0)$, is different from zero. A \emph{transversal Hopf
point of codimension one} is a Hopf point of codimension one for
which the complex eigenvalues depending on the parameters cross the
imaginary axis with nonzero derivative. When $l_1 < 0$ ($l_1 > 0$)
one family of stable (unstable) periodic orbits can be found on the
center manifold and its continuation, shrinking to the Hopf point.

\emph{Hopf point of codimension 2} is an equilibrium
$(x_0,\zeta_0)$ of $f$ that satisfies the definition of Hopf point
of codimension one, except that $l_1(\zeta_0) = 0$, and an
additional condition that the second Lyapunov coefficient,
$l_2(\zeta_0)$, is nonzero. This point is \emph{transversal} if
the sets $\gamma^{-1}(0)$ and $l_1^{-1}(0)$ have transversal
intersection, or equivalently, if the map $\zeta \mapsto
(\gamma(\zeta), l_1(\zeta))$ is regular at $\zeta = \zeta_0$. The
bifurcation diagrams for $l_2 \neq 0$ can be found in \cite[p.
313]{kuznet}. In this bifurcation diagram two families of small
amplitude limit cycles can be found.

\section{Proofs of Theorems \ref{thm:main0} and \ref{thm:main1}}\label{S:3}

\subsection{Proof of Theorem \ref{thm:main0}}\label{S:31}

In this subsection we study Hopf bifurcations that occur at the
equilibrium $E_0$ for parameters in the set $\mathcal{H}_0$.

\begin{theorem}\label{theo:00}
Consider system \eqref{eq:01} with parameter values in
$\mathcal{H}_0$. Then the first Lyapunov coefficient at $E_0$ is
given by
\begin{equation}\label{eq:E01}
l_1(a_c,b)=\frac{2}{1 - 4 b} > 0,
\end{equation}
since $b < 0$. If $\tau_0 = (a_c, b) \in \mathcal{H}_0$
then system \eqref{eq:01} has a transversal Hopf point at $E_0$
for the parameter vector $\tau_0$.
\end{theorem}

\begin{proof}
For parameters on the Hopf curve $\mathcal{H}_0$, the eigenvalues of
the Jacobian matrix of system \eqref{eq:01} at $E_0$ are $\theta_1 =
-1 < 0$, $\theta_{2,3}= \pm i \omega_0$, $\omega_0 = \sqrt{-b}$, $b
< 0$, the eigenvectors $q$ and $p$ defined in \eqref{normalization}
are
\[
q = (i \sqrt{-b}, 1, 0 ), \quad p = \Big( \frac{i}{2
\sqrt{-b}}, \frac{1}{2}, 0 \Big)
\]
and the multilinear symmetric functions $B$ and $C$ can be written
as
\[
B(x,y) = \left( -(x_2y_3+x_3y_2),0,2x_2y_2 \right), \quad C(x,y,z) =
\left( 0, 0, 0 \right).
\]
The complex vectors $h_{11}$ and $h_{20}$ are given by
\[
h_{11}=\left( 0, 0, 2 \right), \quad h_{20}= \Big( 0, 0, \frac{2}{1
+ i 2 \sqrt{-b}} \Big).
\]
By simple calculations, the first Lyapunov coefficient \eqref{coef1}
is given by
\[
l_1(a_c,b)=\frac{2}{1 - 4 b},
\]
which is positive, since $b < 0$. It remains only to verify the
transversality condition of the Hopf bifurcation. In order to do so,
consider the family of differential equations \eqref{eq:01} regarded
as dependent on the parameter $a$. The real part, $\gamma=\gamma
(a)$, of the pair of complex eigenvalues at the critical parameter
$a = a_{c} = 0$ verifies
\[
\gamma'(a_{c}) = \operatorname{Re} \big\langle p,
\frac{dA}{da}\Big|_{a=a_{c}} q  \big\rangle = - \frac{1}{2} < 0.
\]
In the above expression, $A$ is the Jacobian matrix of system
\eqref{eq:01} at $E_{0}$. Therefore, the transversality condition at
the Hopf point holds.
\end{proof}

The proof of Theorem \ref{thm:main0} follows from Theorem
\ref{theo:00}.



From Theorem \ref{theo:00}, the sign of the first Lyapunov
coefficient at $E_0$ is positive for parameters in $\mathcal{H}_0$.
Thus the equilibrium $E_0$ is a weak repelling focus (for the flow
of system \eqref{eq:01} restricted to the center manifold) and there
is one unstable limit cycle near the asymptotically stable
equilibrium $E_0$ for suitable value of the parameters ($a > 0$).
See the pertinent bifurcation diagram in \cite[p. 89]{kuznet}. See
Figures \ref{fig1} and \ref{fig2} where the stability of $E_0$ and
small amplitude limit cycles are depicted.

\subsection{Proof of Theorem \ref{thm:main1}}\label{S:32}

In this subsection we study Hopf bifurcations that occur at the
equilibrium $E_+$ for parameters in the set $\mathcal{H}_+$.

\begin{theorem}\label{theo:01}
Consider system \eqref{eq:01} with parameter values in
$\mathcal{H}_+$. Then the first Lyapunov coefficient at $E_+$ is
given by
\begin{equation}\label{eq:02}
l_1(a,b_c)=\frac{2(a^2-12a-1)}{a (a+1) (a (a+3)+1) (a
(a+6)+1)}.
\end{equation}
If $\zeta_0 = (a, b_c) \in \mathcal{H}_+$ is such that $a \neq a_1$
then system \eqref{eq:01} has a transversal Hopf point at $E_{+}$
for the parameter vector $\zeta_0$.
\end{theorem}

\begin{proof}
For parameters on the Hopf curve $\mathcal{H}_+$, the eigenvalues of
the Jacobian matrix of system \eqref{eq:01} at $E_+$ are $\lambda_1
= -(1+a) < 0$, $\lambda_{2,3}= \pm i \omega_0$, $\omega_0 =
\sqrt{a}$, $a> 0$, the eigenvectors $q$ and $p$ defined in
\eqref{normalization} are
\begin{gather*}
q = \Big(-\frac{\omega_0-i}{\sqrt{2} c},\frac{i
\omega_0+1}{\sqrt{2} c \omega_0},1 \Big), \\
p = \Big(\frac{i c}{\sqrt{2} (c^2-i \omega_0)},\frac{c
(i \omega_0+1) \omega_0}{\sqrt{2} (c^2-i
\omega_0)},\frac{1}{2}-\frac{1}{2 c^2-2 i \omega_0} \Big)
\end{gather*}
where $c=\sqrt{1+a}$, and the multilinear symmetric functions $B$
and $C$ can be written as
\[
B(x,y) = \left( -(x_2y_3+x_3y_2),0,2x_2y_2 \right), \quad C(x,y,z) =
\left( 0, 0, 0 \right).
\]
The complex vectors $h_{11}$ and $h_{20}$ are given by
\begin{gather*}
h_{11}=\Big( 0,-\frac{\omega_0^2+3}{\sqrt{2} c^3
\omega_0^3},-\frac{2}{c^2 \omega_0^2} \Big),\\
\begin{aligned}
 h_{20}&=\bigg( \frac{\sqrt{2} (5 i \omega_0+3) (\omega_0-i)}{c
\omega_0^2 \left(c^2-2 (\omega_0 (2 \omega_0+3
   i)+2)\right)},\frac{\omega_0 (5 \omega_0-8 i)-3}{\sqrt{2} c \omega_0^3 \left(c^2-2 (\omega_0 (2 \omega_0+3
   i)+2)\right)},\\
 &\quad   -\frac{2 i (\omega_0-i) \left(c^2+\omega_0 (\omega_0+i)+2\right)}{c^2 \omega_0^2 \left(c^2-2 (\omega_0
   (2 \omega_0+3 i)+2)\right)}\bigg).
\end{aligned}
\end{gather*}
Therefore, the first Lyapunov coefficient \eqref{coef1} is 
\[
l_1=\frac{D(c,\omega_0)}{2 c^2
   \omega_0^4 \left(c^4+\omega_0^2\right) \left(c^4-8 \left(\omega_0^2+1\right) c^2+4 \left(\omega_0^2+4\right) \left(4
   \omega_0^2+1\right)\right)},
\]
where
\begin{align*} 
D(c,\omega_0)&=\left(7 \omega_0^2-9\right)
c^6+\left(-21 \omega_0^4+76 \omega_0^2+69\right) c^4-6
\omega_0^4 \left(4 \omega_0^4+33 \omega_0^2+66\right)\\
&\quad +2 \left(19 \omega_0^6+75 \omega_0^4-190 \omega_0^2-78\right) c^2+
306 \omega_0^2+96.
\end{align*}
Substituting $\omega_0 = \sqrt{a}$ and $c=\sqrt{1+a}$ into the
expression of $l_1$, it results \eqref{eq:02}.

 It remains only to verify the transversality condition of
the Hopf bifurcation. In order to do so, consider the family of
differential equations \eqref{eq:01} regarded as dependent on the
parameter $b$. The real part, $\gamma=\gamma (b)$, of the pair of
complex eigenvalues at the critical parameter $b=b_{c}$ verifies
\[
\gamma'(b_{c}) = \operatorname{Re} \big\langle p,
\frac{dA}{db}\Big|_{b=b_{c}} q  \big\rangle = \frac{a+2}{a^3+4
a^2+4 a+1}>0,
\]
since $a > 0$. In the above expression $A$ is the Jacobian matrix of
system \eqref{eq:01} at $E_{+}$. Therefore, the transversality
condition at the Hopf point holds.
\end{proof}

The sign of the first Lyapunov coefficient \eqref{eq:02} is
determined by the sign of the numerator of \eqref{eq:02} since the
denominator is positive. If $\zeta_0 = (a,b_c) \in \mathcal{H}_+$,
$a \neq a_1$ then $l_1 (\zeta_0) \neq 0$ and system \eqref{eq:01}
has a transversal Hopf point at $E_{+}$ for the parameter vector
$\zeta_0$. More specifically, if $\zeta_0 = (a, b_c) \in
\mathcal{H}_+$, $0 < a< a_1 $, then $l_1 (\zeta_0) < 0$ and the Hopf
point at $E_+$ is asymptotically stable (weak attracting focus for
the flow of system \eqref{eq:01} restricted to the center manifold)
and for a suitable $\zeta$ close to $\zeta_0$ there exists a stable
limit cycle near the unstable equilibrium $E_+$; if $\zeta_0 = (a,
b_c) \in \mathcal{H}_+$, $a > a_1$, then $l_1 (\zeta_0) > 0$ and the
Hopf point at $E_+$ is unstable (weak repelling focus for the flow
of system \eqref{eq:01} restricted to the center manifold) and for a
suitable $\zeta$ close to $\zeta_0$ there exists an unstable limit
cycle near the asymptotically stable equilibrium $E_+$. See Figures
\ref{fig1} and \ref{fig2} where the stability of $E_+$ and small
amplitude limit cycles are depicted.

In the next theorem we study the stability of the equilibrium $E_+$
for the parameters in $\mathcal{H}_+$ when $a=a_1$.

\begin{theorem}\label{theo:06}
Consider system \eqref{eq:01} with parameters in $\mathcal{H}_+$, $a=a_1$. Then
the second Lyapunov coefficient at $E_+$ is positive.
\end{theorem}

\begin{proof}
Due to the quadratic nature of system \eqref{eq:01}, the multilinear
symmetric functions $D$ and $E$ are $D(x,y,z,w) = E(x,y,z,w,r) = (0,
0, 0)$. The complex vectors  $h_{ij}$ are too long and will be
omitted here. After a long calculation, it follows that the second
Lyapunov coefficient \eqref{coef2} at $E_+$ is given by
\begin{equation}\label{L2E1}
l_2 (a, b_c) = \frac{N(a)}{3 a^3 (1+a)^3 (1+a (3+a))^3 (1+a (6+a))^3
(1+a \ (11+a))},
\end{equation}
where
\begin{align*}
N(a)&=  20 a^{13}+3956 a^{12}+62848 a^{11}+394248 a^{10}+1125116 a^9\\
&\quad 20212 a^8- 8288340 a^7 -16285036 a^6 -11735384 a^5 \\
&\quad - 3575472 a^4- 523708 a^3-44300 a^2-2600 a-72.
\end{align*}
To study the real zeros of $N$ we recall Descartes Theorem: the
number of real positive roots of the real algebraic equation $N=0$,
counted with multiplicities, is at most the number of sign--changes
of terms of $N$. It is easy to see that $N(a)=0$ has at most one
positive real root. Since
\[
N(2)=-\frac{725431}{58852827}<0 \quad \text{and} \quad
N(3)=\frac{341087}{445944744}>0,
\]
the root of the equation $N=0$ is in the open interval $(2,3)$. 
Therefore $N(a_1)>0$. It follows that the sign of the second Lyapunov
coefficient is positive, since the denominator is positive.
\end{proof}

From Theorem \ref{theo:06}, the sign of the second Lyapunov
coefficient at $E_+$ is positive for parameters where $l_1=0$. Thus
the equilibrium $E_+$ is a weak repelling focus (for the flow of
system \eqref{eq:01} restricted to the center manifold) and there
are two limit cycles, one stable and the other unstable, near the
equilibrium $E_+$ for suitable value of the parameters. See the
pertinent bifurcation diagram in \cite[p. 313]{kuznet}. See also
Figures \ref{fig1} and \ref{fig2} where the stability of $E_+$ and
small amplitude limit cycles are depicted.

The proof of Theorem \ref{thm:main1} follows from Theorems
\ref{theo:01} and \ref{theo:06}.

\section{Concluding remarks}\label{S:4}

This paper starts reviewing the stability analysis which accounts
for the characterization, in the plane of parameters, of the
structural as well as Lyapunov stability of the equilibria of system
\eqref{eq:01}. It continues with the extension of the analysis to
the first order, codimension one points, based on the calculation of
the first Lyapunov coefficient for the equilibrium points $E_0$ and
$E_{\pm}$. The bifurcation analysis at the equilibria $E_{\pm}$ of
system \eqref{eq:01} is pushed forward to the calculation of the
second Lyapunov coefficient, which makes possible the determination
of the Lyapunov as well as higher order structural stability.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1}
\end{center}
\caption{Bifurcation diagram of system \eqref{eq:01}. See
also Figure \ref{fig2}}
\label{fig1}
\end{figure}

With the analytic data provided in the analysis performed here, the
bifurcation diagrams of equilibria $E_0$ and $E_+$ are established
and are put together in Figures \ref{fig1} and
\ref{fig2}, without danger of confusion. These figures provide a
qualitative synthesis of the dynamical
conclusions achieved at the parameter values where the system
\eqref{eq:01} has the most complex equilibrium points.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig2}
\end{center}
\caption{Sketch of the local phase portraits of system
\eqref{eq:01} related to the bifurcation diagram of Figure
\ref{fig1}}
\label{fig2}
\end{figure}


In Figure \ref{fig1} the dashed (continuous) curve $\mathcal{H}_0$
($\mathcal{H}_+$) is the Hopf curve of the equilibrium $E_0$ ($E_+$).
The dotted curve $\mathcal S$ represents the curve of
non--hyperbolic periodic orbits. The point $P_1$ has coordinates $a
= a_1$ and $b = b_c$. The phase portraits for the flow of system
\eqref{eq:01} restricted to the center manifold and its
continuations related to the points $P_1, \ldots, P_{10}$ are
illustrated in Figure \ref{fig2} according to the following
convention: linear repelling focus in (a) for the points $P_3$
($E_+$) and $P_9$ ($E_0$); weak repelling focus in (b) for the
points $P_2$ ($E_+$) and $P_8$ ($E_0$); linear attracting focus and
one repelling hyperbolic cycle in (c) for the points $P_7$ ($E_+$)
and $P_{10}$ ($E_0$); weak attracting focus and one repelling
hyperbolic cycle in (d) for the point $P_6$ ($E_+$); linear
repelling focus and two hyperbolic cycles in (e) for the point $P_5$
($E_+$), linear repelling focus and one non--hyperbolic cycle in (f)
for the point $P_4$ ($E_+$); more weak repelling focus in (g) for
the point $P_1$ ($E_+$).


\subsection*{Acknowledgements} 
The first author is supported by grant 2011/01946--0 from
FAPESP.  This article was written during the
postdoctoral program of F. S. Dias at the Instituto de Ci\^encias
Matem\'aticas e de Computa\c c\~ao, USP, S\~ao Carlos, Brazil. The
second author is partially supported by CNPq grant 304926/2009--4
and by FAPEMIG grant PPM--00204--11.

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\end{document}