\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 161, pp. 1--25.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/161\hfil Analysis of a quadratic system] {Analysis of a quadratic system obtained from a scalar third order differential equation} \author[F. S. Dias, L. F. Mello \hfil EJDE-2010/161\hfilneg] {Fabio Scalco Dias, Luis Fernando Mello} % in alphabetical order \address{Fabio Scalco Dias \newline Instituto de Ci\^encias Exatas, 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 Ci\^encias Exatas, Universidade Federal de Itajub\'a\\ Avenida BPS 1303, Pinheirinho, CEP 37.500-903, Itajub\'a, MG, Brazil} \email{lfmelo@unifei.edu.br, Tel 00-55-35-36291217, Fax 00-55-35-36291140} \thanks{Submitted February 15, 2010. Published November 10, 2010.} \subjclass[2000]{70K50, 70K20, 34C60} \keywords{Bifurcation; Hopf bifurcation; Bogdanov-Takens bifurcation; \hfill\break\indent fold-Hopf bifurcation; limit cycle; stability; quadratic system} \begin{abstract} In this article, we study the nonlinear dynamics of a quadratic system in the three dimensional space which can be obtained from a scalar third order differential equation. More precisely, we study the stability and bifurcations which occur in a parameter dependent quadratic system in the three dimensional space. We present an analytical study of codimension one, two and three Hopf bifurcations, generic Bogdanov-Takens and fold-Hopf bifurcations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \allowdisplaybreaks \section{Introduction}\label{S:1} In this paper we study the stability and bifurcations in the dynamics of the third order differential equation \begin{equation}\label{eq:01} x''' + f(x) x'' + g(x) x' + h(x) = 0, \end{equation} where $f, g, h : \mathbb{R} \to \mathbb{R}$ are \begin{equation}\label{eq:02} f(x) = a_1 x + a_0, \quad g(x) = b_1 x + b_0, \quad h(x) = c_2 x^2 + c_1 x + c_0, \end{equation} with $a_1, a_0, b_1, b_0, c_2, c_1, c_0 \in \mathbb{R}$, $c_2 \neq 0$. By defining of the variables $y = x'$ and $z = x''$, differential equation \eqref{eq:01} can be written as the system of nonlinear differential equations \begin{equation}\label{eq:03} \begin{gathered} x' = y,\\ y' = z,\\ z' = - \big( (a_1 x + a_0) z + (b_1 x + b_0) y + c_2 x^2 + c_1 x + c_0 \big), \end{gathered} \end{equation} where $(x,y,z) \in \mathbb{R}^3$ are the state variables and $(a_0, a_1, b_0, b_1, c_0, c_1, c_2) \in \mathbb{R}^7$, $c_2 \neq 0$, are real parameters. The choice of real affine functions to $f$ and $g$ and a quadratic function to $h$ imply that the vector field that defines \eqref{eq:03}, \begin{equation}\label{eq:04} F(x,y,z) = \left( y, z, - \left( (a_1 x + a_0) z + (b_1 x + b_0) y + c_2 x^2 + c_1 x + c_0 \right) \right), \end{equation} is a quadratic vector field. So, system \eqref{eq:03} is a quadratic system of differential equations in $\mathbb{R}^3$. 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 \cite{lo}, the Chen system \cite{chen}, the Liu system \cite{liu}, the R\"ossler system \cite{rossler}, the Rikitake system \cite{rikitake}, the L\"u system \cite{lc}, the Genesio system \cite{genesio} among several others. An interesting problem related to quadratic systems defined in $\mathbb{R}^n$ 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{libre} provided an example of quadratic vector field in $\mathbb{R}^3$ and an analytical proof that it has infinitely many limit cycles. As far as we know, differential equation \eqref{eq:01}, or equivalently system \eqref{eq:03}, was analyzed in two particular cases: \begin{enumerate} \item [(a)] When $a_1 = b_1 = c_0 = 0$, $c_1 = 1$ and $c_2 = -1$ differential equation \eqref{eq:01} is a feedback control system of Lur'e type. The Hopf bifurcations of codimension one of the equivalent system \eqref{eq:03} were studied in \cite{kuznet}; \item [(b)] When $a_1 = b_1 = c_0 = 0$ and $c_2 = -1$ differential equation \eqref{eq:01} is an extension of the above feedback control system of Lur'e type. The equivalent system \eqref{eq:03} was studied in \cite{genesio} from the chaotic behavior point of view and in \cite{zc} were studied its Hopf bifurcations of codimension one and homoclinic connections. \end{enumerate} On the other hand, differential equation \eqref{eq:01}, or equivalently system \eqref{eq:03}, can be seen as a particular case of a more general quadratic third order differential equation \cite{igg}. In \cite{igg} the authors studied oscillations that appear from codimension one Hopf bifurcations. The study was made using an approach based on harmonic balance techniques. However there exist more degenerate cases that were not analyzed by them. Despite the simplicity, system \eqref{eq:03} has a rich local dynamical behavior presenting several degenerate bifurcations. The study carried out in the present paper may contribute to understand analytically the stability and some bifurcations of system \eqref{eq:03}. For this purpose the paper is organized as follows. After some general results the linear analysis of the equilibria of system \eqref{eq:03} is presented in Section \ref{S:2}. A brief review of the methods used to study Hopf, Bogdanov-Takens and fold-Hopf bifurcations are presented in Section \ref{S:3}. These methods are used in Section \ref{S:4} to prove the main results of this paper. More specifically, in subsections \ref{S:41} and \ref{S:42} we study all the possible Hopf bifurcations (generic and degenerate ones) which occur in the equilibria of system \eqref{eq:03}. An application of these results is made in subsection \ref{S:43} for a particular case of system \eqref{eq:03}. In subsection \ref{S:44} we present the study of a Bogdanov-Takens bifurcation which occurs at an equilibrium point of system \eqref{eq:03} for a suitable choice of the parameters. This study leads to the existence of homoclinic connections and global bifurcations in system \eqref{eq:03}. Other global bifurcations in system \eqref{eq:03} can be determined by the existence of a fold-Hopf bifurcation at an equilibrium point for a suitable choice of the parameters. The study of this bifurcation is presented in subsection \ref{S:45}. In Section \ref{S:5} we make some concluding comments. \section{Linear analysis of system \eqref{eq:03}}\label{S:2} The equilibria of system \eqref{eq:03} are $E_{\ast} = (x_{\ast}, 0, 0)$, where $x_{\ast}$ is a real zero of the function $h$, that is $h(x_{\ast}) = 0$. By assumption $h$ is a quadratic function, so it may have 0, 1 or 2 real zeros. This implies that system \eqref{eq:03} has 0, 1 or 2 equilibrium points. The local behavior of the flow of system \eqref{eq:03} is trivial when there is no equilibrium point. Nevertheless the global behavior of the flow can be very interesting with the study, for example, of large amplitude limit cycles, that is limit cycles out of compact parts of $\mathbb{R}^3$ \cite{llibre1}. In this paper we only study the cases with 1 or 2 equilibria. Suppose that system \eqref{eq:03} has only one equilibrium point. Without loss of generality, we can consider $h(x) = x^2$, that is $c_2 = 1$ and $c_1 = c_0 = 0$. This implies that the equilibrium point $E_{\ast}$ is at the origin. The linear part of system \eqref{eq:03} at the origin has the form \[ A = DF ( E_{\ast} ) = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -b_0 & -a_0 \end{pmatrix}, \] and its characteristic polynomial is \begin{equation}\label{eq:05} p(\lambda) = -\lambda \left( \lambda^2 + a_0 \lambda + b_0 \right). \end{equation} It follows that one eigenvalue is $\lambda_1 = 0$ and this implies that the origin is a non-hyperbolic equilibrium point. A more detailed study of the stability of this equilibrium point is presented in subsections \ref{S:44} and \ref{S:45}. Now suppose that system \eqref{eq:03} has two equilibrium points. Thus the function $h$ has the form $h(x) = c_2 (x-x_0)(x-x_1)$, $c_2 \neq 0$. By the following change of coordinates and a reparametrization in time \[ x = X, \quad y = c_2^{1/3} Y, \quad z = c_2^{2/3} Z, \quad t = c_2^{1/3} \tau, \] system \eqref{eq:03} can be written with a function $h$ of the form $h(x) = (x-x_0)(x-x_1)$. Without loss of generality, we can consider $x_0 = 0$ and $x_1 = -1$. It follows that system \eqref{eq:03} has the equilibria $E_0=(0,0,0)$ and $E_1=(-1,0,0)$ and can be written as \begin{equation}\label{syst2} \begin{gathered} x'=y,\\ y'=z,\\ z'=-\big( (a_1x+a_0)z+ (b_1x+b_0)y+x(x+1)\big), \end{gathered} \end{equation} where $(x,y,z) \in \mathbb{R}^3$ are the state variables and $(a_0, b_0, a_1, b_1) \in \mathbb{R}^4$ are real parameters. A useful tool for the linear analysis of an equilibrium point is the following Routh-Hurwitz stability criterion whose proof can be found in \cite[p. 58]{pon}. \begin{lemma}\label{R-H} The polynomial $L(\lambda) = \lambda ^3 + p_1 \lambda ^2 + p_2 \lambda + p_3$ with real coefficients has all roots with negative real parts if and only if the numbers $p_1$, $p_2$, $p_3$ are positive and the inequality $p_1 p_2 > p_3$ is satisfied. \end{lemma} \subsection{Linear analysis at $E_0$}\label{S:21} In this subsection we study the stability of the equilibrium $E_0=(0,0,0)$ of system \eqref{syst2} from the linear point of view. Consider the set of parameters \[ \mathcal{W} = \{ (a_0, b_0, a_1, b_1) \in \mathbb{R}^4 \}. \] We have the following proposition. \begin{proposition}\label{laE0} Define the following subsets of $\mathcal{W}$: \begin{gather*} \mathcal{W}_1 = \{(a_0, b_0, a_1, b_1) \in \mathcal{W} : a_0 \leq 0 \} \cup \{(a_0, b_0, a_1, b_1) \in \mathcal{W} : b_0 \leq 0 \}, \\ \mathcal{W}_2 = \{(a_0, b_0, a_1, b_1) \in \mathcal{W} : a_0 > 0,\; b_0 > 0, \; a_0 b_0 < 1 \}, \\ \mathcal{W}_3 = \{(a_0, b_0, a_1, b_1) \in \mathcal{W} : a_0 > 0,\; b_0 > 0,\; a_0 b_0 > 1 \}. \end{gather*} Then the following statements hold: \begin{enumerate} \item If $(a_0, b_0, a_1,b_1) \in \mathcal{W}_1$ then the equilibrium $E_0$ is unstable; \item If $(a_0, b_0, a_1,b_1) \in \mathcal{W}_2$ then the equilibrium $E_0$ is unstable; \item If $(a_0, b_0, a_1,b_1) \in \mathcal{W}_3$ then the equilibrium $E_0$ is locally asymptotically stable. \end{enumerate} \end{proposition} \begin{proof} The characteristic polynomial of the Jacobian matrix of system \eqref{syst2} at $E_{0}$ is \[ p(\lambda) = \lambda^3 + a_0 \lambda^2 + b_0\lambda + 1. \] If $(a_0, b_0, a_1, b_1) \in \mathcal{W}_1$ then the coefficients $a_0$ and $b_0$ of $p(\lambda)$ are non-positive. From Lemma \ref{R-H} it follows that the equilibrium $E_{0}$ is unstable. This proves item 1 of the proposition. From Lemma \ref{R-H} the equilibrium $E_{0}$ is locally asymptotically stable if the coefficients of the characteristic polynomial satisfy \begin{equation}\label{R-H-1} a_0 > 0, \quad b_0 > 0, \quad a_0b_0 > 1. \end{equation} So if $(a_0, b_0, a_1, b_1) \in \mathcal{W}_2$ then $E_{0}$ is unstable and if $(a_0, b_0, a_1, b_1) \in \mathcal{W}_3$ then $E_{0}$ is locally asymptotically stable. This proves item 2 and 3 of the proposition. \end{proof} Define the set \begin{equation}\label{H0} \mathcal{H}_0 = \{(a_0, b_0, a_1, b_1) \in \mathcal{W} : a_0 > 0, \; b_0 > 0, \; a_0 b_0 = 1 \}. \end{equation} Thus $\mathcal{W} = \mathcal{W}_1 \cup \mathcal{W}_2 \cup \mathcal{W}_3 \cup \mathcal{H}_0$. If $(a_0, b_0, a_1, b_1) \in \mathcal{H}_0$ then the equilibrium $E_0$ is non-hyperbolic, that is the Jacobian matrix of system \eqref{syst2} at $E_{0}$ has one negative real eigenvalue and a pair of purely imaginary eigenvalues \[ \lambda_1 = - \frac{1}{b_0} < 0, \quad \lambda_{2,3} = \pm i \sqrt{b_0}. \] The set $\mathcal{H}_0$ is called the Hopf hypersurface of the equilibrium $E_{0}$. 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{syst2} and can be continued with arbitrary high class of differentiability to nearby parameter values (see \cite[p. 152]{kuznet}). This center manifold is attracting since $\lambda_1 < 0$. So it is enough to study the stability of $E_{0}$ for the flow restricted to the family of parameter-dependent continuations of the center manifold. A detailed analysis of this case will be presented in subsection \ref{S:41}. \subsection{Linear analysis at $E_1$}\label{S:22} In this subsection, we study the stability of the equilibrium $E_1=(-1,0,0)$ of system \eqref{syst2} from the linear point of view. The characteristic polynomial of the Jacobian matrix of system \eqref{syst2} at $E_1$ is \[ p(\lambda) = \lambda^3 + (a_0-a_1) \lambda^2 + (b_0-b_1) \lambda - 1. \] The coefficient $-1$ of $p(\lambda)$ is negative. From Lemma \ref{R-H} it follows that the equilibrium $E_1$ is unstable for all parameters $(a_0, b_0, a_1, b_1) \in \mathcal{W} $. Define the set \begin{equation}\label{H1} \mathcal{H}_1 = \{(a_0, b_0, a_1, b_1) \in \mathcal{W} : (a_0 - a_1) < 0, \; (a_0-a_1)(b_0-b_1) = -1 \}. \end{equation} If $(a_0, b_0, a_1, b_1) \in \mathcal{H}_1$ then the Jacobian matrix of system \eqref{syst2} at $E_1$ has eigenvalues \[ \lambda_1 = (a_1-a_0) > 0, \quad \lambda_{2,3} = \pm i \frac{1}{\sqrt{a_1-a_0}}. \] The set $\mathcal{H}_1$ is called the Hopf hypersurface of the equilibrium $E_1$. 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{syst2} and can be continued with arbitrary high class of differentiability to nearby parameter values (see \cite[p. 152]{kuznet}). This center manifold is repelling since $\lambda_1 > 0$. We are interested in the study the stability of $E_1$ for the flow restricted to the family of parameter-dependent continuations of the center manifold. A detailed analysis of this case will be presented in subsection \ref{S:42}. \section{Generalities on Hopf, Bogdanov-Takens and fold-Hopf bifurcations}\label{S:3} \subsection{Hopf bifurcations}\label{S:31} In this subsection 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, denoted by $l_1$ and $l_2$ respectively. 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 $\zeta \in \mathbb{R}^n$ are respectively vectors representing phase variables and control parameters. Assume that $f$ is of class $C^{\infty}$ in $\mathbb{R}^3 \times \mathbb{R}^n$. 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) + \frac{1}{120} E(x, x, x, x, x) \\ &\quad + \frac{1}{720} K(x, x, x, x,x , x)+ \frac{1}{5040} L(x, x, x, x, x, x, x) + O(\| x \|^8), \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 $D_i$, $E_i$, $K_i$ and $L_i$. 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 eigenvalues $\lambda_1 \neq 0$. 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^T p = -i \omega_0 p, \quad \langle p,q \rangle = \sum_{i=1}^3 \bar{p}_i q_i = 1, \end{equation} where $A^T$ 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 7} \frac{1}{j!k!} h_{jk}w^j{\bar w}^k + O(|w|^8), \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{smb1}, p. 27), so that system \eqref{ku}, on the chart $w$ for a central manifold, writes as \[ w'= i \omega_0 w + \frac{1}{2} G_{21} w |w|^2 + \frac{1}{12} G_{32} w |w|^4 + \frac{1}{144} G_{43} w |w|^6 + O(|w|^8), \] with $G_{jk} \in \mathbb{C}$. The \emph{first Lyapunov coefficient} $l_1$ is \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 \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*} The \emph{third Lyapunov coefficient} is \begin{equation} \label{coef3} l_3= \frac{1}{144} \operatorname{Re} G_{43}, \end{equation} where $G_{43} = \langle p, \mathcal{H}_{43} \rangle$. The expression for $\mathcal{H}_{43}$ is too large to be put in print and can be found in \cite[eq. (44)]{smb1}. 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. As $l_1 < 0$ ($l_1 > 0$) 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 point $(x_0,\zeta_0)$ 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}, and in \cite{takens}. A {\it Hopf point of codimension 3} is a Hopf point of codimension 2 where $l_2$ vanishes but $l_3 \neq 0$. A Hopf point of codimension 3 is called {\it transversal} if the sets $\gamma^{-1} (0)$, $l_1^{-1} (0)$ and $l_2^{-1} (0)$ have transversal intersections. The bifurcation diagram for $l_3 \neq 0$ can be found in \cite{smb1} and in Takens \cite{takens}. \subsection{Bogdanov-Takens bifurcations}\label{S:32} In this subsection we present an approach based on \cite[p. 321]{kuznet}, and \cite{kuznet1} for the Bogdanov-Takens bifurcation. Consider a system $x' = f (x,\alpha)$, $x \in \mathbb{R}^3$, $\alpha \in \mathbb{R}^n$ and assume that $f$ is of class $C^{\infty}$ in $\mathbb{R}^3 \times \mathbb{R}^n$. Suppose that for $\alpha = \alpha_0$ there is an equilibrium point $x = x_0$ such that the Jacobian matrix $A$ of $f$ at $x_0$ has a double zero eigenvalue; that is, $\lambda_{2,3} = 0$ and the other eigenvalue $\lambda_1 \neq 0$. Denoting the variable $x-x_0$ also by $x$ we consider \[ F(x) = f(x, \alpha_0) = A x + \frac{1}{2} B(x,x) + O(\|x\|^3), \] where, for $i = 1,2,3$, \[ B_i (x,y) = \sum_{j,k=1}^3 \frac{\partial ^2 F_i(\xi)}{\partial \xi_j \partial \xi_k} \Big|_{\xi=0} x_j y_k. \] Let $q_0,q_1,p_0,p_1 \in \mathbb{R}^3$ be vectors such that $A q_0 = 0$, $A q_1 = q_0$, $A^T p_1 = 0$, $A^T p_0 = p_1$, where $A^T$ is the transpose of the matrix $A$, satisfying the conditions $\langle q_0,p_1 \rangle = 0$, $\langle q_1,p_0 \rangle =0$, $\langle q_0,p_0 \rangle = 1$ and $\langle q_1,p_1 \rangle = 1$. Write the polynomial characteristic of the Jacobian matrix of $f$ at $(x, \alpha)$ as $p(\lambda) = \lambda^3 + R(x, \alpha) \lambda^2 + T(x, \alpha) \lambda + D(x, \alpha)$ and assume that the following conditions hold: \begin{itemize} \item[(BT1)] The Jacobian matrix satisfies $A \neq 0$; \item[(BT2)] \begin{equation}\label{bt2} a (\alpha_0) = \frac{1}{2} \langle p_1,B(q_0,q_0) \rangle \neq 0; \end{equation} \item [(BT3)] \begin{equation}\label{bt3} b (\alpha_0) = \langle p_0,B(q_0,q_0) \rangle + \langle p_1,B(q_0,q_1) \rangle \neq 0; \end{equation} \item[(BT4)] The map $G: (x, \alpha) \to (f (x,\alpha), T(x,\alpha), D(x,\alpha))$ is regular at $(x_0,\alpha_0)$. \end{itemize} Under the above assumptions the system undergoes a Bogdanov-Takens bifurcation at $x_0$ for parameters at $\alpha_0$. The bifurcation diagram of the Bogdanov-Takens bifurcation can be found in \cite[p. 322]{kuznet}. The assumption (BT4) is called transversality condition for the Bogdanov-Takens bifurcation while the assumptions (BT1)-(BT3) are the non-degenerescence conditions. Define $s = \operatorname{sign} a(\alpha_0) b(\alpha_0) = \pm 1$. If $s = -1$ ($s = 1$, resp.) then the limit cycle bifurcating from the Hopf point or from the homoclinic loop is attracting (repelling, resp.). \subsection{Fold-Hopf bifurcations}\label{S:33} In this subsection a review of the fold-Hopf bifurcation is presented based on \cite{kuznet} and \cite{kuznet1}. This kind of bifurcation is also called zero-Hopf bifurcation. Consider the differential equation \eqref{diffequat}, where $x \in \mathbb{R}^3$ and $\zeta \in \mathbb{R}^n$ are respectively vectors representing phase variables and control parameters. Assume that $f$ is of class $C^{\infty}$ in $\mathbb{R}^3 \times \mathbb{R}^n$. Suppose that \eqref{diffequat} has an equilibrium point $x = x_0$ at $\zeta = \zeta_0 = 0$. Denoting the variable $x-x_0$ also by $x$, we can write \eqref{Fhomo} as \[ F(x) = f (x, 0) \] where \[ F(x) = Ax + \frac{1}{2} B(x,x) + \frac{1}{6} C(x, x, x) + O(\| x \|^4), \] $A = f_{x}(0,0)$ and, for $i = 1,2,3$, \[ B_i (x,y) = \sum_{j,k=1}^3 \frac{\partial ^2 F_i(\xi)}{ \partial \xi_j \partial \xi_k} \Big|_{\xi=0} x_j y_k, \quad C_i ( x,y,z) = \sum_{j,k,l=1}^3 \frac{\partial ^3 F_i(\xi)}{\partial \xi_j \partial \xi_k \partial \xi_l} \Big|_{\xi=0} x_j y_k z_l. \] Suppose that $(x_0, \zeta_0) = (0,0)$ is an equilibrium point of \eqref{diffequat} where the Jacobian matrix $A$ has a zero eigenvalue $\lambda_1 = 0$ and a pair of purely imaginary eigenvalues $\lambda_{2,3} = \pm i \omega_0$, $\omega_0 > 0$. Let $p_0, q_0 \in \mathbb{R}^3$ be vectors such that \begin{equation}\label{norm:fh1} A q_0 = 0, \quad A^T p_0 = 0, \quad \langle p_0,q_0 \rangle = 1, \end{equation} and let $p_1, q_1 \in \mathbb{C}^3$ be vectors such that \begin{equation}\label{norm:fh2} A q_1 = i \omega_0 q_1, \quad A^T p_1 = -i \omega_0 p_1, \quad \langle p_1,q_1 \rangle = 1, \end{equation} where $A^T$ is the transpose of the matrix $A$. From the above assumptions, it follows that \[ \langle p_1,q_0 \rangle = \langle p_0,q_1 \rangle = 0. \] Consider the complex numbers \begin{gather}\label{G200} G_{200} = \langle p_0,B(q_0, q_0) \rangle, \\ \label{G110} G_{110} = \langle p_1,B(q_0, q_1) \rangle, \\ \label{G011} G_{011} = \langle p_0,B(q_1, \bar{q}_1) \rangle, \end{gather} the complex vectors, in $\mathbb{C}^3$, \begin{equation}\label{h020} h_{020} = (2 i \omega_0 I_3 - A)^{-1} B(q_1,q_1), \end{equation} $h_{200}$ the solution of \begin{equation}\label{h200} \begin{pmatrix} A & q_0 \\ p_0 & 0 \end{pmatrix} \begin{pmatrix} h_{200}\\ \\ s \end{pmatrix} = \begin{pmatrix} -B(q_0, q_0) + \langle p_0,B(q_0, q_0) \rangle q_0 \\ \\ 0 \end{pmatrix}, \end{equation} $h_{011}$ the solution of \begin{equation}\label{h011} \begin{pmatrix} A & q_0 \\ p_0 & 0 \end{pmatrix} \begin{pmatrix} h_{011}\\ \\ s \end{pmatrix} = \begin{pmatrix} -B(q_1, {\bar q}_1) + \langle p_0,B(q_1, {\bar q}_1) \rangle q_0 \\ \\ 0 \end{pmatrix}, \end{equation} and the vector $h_{110}$ which is solution of \begin{equation}\label{h110} \begin{pmatrix} i \omega_0 I_3 - A & q_1 \\ {\bar{p}}_1 & 0 \end{pmatrix} \begin{pmatrix} h_{110}\\ \\ s \end{pmatrix} = \begin{pmatrix} B(q_0, q_1) - \langle p_1,B(q_0, q_1) \rangle q_1 \\ \\ 0 \end{pmatrix}. \end{equation} From the above complex vectors define the complex numbers \begin{gather}\label{G300} G_{300} = \langle p_0,C(q_0,q_0,q_0) + 3 B(q_0, h_{200}) \rangle,\\ \label{G111} G_{111} = \langle p_0,C(q_0,q_1,{\bar q}_1) + B(q_0, h_{011}) + B(q_1, {\bar h}_{110}) + B({\bar q}_1, h_{110}) \rangle,\\ \label{G210} G_{210} = \langle p_1,C(q_0,q_0,q_1) + 2 B(q_0, h_{110}) + B(q_1, h_{200}) \rangle, \\ \label{G021} G_{021} = \langle p_1,C(q_1,q_1,{\bar q}_1) + 2 B(q_1, h_{011}) + B({\bar q}_1, h_{020}) \rangle. \end{gather} The theorem about the fold-Hopf bifurcation states that if \begin{enumerate} \item [(FH1)] $b(0) c(0) e(0) \neq 0$, \item [(FH2)] The map $G: (x, \zeta) \mapsto (f(x,\zeta), \operatorname{Tr} (f_x(x, \zeta)), \det (f_x(x, \zeta)))$ is regular at $(x_0,\zeta_0) = (0,0)$, \end{enumerate} then \eqref{diffequat} is locally orbitally smoothly equivalent near the origin to the complex normal form \begin{gather*} \xi' = \beta_1 + b(\beta) \xi^2 + c(\beta) |\chi|^2 + O(\|(\xi, \chi)\|^4), \\ \chi' = (\beta_2 + i \omega(\beta)) \chi + d(\beta) \xi \chi + e(\beta) \xi^2 \chi + O(\|(\xi, \chi)\|^4), \end{gather*} where $\beta = (\beta_1, \beta_2)$, $\omega (0) = \omega_0$, \begin{equation}\label{b0} b(0) = \frac{G_{200}}{2}, \quad c(0) = G_{011}, \quad d(0) = G_{110} - i \omega_0 \frac{G_{300}}{3 G_{200}} \end{equation} and \begin{equation}\label{e0} e(0) = \frac{1}{2} \operatorname{Re} \Big( G_{210} + G_{110} \Big( \frac{\operatorname{Re} G_{021}}{G_{011}} - \frac{G_{300}}{G_{200}} + \frac{G_{111}}{G_{011}} \Big) - \frac{G_{021} G_{200}}{2 G_{011}} \Big). \end{equation} In general the $O$-terms cannot be truncated. See \cite[p. 336.]{kuznet}, Depending upon the coefficients $b(0)$, $c(0)$, $d(0)$ and $e(0)$ the system can have two-dimensional invariant tori and even chaotic motions. Define \begin{equation}\label{stheta} s = \operatorname{sign} b(0) c(0), \quad \theta(0) = \frac{\operatorname{Re} d(0)}{G_{200}}. \end{equation} For example, if $s = 1$ and $\theta (0) < 0$ then the system exhibits Hopf bifurcations and torus ``heteroclinic destruction'' (see \cite[p. 341]{kuznet}), giving rise to chaotic invariant sets. The bifurcation diagrams for the fold-Hopf bifurcation can be found in \cite[pp. 339--343]{kuznet}. \section{Bifurcation analysis of system \eqref{eq:03}}\label{S:4} \subsection{Hopf bifurcation analysis at $E_0$}\label{S:41} In this subsection we study the Hopf bifurcations that occur at the equilibrium $E_0$ for parameters in the set $\mathcal{H}_0$ defined in \eqref{H0}. Define the critical parameter \[ a_{0_c}=\frac{1}{b_0} > 0. \] \begin{theorem}\label{theo:01} Consider system \eqref{syst2}. The first Lyapunov coefficient at $E_{0}$ for parameter values in $\mathcal{H}_0$ is \begin{equation}\label{coefL1} l_1 (a_{0_c}, b_0, a_1, b_1) = \frac{N(a_{0_c},b_0,a_1,b_1)}{2(b_0 +5b_0^4 + 4b_0^7)}, \end{equation} where \begin{align*} N(a_{0_c},b_0,a_1,b_1) &= b_1-b_0 \Big(2+b_0 \Big(16b_0^2 + a_1^2b_0 (-3 + 8b_0^3) \\ &\quad - 10b_0b_1 + b_1^2 + a_1 \big(1 + 12b_0^2(-2b_0 + b_1)\big)\Big)\Big). \end{align*} If $\zeta_0 = (a_{0_c}, b_0, a_1, b_1) \in \mathcal{H}_0$ is such that $l_1 (\zeta_0) \neq 0$ then system \eqref{syst2} has a transversal Hopf point at $E_{0}$ for the parameter vector $\zeta_0$. \end{theorem} \begin{proof} For parameters on the Hopf hypersurface $\mathcal{H}_0$ \eqref{H0}, the eigenvalues of the Jacobian matrix of system \eqref{syst2} at $E_0$ are \[ \lambda_1 = -\frac{1}{b_0}, \quad \lambda_{2,3}= \pm i \omega_0, \quad \omega_0 = \sqrt{b_0}, \quad b_0> 0, \] the eigenvectors $q$ and $p$ defined in \eqref{normalization} are \[ q = \Big( -\frac{1}{b_0},\frac{-i}{\sqrt{b_0}},1 \Big), \quad p = \Big(\frac{-i b_0}{2(b_0^{3/2}+i)}, \frac{-i\sqrt{b_0}}{2}, \frac{b_0^{3/2}}{2(b_0^{3/2}+i)}\Big) \] and the multilinear symmetric functions $B$ and $C$ write as {\small \[ B(x,y) = \left( 0,0, - a_1 (x_1 y_3 + x_3 y_1) -b_1 (x_1 y_2 + x_2 y_1) - 2 x_1 y_1 \right), \quad C(x,y,z) = \left( 0, 0, 0 \right). \] } The complex vectors $h_{11}$ and $h_{20}$ are \[ h_{11}=\Big( \frac{2 (-1+a_1 b_0)}{b_0^2},0,0 \Big), \] \[ h_{20}=\Big( \frac{2(-i+ia_1b_0 + \sqrt{b_0}b_1)}{3b_0^2(-i + 2b_0^{3/2})}, \frac{ 4(1 - a_1b_0 + i\sqrt{b_0}b_1)}{3b_0^{3/2}(-i+ 2b_0^{3/2})}, \frac{8(i - ia_1b_0 - \sqrt{b_0}b_1}{3b_0(-i + 2b_0^{3/2})}\Big). \] The complex number $G_{21}$ defined in \eqref{coef1} has the form \begin{align*} G_{21}&=\Big(a_1^2b_0^2 (i-12b_0^{2/3})-i \big(-5i + \sqrt{b0} (12b_0-b_1)\big )(-2i + \sqrt{b_0}b_1) \\ &\quad + a_1b_0 (-11i + 36b_0^{2/3} + 12ib_0^2b_1)\Big)/ \Big(-3b_0^{2/3}-9ib_0^3+6b_0^{9/2}\Big). \end{align*} Performing the calculations in \eqref{coef1}, the first Lyapunov coefficient is given by \eqref{coefL1}. It remains only to verify the transversality condition of the Hopf bifurcation. In order to do so, consider the family of differential equations \eqref{syst2} regarded as dependent on the parameter $a_0$. The real part, $\gamma=\gamma (a_0)$, of the pair of complex eigenvalues at the critical parameter $a_0=a_{0_c}$ verifies \[ \gamma'(a_{0_c}) = \operatorname{Re} \langle p, \frac{dA}{da_0}\Big|_{a_0=a_{0_c}} q \rangle = - \frac{b_0^3}{2( b_0^3 +1)} < 0, \] since $b_0 > 0$. In the above expression $A$ is the Jacobian matrix of system \eqref{syst2} at $E_{0}$. Therefore, the transversality condition at the Hopf point holds. \end{proof} The sign of the first Lyapunov coefficient \eqref{coefL1} is determined by the sign of the numerator of \eqref{coefL1}, $N(a_{0_c},b_0,a_1,b_1)$, since the denominator is positive. If $\zeta_0 = (a_{0_c}, b_0, a_1, b_1) \in \mathcal{H}_0$ is such that $l_1 (\zeta_0) \neq 0$ then system \eqref{syst2} has a transversal Hopf point at $E_{0}$ for the parameter vector $\zeta_0$. More specifically, if $\zeta_0 = (a_{0_c}, b_0, a_1, b_1) \in \mathcal{H}_0$ is such that $l_1 (\zeta_0) < 0$ then the Hopf point at $E_0$ is asymptotically stable (weak attracting focus for the flow of system \eqref{syst2} restricted to the center manifold) and for a suitable $\zeta$ close to $\zeta_0$ there exists a stable limit cycle near the unstable equilibrium point $E_0$; if $\zeta_0 = (a_{0_c}, b_0, a_1, b_1) \in \mathcal{H}_0$ is such that $l_1 (\zeta_0) > 0$ then the Hopf point at $E_0$ is unstable (weak repelling focus for the flow of system \eqref{syst2} 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 point $E_0$. In the rest of this subsection we study the stability of the equilibrium $E_0$ with the restriction $a_1 = 0$. This makes the analysis of the sign as well as the analysis of the zero set of the first Lyapunov coefficient \eqref{coefL1} more simple. See Remark \ref{signL1}. Define the following subset $\mathcal{H}_{00}$ of the Hopf hypersurface $\mathcal{H}_0$ \[ \mathcal{H}_{00}=\{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{0} : a_1=0 \}. \] \begin{corollary}\label{cor:01} Consider system \eqref{syst2} with parameter values in $\mathcal{H}_{00}$. If either \[ b_1 = b_{11}= \frac{1+8 b_0^3}{b_0^2} \quad\text{or}\quad b_1 = b_{12}= 2 b_0, \] then the first Lyapunov coefficient at $E_{0}$ vanishes; that is, \[ l_1 (a_{0_c}, b_0, 0, b_{11}) = l_1 (a_{0_c}, b_0, 0, b_{12}) = 0. \] \end{corollary} \begin{proof} Substituting $a_1 = 0$ into the expression of $G_{21}$ in the proof of Theorem \ref{theo:01} results \begin{align*} G_{21}&=-\frac{(2b_0-b_1)(1 + 8b_0^3 - b_0^2b_1)}{b_0 + 5b_0^4 + 4b_0^7} \\ &\quad + i \frac{-10 + b_0(-52b_0^2 + 3b_0b_1( 1 - 8b_0^3) + b_1^2(-1 + 2b_0^3))}{3b_0^{3/2} (1+ 5b_0^3 + 4b_0^6)}. \end{align*} If $b_1 = b_{11}$ then the second parenthesis in the numerator of the real part of $G_{21}$ vanishes. Then $l_1(a_{0_c}, b_0, 0, b_{11})=0$. On the other hand, if $b_1 = b_{12}$ then the first parenthesis in the numerator of the real part of $G_{21}$ vanishes. Then $l_1(a_{0_c}, b_0, 0, b_{12})=0$. \end{proof} From Corollary \ref{cor:01} the first Lyapunov coefficient vanishes on the curves \[ \mathcal{L}_1= \big\{(a_0,b_0,b_1) \in \mathcal{H}_{00} : a_0=\frac{1}{b_0}, \quad b_1 = \frac{1+8b_0^3}{b_0^2} \big\} \] and \[ \mathcal{L}_2= \big\{(a_0,b_0,b_1) \in \mathcal{H}_{00} : a_0=\frac{1}{b_0}, \quad b_1 = 2b_0 \big\}. \] See Figure \ref{fig1}. It is simple to see that the curves $\mathcal{L}_1$ and $\mathcal{L}_2$ have no intersection and divide the Hopf surface $\mathcal{H}_{00}$ into three connected components \begin{gather*} \mathcal{H}_{01}= \{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{00} : b_1 > \frac{1+8b_0^3}{b_0^2} \}, \\ \mathcal{H}_{02}= \{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{00} : 2b_0 < b_1 < \frac{1+8b_0^3}{b_0^2} \}, \\ \mathcal{H}_{03}= \{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{00} : b_1 < 2b_0 \}, \end{gather*} where the sign of the first Lyapunov coefficient at $E_0$ is fixed: $l_1 (a_{0_c}, b_0, 0, b_1)>0 $ on $\mathcal{H}_{02}$ and $l_1 (a_{0_c}, b_0, 0, b_1)<0 $ on $\mathcal{H}_{01} \cup \mathcal{H}_{03}$. See Figure \ref{fig1}. The bifurcation diagram for $l_1 < 0$ can be found in \cite[p. 161]{kuznet}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig1} \end{center} \caption{The Hopf surface $\mathcal{H}_{00} = \mathcal{H}_0 \cap \{ a_1 = 0 \}$ for $E_0$, the sets $\mathcal{H}_{01}$, $\mathcal{H}_{02}$ and $\mathcal{H}_{03}$ and the curves $\mathcal{L}_1$ and $\mathcal{L}_2$} \label{fig1} \end{figure} \begin{remark}\label{signL1} \rm It is well known that the first Lyapunov coefficient is a continuous function of the parameters. Thus if $\zeta_{00} = (a_{0_c}, b_0, 0, b_1) \in \mathcal{H}_{01}$ then there exists a neighborhood $\mathcal V_{\zeta_{00}}$ of $\zeta_{00}$ in the Hopf hypersurface $\mathcal{H}_0$ such that $l_1 (\zeta_{0}) < 0$ for all $\zeta_0 \in \mathcal V_{\zeta_{00}}$, since $l_1 (\zeta_{00}) < 0$. Analogous conclusions hold for the other subsets $\mathcal{H}_{02}$ and $\mathcal{H}_{03}$. \end{remark} In the next theorem we give the stability of the equilibrium $E_0$ for parameters in the curve $\mathcal{L}_1$. \begin{theorem}\label{theo:02} Consider system \eqref{syst2} with parameter values in $\mathcal{L}_1$. Then the second Lyapunov coefficient at $E_{0}$ is \begin{equation}\label{l2E0} l_2 (a_{0_c}, b_0, 0, b_{11})= -\frac{9+121b_0^3 + 570b_0^6 + 1008b_0^9}{3b_0^5 (1 + 14b_0^3 + 49b_0^6 + 36b_0^9)}. \end{equation} As $b_0 > 0$ then $l_2 (a_{0_c}, b_0, 0, b_{11}) < 0$ and system \eqref{syst2} has a transversal Hopf point of codimension 2 at $E_{0}$ which is a stable equilibrium point. The bifurcation diagram of system \eqref{syst2} at a typical point on the curve $\mathcal{L}_1$ can be found in \cite[p. 313]{kuznet}. \end{theorem} \begin{proof} By Corollary \ref{cor:01}, for parameters in $\mathcal{L}_1$, $l_1(a_{0_c}, b_0, 0, b_{11})=0$. Due to the quadratic nature of the system, the multilinear symmetric functions $D$ and $E$ are \[ D(x,y,z,w) = (0, 0, 0), \quad E(x,y,z,w,r) = (0, 0 ,0). \] The complex vectors $h_{11}$, $h_{20}$, $h_{21}$, $h_{22}$, $h_{30}$ and $h_{31}$ are \begin{gather*} h_{11}=\big(-\frac{2}{b_0^2},0,0 \big), \\ h_{20}=\Big(\frac{16 b_0^3-2 i b_0^{3/2}+2}{6 b_0^5-3 i b_0^{7/2}},\frac{4 \big(8 i b_0^3+b_0^{3/2}+i\big)}{6 b_0^{9/2}-3 i b_0^3},\frac{-64 b_0^3+8 i b_0^{3/2}-8}{6 b_0^4-3 i b_0^{5/2}}\Big), \\ \begin{aligned} h_{21}&=\Big( \frac{i-3b_0^{3/2}+16ib_0^3-48 b_0^{9/2}}{6b_0^6 (-i+ b_0^{3/2})},\frac{1-ib_0^{3/2}+16b_0^3-16ib_o^{9/2}} {6(-ib_0^{11/2}+b_0^7)},\\ &\quad -\frac{-3i + b_0^{3/2}-48ib_0^3 + 16b_0^{9/2}} {6b_0^5(-i+b_0^{3/2})} \Big), \end{aligned} \\ h_{22}=\Big(-\frac{4(5 + 141b_0^3 + 714b_0^6 + 848b_0^9)}{9b_0^7 (1 +5b_0^3 + 4b_0^6)},0,0\Big), \\ \begin{aligned} h_{30}&= \Big( \frac{3- 5ib_0^{3/2} +46b_0^3 -40ib_0^{9/2} +192b_0^6}{4b_0^6 + 20ib_0^{15/2}-24b_0^9},\\ &\quad -\frac{3i(3 -5ib_0^{3/2} + 46b_0^3 -40ib_0^{9/2}+ 192b_0^6)}{4b_0^{11/2} (-1-5ib_0^{3/2}+6b_0^3)}, \\ &\quad \frac{9(3-5ib_0^{3/2}+ 46b_0^3 - 40ib_0^{9/2}+ 192b_0^6)}{4b_0^5(-1-5ib_0^{3/2}+6b_0^3)} \Big) \end{aligned} \end{gather*} \and \begin{align*} h_{31} &=\bigg(\Big(7i - 65b_0^{3/2}+ 133ib_0^3 -1897b_0^{9/2}-2050ib_0^6 -12056b_0^{15/2}\\ &\quad - 31744ib_0^9 +21504b_0^{21/2}\Big)/ \Big(18b_0^{17/2} (i-2b_0^{3/2})^2(-1-4ib_0^{3/2}+3b_0^3)), \\ &\quad -\Big(1 + 35ib_0^{3/2}+ 19b_0^3 +1099ib_0^{9/2}-1774b_0^6 +7256ib_0^{15/2}\\ &\quad - 22720b_0^9 -16896ib_0^{21/2}\Big)/\Big(9b_0^8(i-2b_0^{3/2})^2 (-1-4ib_0^{3/2}+3b_0^3)\Big),\\ &\quad \Big(2(5i +5b_0^{3/2} +95ib_0^3 +301b_0^{9/2}+1498ib_0^6 + 2456b_0^{15/2} +13696ib0^9 \\ &\quad -12288b_0^{21/2})\Big) /\Big(9b_0^{15/2}(i-2b_0^{3/2})^2(-1-4ib_0^{3/2}+3b_0^3)\Big)\bigg), \end{align*} respectively. From \eqref{coef2}, \begin{align*} G_{32}&=-\frac{4( 9 + 121b_0^3 + 570b_0^6 + 1008b_0^9)}{b_0^5(1 + 14b_0^3 + 49b_0^6 +36b_0^9)} \\ &\quad -i \frac{(17+1214b_0^3 +21105b_0^6 +155492b_0^9 + 463040b_0^{12} + 377856b_0^{15})}{36b_0^{19/2}(1 + 14b_0^3 +49b_0^6 +36b_0^9)}. \end{align*} Thus, the second Lyapunov coefficient \eqref{coef2} is \[ l_2 (a_{0_c}, b_0, 0, b_{11})=\frac{1}{12} \operatorname{Re} G_{32} = -\frac{9+121b_0^3 + 570b_0^6 + 1008b_0^9}{3b_0^5 (1 + 14b_0^3 + 49b_0^6 + 36b_0^9)}. \] The proof is complete. \end{proof} From Theorem \ref{theo:02}, the sign of the second Lyapunov coefficient at $E_0$ is always negative on $\mathcal{L}_1$. Thus the equilibrium $E_0$ is a weak attracting focus (for the flow of system \eqref{syst2} restricted to the center manifold) and there are two limit cycles, one stable and the other unstable, near the equilibrium $E_0$ for suitable value of the parameters. See the pertinent bifurcation diagram in \cite[p. 313]{kuznet}. In the next theorem we study the stability of the equilibrium $E_0$ for parameters in the curve $\mathcal{L}_2$. \begin{theorem}\label{theo:03} Consider system \eqref{syst2} with parameter values in $\mathcal{L}_2$. See Figure \ref{fig1}. Then the second and third Lyapunov coefficients at $E_{0}$ vanish; that is, \[ l_2 (a_{0_c}, b_0, 0, b_{12}) = l_3 (a_{0_c}, b_0, 0, b_{12}) = 0. \] \end{theorem} \begin{proof} By Corollary \ref{cor:01}, for parameters in $\mathcal{L}_2$, $l_1(a_{0_c}, b_0, 0, b_{12}) = 0$. Due to the quadratic nature of the system, the multilinear symmetric functions $D$, $E$, $K$ and $L$ are \[ D(x,y,z,w) = E(x,y,z,w,r) = K(x,y,z,w,r,s) = L(x,y,z,w,r,s,t) = (0, 0, 0). \] The complex vectors $h_{11}$, $h_{20}$, $h_{21}$, $h_{22}$, $h_{30}$ and $h_{31}$ are \begin{gather*} h_{11}=\Big(-\frac{2}{b_0^2},0,0 \Big), \quad h_{20}= \Big(\frac{2}{3 b_0^2},\frac{4 i}{3 b_0^{3/2}}, -\frac{8}{3 b_0}\Big),\\ h_{21}=\Big(-\frac{5(-i+3b_0^{3/2})}{3b_0^3(-i+b_0^{3/2})}, \frac{5(1-ib_0^{3/2})}{3(-ib_0^{5/2}+b_0^4)}, -\frac{5(-3i+ib_0^{3/2})}{3b_0^2(-i+b_0^{3/2})} \Big), \\ h_{22}=\Big(-\frac{16(17+32b_0^3)}{9(b_0^4+b_0^7)},0,0\Big), \quad h_{30}= \Big(-\frac{1}{2 b_0^3},\frac{3i}{2b_0^{5/2}}, \frac{9}{2b_0^2} \Big), \\ h_{31}=\Big(\frac{89i -149b_0^{3/2}}{9ib_0^4-9b_0^{11/2}}, \frac{118+238ib_0^{3/2}}{9(-ib_0^{7/2}+b_0^5)}, -\frac{4(-29i+89b_0^{3/2})}{9b_0^3(-i+b_0^{3/2})}\Big), \end{gather*} respectively. From the above results the complex number $G_{32}$ \eqref{coef2} can be written as \[ G_{32}=-\frac{5i(157+277 b_0^3)}{9b_0^{7/2}(1+b_0^3)}. \] By the above expression of $G_{32}$, $ l_2 (a_{0_c}, b_0, 0, b_{12})= \operatorname{Re} G_{32}/12 = 0$. The complex vectors $h_{32}$, $h_{33}$, $h_{40}$, $h_{41}$ and $h_{42}$ are, respectively, \begin{gather*} \begin{aligned} h_{32}&=\Big(-\frac{5(-187i+561b_0^{3/2}-187ib_0^3+1041b_0^{9/2})} {18b_0^5(-i+b_0^{3/2})^2(i+b_0^{3/2})},\\ &\quad -\frac{5i(187+120ib_0^{3/2}+307b_0^3)}{18b_0^{9/2} (-i+b_0^{3/2})^2}, -\frac{5(-441i+307(b_0^{3/2}-3ib_0^3+b_0^{9/2}))} {18b_0^4(-i+b_0^{3/2})^2(i+b_0^{3/2})}\Big), \end{aligned} \\ h_{33}=\Big(-\frac{33137 + 114154b_0^3 + 109817b_0^6}{18b_0^6(1 + b_0^3)^2}, 0, 0\Big), \\ h_{40}=\Big( \frac{4}{9b_0^4},\frac{16i}{9b_0^{7/2}}, -\frac{64}{9b_0^3}\Big), \\ h_{41}=\Big(\frac{109i-169b_0^{3/2}}{6b_0^5(-i+b_0^{3/2})}, \frac{89+149ib_0^{3/2}}{2b_0^{9/2}(i-b_0^{3/2})}, \frac{9(-23i+43b_0^{3/2})}{2b_0^4(-i+b_0^{3/2})}\Big), \\ h_{42}=(h_{{42}_1},h_{{42}_2},h_{{42}_3}), \end{gather*} where \begin{gather*} h_{{42}_1}=\frac{2(-8001i + 14701b_0^{3/2}-9761ib_0^3 + 22461b_0^{9/2})}{27b_0^6(-i+b_0^{3/2})^2(i+b_0^{3/2})}, \\ h_{{42}_2}=\frac{4(4651 + 10151b_0^{3/2}+5211b_0^3 + 16711b_0^{9/2})}{27b_0^{11/2}(-i+b_0^{3/2})^2(i+b_0^{3/2})}, \\ h_{{42}_3}=\frac{8(1901i-6201b_0^{3/2}+1261ib_0^3 -11561b_0^{9/2})}{27b_0^5(-i+b_0^{3/2})^2(i+b_0^{3/2})}. \end{gather*} Substituting the above results into the expression of the complex number $G_{43}$ \eqref{coef3} and making the simplifications it follows that \[ G_{43}=-\frac{5i(13099 + 43838b_0^3 + 40339b_0^6)}{9b_0^{11/2}(1+b_0^3)^2}, \] and, by \eqref{coef3}, $l_3 (a_{0_c}, b_0, 0, b_{12})=\frac{1}{144} \operatorname{Re} G_{43}= 0$. \end{proof} Based on the above theorem we have the following question. \begin{question}\label{q:01}\rm Consider system \eqref{syst2} with parameters in $\mathcal{L}_2$. Is the equilibrium $E_0$ a center for the flow of system \eqref{syst2} restricted to the center manifold? \end{question} This question is related with the planar center-focus problem. In his seminal paper Bautin \cite{bautin} solves the center-focus problem for quadratic systems in the plane: If the three first Lyapunov coefficients are zero at the equilibrium point then it is a center. It is not known an extension of the Bautin's theorem for quadratic systems in $\mathbb{R}^3$. We have calculated the following Lyapunov coefficient, $l_4$, at $E_0$ for parameters in $\mathcal{L}_2$ and it vanishes too. These calculations are not presented here. Based on this information and Theorem \ref{theo:03} we have the following question. \begin{question}\label{q:02} \rm How many limit cycles can bifurcate from $E_0$ for a suitable perturbation of a parameter vector in $\mathcal{L}_2$? \end{question} \subsection{Hopf bifurcation analysis at $E_1$}\label{S:42} In this subsection we study the Hopf bifurcations that occur at the equilibrium $E_1$ for parameters in the set $\mathcal{H}_1$ defined in \eqref{H1}. Define the critical parameter \[ b_{0_c}=\frac{1}{a_1-a_0}+ b_1. \] \begin{theorem}\label{theo:04} Consider system \eqref{syst2}. The first Lyapunov coefficient at $E_1$ for parameter values in $\mathcal{H}_1$ is \begin{equation}\label{L1E1} l_1 (a_0,b_{0_c},a_1,b_1) = \frac{D(a_0,b_{0_c},a_1,b_1)}{2(-4+(a_0-a_1)^3) (-1+(a_0-a_1)^3)}, \end{equation} where \begin{align*} &D(a_0,b_{0_c},a_1,b_1)\\ &= a_0 (a_0-a_1) \big(2a_0 (-8+a_0^3)+8 a_1 - 11 a_0^3 a_1 + 21 a_0^2 a_1^2-17 a_0 a_1^3 + 5 a_1^4\big) \\ &\quad - (a_0-a_1)^3 \big((a_0-a_1)^4- 2a_1-10a_0\big)b_1 -(a_0-a_1)^5 b_1^2. \end{align*} If $\zeta_1 = (a_0, b_{0_c}, a_1, b_1) \in \mathcal{H}_1$ is such that $l_1 (\zeta_1) \neq 0$ then system \eqref{syst2} has a transversal Hopf point at $E_1$ for the parameter vector $\zeta_1$. \end{theorem} \begin{proof} For parameters on the Hopf hypersurface $\mathcal{H}_1$ we have \begin{gather*} \lambda_1 = a_1-a_0, \quad \lambda_{2,3}= \pm i \omega_0, \quad \omega_0 = \frac{1}{\sqrt{a_1-a_0}}, \quad a_1-a_0 > 0, \\ q=(a_0-a_1,-i\sqrt{a_1-a_0},1), \\ p= \Big( \frac{\sqrt{a_1-a_0}}{2(-i -(a_1-a_0)^{3/2})},\frac{-i}{2\sqrt{a_1-a_0}}, \frac{-i}{2(-i-(a_1-a_0)^{3/2})}\Big), \\ B(x,y) = \left( 0,0, - a_1 (x_1 y_3 + x_3 y_1) -b_1 (x_1 y_2 + x_2 y_1) - 2 x_1 y_1 \right), \\ C (x,y,z) = ( 0, 0, 0 ). \end{gather*} The complex vectors $h_{11}$ and $h_{20}$ are \begin{gather*} h_{11}=\Big( 2a_0(a_0-a_1),0,0 \Big), \\ \begin{aligned} h_{20}&=\Big(- \frac{2(a_0-a_1)^2(a_0(\sqrt{a_1-a_0}+ib_1) -ia_1b_1)}{6i-3(a_1-a_0)^{3/2}}, \\ &\quad -\frac{4(a_0-a_1)^2 (ia_0+b_1\sqrt{a_1-a_0})}{6i-3(a_1-a_0)^{3/2}},\\ &\quad -\frac{8(a_0-a_1)(a_0(\sqrt{a_1-a_0}+ib_1) -ia_1b_1}{6i-3(a_1-a_0)^{3/2}}\Big). \end{aligned} \end{gather*} Substituting the above expressions into \eqref{coef1} and making the simplifications, results that the complex number $G_{21}$ is \[ G_{21}=\frac{D^{*}(a_0,b_{0_c},a_1,b_1)}{3(2+a_0^3-3a_0^2a_1-a_1^3 +3ia_1\sqrt{a_1-a_0} +3a_0(a_1^2-i\sqrt{a_1-a_0}))}, \] where \begin{align*} &D^{*}(a_0,b_{0_c},a_1,b_1)\\ &=(a_0-a_1)\Big(a_0^3(10i\sqrt{a_1-a_0}-3b_1) +a_1^2b_1(3a_1-ib_1\sqrt{a_1-a_0})\\ &\quad +a_0^2 (-24-19ia_1\sqrt{a_1-a_0}+9a_1b_1-ib_1^2\sqrt{a_1-a_0})\\ &\quad + a_0\big(9ia_1^2(\sqrt{a_1-a_0}+ib_1) + 12ib_1\sqrt{a_1-a_0} +2a_1(6+ib_1^2\sqrt{a_1-a_0})\big)\Big). \end{align*} Performing the calculations in \eqref{coef1}, the first Lyapunov coefficient is given by \eqref{L1E1}. It remains only to verify the transversality condition of the Hopf bifurcation. In order to do so, consider the family of differential equations \eqref{syst2} regarded as dependent on the parameter $b_0$. The real part, $\gamma=\gamma (b_0)$, of the pair of complex eigenvalues at the critical parameter $b_0=b_{0_c}$ verifies \[ \gamma'(b_{0_c}) = \operatorname{Re} \big\langle p, \frac{dA}{db_0}\Big|_{b_0=b_{0_c}} q \big\rangle = \frac{(a_1 - a_0)^2}{2\big( (a_1 - a_0)^3 +1 \big)} > 0, \] since $a_1 - a_0 > 0$. In the above expression $A$ is the Jacobian matrix of system \eqref{syst2} at $E_1$. Therefore, the transversality condition at the Hopf point holds. \end{proof} Note that the sign of the first Lyapunov coefficient \eqref{L1E1} in Theorem \ref{theo:04} is determined by the sign of the function $D(a_0, b_{0_c}, a_1, b_1)$, the numerator of $l_1$, since the denominator is positive. In the rest of this subsection we study the stability of the equilibrium $E_1$ with the restriction $a_0 = 0$. This makes the analysis of the sign as well as the analysis of the zero set of the first Lyapunov coefficient \eqref{L1E1} simpler. See Remark \ref{signL1}. Define the following subset of the Hopf hypersurface $\mathcal{H}_1$ for $E_1$ \[ \mathcal{H}_{10}=\{(a_0,b_0,a_1,b_1) \in \mathcal{H}_1 : a_0=0\}. \] \begin{corollary}\label{cor:04} Consider system \eqref{syst2} with parameter values in $\mathcal{H}_{10}$. Then the first Lyapunov coefficient at $E_1$ is \[ l_1 (0, b_{0_c}, a_1, b_1)= \frac{a_1^4b_1(-2+a_1^3+a_1b_1)}{2(4+5a_1^3 + a_1^6)}. \] If either \[ b_1 = b_{13} = 0, \quad\text{or} \quad b_1 = b_{14} = \frac{2-a_1^3}{a_1}, \] then the first Lyapunov coefficient at $E_1$ vanishes; that is, \[ l_1 (0, b_{0_c}, a_1, b_{13}) = l_1 (0, b_{0_c}, a_1, b_{14}) = 0. \] \end{corollary} \begin{proof} Substituting $a_0 = 0$ into the expression of $G_{21}$ in the proof of Theorem \ref{theo:04} results \[ G_{21}=\frac{a_1^4 b_1 (-2+a_1^3+a_1b_1)}{4+5a_1^3+a_1^6}+ i \frac{ a_1^{7/2} b_1(2b_1+a_1^2(9-a_1b_1))}{3(4+5a_1^3+a_1^6)}. \] If $b_1 = b_{13}$, then the numerator of the real part of $G_{21}$ vanishes. Then the first Lyapunov coefficient $l_1(0, b_{0_c}, a_1, b_{13})=0$. On the other hand, if $b_1 = b_{14}$ then the parenthesis in the numerator of the real part of $G_{21}$ vanishes. Then $l_1(0, b_{0_c}, a_1, b_{14})=0$. \end{proof} From Corollary \ref{cor:04} the first Lyapunov coefficient vanishes on the curves \begin{gather*} \mathcal{L}_3= \{(b_0,a_1,b_1) \in \mathcal{H}_{10} : b_0=\frac{1}{a_1}, \quad b_1 = 0 \},\\ \mathcal{L}_4= \{(b_0,a_1,b_1) \in \mathcal{H}_{10} : b_0=\frac{3-a_1^3}{a_1}, \quad b_1 = \frac{2-a_1^3}{a_1} \}. \end{gather*} See Figure \ref{fig2}. These curves have only one intersection point $P_1= \left( (\sqrt[3]{2})^{-1},\sqrt[3]{2},0 \right)$ and divide the Hopf surface $\mathcal{H}_{10}$ into four connected components \begin{gather*} \mathcal{H}_{11}= \{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{10} : b_1>0, \quad b_0 > \frac{3-a_1^3}{a_1} \},\\ \mathcal{H}_{12}= \{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{10} : b_1>0 ,\quad b_0 < \frac{3-a_1^3}{a_1} \}, \\ \mathcal{H}_{13}= \{ (a_0,b_0,a_1,b_1) \in \mathcal{H}_{10} : b_1<0, \quad b_0 < \frac{3-a_1^3}{a_1} \}, \\ \mathcal{H}_{14}= \{(a_0,b_0,a_1,b_1) \in \mathcal{H}_{10} : b_1<0, \quad b_0 > \frac{3-a_1^3}{a_1} \}, \end{gather*} where the first Lyapunov coefficient at $E_1$ has fixed sign: $l_1 (0,b_{0_c},a_1,b_1)>0$ on $\mathcal{H}_{11} \cup \mathcal{H}_{13}$ and $l_1 (0,b_{0_c},a_1,b_1) <0 $ on $\mathcal{H}_{12} \cup \mathcal{H}_{14}$. See Figure \ref{fig2}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig2} \end{center} \caption{The Hopf surface $\mathcal{H}_{10} = \mathcal{H}_1 \cap \{ a_0 = 0 \}$ for $E_1$, the sets $\mathcal{H}_{11}$, $\mathcal{H}_{12}$, $\mathcal{H}_{13}$, $\mathcal{H}_{14}$ and the curves $\mathcal{L}_3$ and $\mathcal{L}_4$.} \label{fig2} \end{figure} In the next theorem we give the stability of the equilibrium $E_1$ for parameters in the curve $\mathcal{L}_3$. \begin{theorem}\label{theo:05} Consider system \eqref{syst2} with parameter values in $\mathcal{L}_{3}$. Then the second and third Lyapunov coefficients at $E_1$ vanish; that is, \[ l_2 (0, b_{0_c}, a_1, b_{13}) = l_3 (0, b_{0_c}, a_1, b_{13}) = 0. \] \end{theorem} \begin{proof} By Corollary \ref{cor:04}, $l_1(0,b_{0_c}, a_1, b_{13}) = 0$. Due to the quadratic nature of the system the multilinear symmetric functions $D$, $E$, $K$ and $L$ satisfy \[ D(x,y,z,w) = E(x,y,z,w,r) = K(x,y,z,w,r,s) = L(x,y,z,w,r,s,t) = (0,0,0). \] For $a_0 = 0$ and $b_1 = b_{13} = 0$ all the complex vectors $h_{11}$, $h_{20}$, $h_{21}$, $h_{22}$, $h_{30}$, $h_{31}$, $h_{32}$, $h_{33}$, $h_{40}$, $h_{41}$ and $h_{42}$ are the zero vector. Therefore, from \eqref{coef2} and \eqref{coef3}, $G_{32}=G_{43}=0$ and we have $ l_2(0, b_{0_c}, a_1, b_{13})=l_3(0, b_{0_c}, a_1, b_{13})=0$. \end{proof} Based on the above theorem we have a question analogous to Question \ref{q:01} about the stability of the equilibrium point $E_1$ for the flow of system \eqref{syst2} restricted to the center manifold. Moreover, we can formulate a similar question to Question \ref{q:02} about the number of limit cycles that can bifurcate from $E_1$ for a suitable perturbation of the parameters. In the next three theorems we study the stability of the equilibrium $E_1$ for parameters in the curve $\mathcal{L}_4$. \begin{theorem}\label{theo:06} Consider system \eqref{syst2} with parameter values in $\mathcal{L}_4$. Then the second Lyapunov coefficient at $E_1$ is \begin{equation}\label{L2E1} l_2 (0, b_{0_c}, a_1, b_{14}) = -\frac{2a_1^5(a_1^3-2)(a_1^6+22a_1^3-105)}{3( 36+a_1^3 (7+a_1^3)^2)} . \end{equation} \end{theorem} \begin{proof} By Corollary \ref{cor:04}, $l_1(0,b_{0_c}, a_1, b_{14})=0$. Due to the quadratic nature of the system the multilinear symmetric functions $D$, $E$, $K$ and $L$ satisfy \[ D(x,y,z,w) = E(x,y,z,w,r) = (0,0,0). \] The complex vectors $h_{11}$, $h_{20}$, $h_{21}$, $h_{22}$, $h_{30}$ and $h_{31}$ are \begin{gather*} h_{11}=(0,0,0), \quad h_{20}= \Big( \frac{2 i a_1^2 (a_1^3-2)}{3 (a_1^{3/2}-2 i)},-\frac{4 a_1^{3/2} (a_1^3-2)}{3 (a_1^{3/2}-2 i)},-\frac{8 i a_1 (a_1^3-2)}{3 (a_1^{3/2}-2 i)}\Big),\\ \begin{aligned} h_{21}&=\Big(-\frac{a_1^3(a_1^{3/2}-3i)(a_1^3-2)}{6(a_1^{3/2}-i)},\ \frac{i(-2ia_1^{5/2}- 2a_1^4 +ia_1^{11/2}+a_1^7)}{6(a_1^{3/2}-i)},\\ &\quad -\frac{(a_1^2 (3a_1^{3/2}-i)(a_1^3-2)}{6(a_1^{3/2}-i) } \Big), \end{aligned} \\ h_{22}=\Big( \frac{16a_1^4(a_1^3-2)}{4+a_1^3},0,0\Big), \\ \begin{aligned} h_{30}&= \Big( \frac{(3a_1^3(a_1^3-2)(a_1^3-2-ia_1^{3/2})}{4(-6 -5ia_1^{3/2}+a_1^3)},\; \frac{9a_1^{5/2}( ia_1^3-2i+a_1^{3/2})(a_1^3-2)}{4(-6 -5ia_1^{3/2}+a_1^3)}, \\ &\quad -\frac{ 27a_1^2(a_1^3-2)(-2-ia_1^{3/2}+a_1^3)} {4(a_1^3-6 -5ia_1^{3/2}) } \Big) \end{aligned} \end{gather*} and \begin{align*} h_{31} &=\Big(\frac{(a_1^4(a_1^3-2)(372+370ia_1^{3/2}-150a_1^3-127ia_1^{9/2} +54 a_1^6+7ia_1^{15/2})}{18(a_1^{3/2}-2i)^2(-3-4ia_1^{3/2}+a_1^3)},\\ &\quad -\frac{(a_1^{7/2}(a_1^3-2)(-300i+238a_1^{3/2} +42ia_1^3-49a_1^{9/2}-18ia_1^{6}+a_1^{15/2})}{9(a_1^{3/2}-2i)^2 (-3-4ia_1^{3/2}+a_1^3)},\\ &\quad \frac{2a_1^{3}(a_1^3-2)(-228-106ia_1^{3/2} -66a_1^3-29ia_1^{9/2}+18a_1^{6}+5ia_1^{15/2})}{9(a_1^{3/2}-2i)^2 (-3-4ia_1^{3/2}+a_1^3)}\Big), \end{align*} respectively. Substituting the above expressions into \eqref{coef2} and making the simplifications it follows that \begin{align*} G_{32}&=-\frac{8a_1^5(a_1^3-2)(a_1^6+22a_1^3-105)}{36+a_1^3 (7+a_1^3)^2} \\ &\quad - i \frac{(a_1^{7/2} (a_1^3-2)(-20232 + 17714a_1^3 + 93a_1^6 +180a_1^9 +17a_1^{12})}{36( 36 +a_1^3(7+a_1^3)^2)}. \end{align*} From the expression of $G_{32}$ and \eqref{coef2} we have \[ l_2 (0, b_{0_c}, a_1, b_{14})=\frac{1}{12} \operatorname{Re} G_{32} = -\frac{2a_1^5(a_1^3-2)(a_1^6+22a_1^3-105)}{3(36+a_1^3 (7+a_1^3)^2)}. \] The proof is complete. \end{proof} \begin{remark}\label{Re:01E1} \rm When $a_0 = 0$ we have $a_1 > 0$, since $a_1 - a_0 > 0$ in $\mathcal{H}_1$. So $l_2 (0, b_{0_c}, a_1, b_{14})=0$ if and only if $a_1=a_{11}=\sqrt[3]{2}$ or $a_1=a_{12}=\sqrt[3]{\sqrt{226} -11}$. \end{remark} From Theorem \ref{theo:06} and Remark \ref{Re:01E1} it follows that the sets \begin{gather*} \mathcal{L}_{41} = \{ (b_0,a_1,b_1) \in \mathcal{L}_4 : 0 < a_1 < \sqrt[3]{2} \}, \\ \mathcal{L}_{42} = \{ (b_0,a_1,b_1) \in \mathcal{L}_4 : \sqrt[3]{2}\sqrt[3]{\sqrt{226} - 11} \} \end{gather*} are arcs of the curve $\mathcal{L}_{4}$ where the second Lyapunov coefficient at $E_1$ is nonzero. More specifically, $l_2(0,b_{0_c},a_1,b_1)<0$ on $\mathcal{L}_{41} \cup \mathcal{L}_{43}$ and $l_2(0,b_{0_c},a_1,b_1)>0$ on $\mathcal{L}_{42}$. See Figure \ref{fig2}. At the points \begin{gather*} P_1= \left( (\sqrt[3]{2})^{-1}, \sqrt[3]{2}, 0 \right), \\ P_2=\Big( {\frac {\sqrt {226} - 14}{\sqrt [3]{\sqrt {226} - 11}}}, {\frac {13-\sqrt {226}}{\sqrt [3]{\sqrt {226} - 11}}},\sqrt [3]{\sqrt {226} - 11} \Big) \end{gather*} the second Lyapunov coefficient at $E_1$ vanishes. From Theorem \ref{theo:06} it follows that the sign of the second Lyapunov coefficient at $E_1$ is negative on $\mathcal{L}_{41} \cup \mathcal{L}_{43}$. Thus the equilibrium $E_1$ is a weak attracting focus (for the flow of system \eqref{syst2} restricted to the center manifold) and there are two limit cycles, one stable and the other unstable, near the equilibrium $E_1$ for suitable values of the parameters. On the other hand, the sign of the second Lyapunov coefficient at $E_1$ is positive on $\mathcal{L}_{42}$. Thus the equilibrium $E_1$ is a weak repelling focus (for the flow of system \eqref{syst2} restricted to the center manifold) and there are two limit cycles, one unstable and the other stable, near the equilibrium $E_1$ for suitable values of the parameters. See the pertinent bifurcation diagrams in \cite[p. 313]{kuznet}. In the next two theorems we study the stability of the equilibrium $E_1$ for the parameters at $P_1$ and $P_2$, respectively. \begin{theorem}\label{theo:07} Consider system \eqref{syst2} with parameter values at $P_1$. Then the second and third Lyapunov coefficients at $E_1$ vanish, that is \[ l_2(P_1) = l_3(P_1) = 0. \] \end{theorem} \begin{proof} Substituting $a_1=a_{11}=\sqrt[3]{2}$ into \eqref{L2E1} results $l_2(P_1)=0$. The calculations to find $l_3(P_1)$ follow the same steps presented in the proof of Theorem \ref{theo:05} and will be omitted here. \end{proof} \begin{theorem}\label{theo:08} Consider system \eqref{syst2} with the parameter values at $P_2$. Then the second and third Lyapunov coefficients at $E_1$ are $l_2(P_2) = 0$ and \begin{equation*} l_3(P_2)= \frac{1728 \left(\sqrt{226}-11\right)^{7/3} \left(1775502296303 \sqrt{226}-26691643307570\right)}{144\left(430054-28843 \sqrt{226}\right)^2 \left(72+\sqrt{226}\right)} > 0. \end{equation*} \end{theorem} \begin{proof} Substituting $a_1=a_{12}$ into expression \eqref{L2E1} results $l_2(P_2)=0$. The value of $l_3(P_2)$ is obtained following the same steps as presented in the proof of Theorem \ref{theo:03} and will be omitted here. The value of $l_3(P_2)$ is approximately $2.528833 > 0$ with five decimal round-off coordinates. \end{proof} From Theorem \ref{theo:08} it follows that the equilibrium $E_1$ is a weak repelling focus for the flow of system \eqref{syst2} restricted to the center manifold and there are three limit cycles, one stable and two unstable, near the equilibrium $E_1$ for suitable values of the parameters. See the pertinent bifurcation diagram in \cite{smb1, takens}. \subsection{Genesio system}\label{S:43} Consider the system of quadratic differential equations \begin{equation}\label{gen} \begin{gathered} x' = y,\\ y' = z,\\ z' = cz+by+ax+x^2, \end{gathered} \end{equation} where $(x, y, z)$ are the state variables and $a < 0$, $b < 0$, $c < 0$ are parameters. System \eqref{gen} is called Genesio system and was studied in \cite{genesio} from the point of view of its chaotic behavior. In \cite{zc} the Hopf bifurcations of system \eqref{gen} were analyzed, but there are errors in the signs of the first Lyapunov coefficient. System \eqref{gen} can be obtained from system \eqref{syst2} taking the following parameters values \[ a_1=b_1=0, \quad a_0= \frac{c}{\sqrt[3]a}, \quad b_0= -\frac{b}{\sqrt[3]{a^2}} \] and performing the following change of coordinates and a reparametrization in time \[ x = \frac{X}{a}, \quad y = - \frac{Y}{\sqrt[3]{a^4}}, \quad z = \frac{Z}{\sqrt[3]{a^5}}, \quad t = -\sqrt[3]a \tau. \] Therefore, all the calculations and results obtained in subsections \ref{S:41} and \ref{S:42} for system \eqref{syst2} can be applied to system \eqref{gen}. In what follows we will concentrate our attention only in the Hopf bifurcations of system \eqref{gen}. It is simple to see that system \eqref{gen} has a Hopf point at $\mathcal E_0 = (0, 0, 0)$ for parameters on the surface \[ \mathcal{H} = \{ a = a_c = - bc, b < 0, c < 0 \}. \] By the above change of coordinates and reparametrization in time, in order to study the Hopf point at $\mathcal E_0 = (0, 0, 0)$ for parameters in $\mathcal{H}$ of system \eqref{gen} it is sufficient to study the Hopf point at $E_0 = (0, 0, 0)$ for parameters in $\mathcal{H}_{00}$ of system \eqref{syst2}. The following corollary gives the corrected sign of the first Lyapunov coefficient at $\mathcal E_0$ for parameters in $\mathcal{H}$. \begin{corollary}\label{cor:05} Consider system \eqref{gen} with parameters in $\mathcal{H}$. Then the first Lyapunov coefficient at $\mathcal E_{0}$ is negative and system \eqref{gen} has a transversal Hopf point at $\mathcal E_0$ for all parameters in $\mathcal{H}$. More specifically, the Hopf point at $\mathcal E_{0}$ is stable (weak attracting focus) and for each $a < a_c$, but close to $a_c$, there exists a stable limit cycle near the unstable equilibrium point $\mathcal E_{0}$. \end{corollary} \begin{proof} It is sufficient to study the sign of the first Lyapunov coefficient at $E_0$ for parameters in $\mathcal{H}_{00}$ of system \eqref{syst2}. Now, the expression \begin{equation}\label{coefL11} l_1 (a_{0_c}, b_0) = - \frac{1+8 b_0^3}{1+5b_0^3 + 4b_0^6} \end{equation} of this first Lyapunov coefficient follows directly from the general expression \eqref{coefL1} obtained in Theorem \ref{theo:01} taking into account $a_1 = b_1 = 0$. The transversality condition is also a consequence of Theorem \ref{theo:01}. As $b_0 > 0$ then $l_1 (a_{0_c},b_0) < 0$ and system \eqref{syst2} has a transversal Hopf point at $E_{0}$ for all critical parameters. The corollary is proved. \end{proof} \subsection{Bogdanov-Takens bifurcation analysis at $E_{\ast}$}\label{S:44} In this subsection we analyze the Bogdanov-Takens bifurcation at the equilibrium point $E_{\ast}=(0,0,0)$ of system \eqref{eq:03} when the quadratic function $h$ has only one real zero. Without loss of generality, we consider $h(x) = x^2 + c_0$ at $c_0 = 0$. Thus system \eqref{eq:03} has the form \begin{equation}\label{bt:01} \begin{gathered} x' = y,\\ y' = z,\\ z' = - \left( (a_1 x + a_0) z + (b_1 x + b_0) y + x^2 + c_0 \right). \end{gathered} \end{equation} We have the following theorem. \begin{theorem}\label{theo:bt} System \eqref{bt:01} undergoes a Bogdanov-Takens bifurcation at equilibrium point $E_{\ast} = (0,0,0)$ for parameter values $b_0 = c_0 = 0$, $a_0 \neq 0$, $b_1 \neq 2/a_0$ and $a_1 \in \mathbb{R}$. \end{theorem} \begin{proof} It is simple to see that $E_{\ast}=(0,0,0)$ is the only equilibrium point of system \eqref{bt:01} when $c_0 = 0$. Take the parameter values $b_0 = c_0 = 0$, $a_0 \neq 0$. The Jacobian matrix of system \eqref{bt:01} at $E_{\ast}$ is written as \[ A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & -a_0 \end{pmatrix} \] and its characteristic polynomial is $p(\lambda) = \lambda^2 (\lambda + a_0)$. Thus we have the following eigenvalues $\lambda_1 = -a_0 \neq 0$ and $\lambda_{2,3} = 0$. Consider the vectors \[ q_0 = \big( \frac{1}{a_0}, 0, 0 \big), \quad q_1 = \big( 0, \frac{1}{a_0}, 0 \big), \quad p_0 = \big( a_0, 0, -\frac{1}{a_0} \big), \quad p_1 = \big( 0, a_0, 1 \big). \] It follows that \begin{gather*} A q_0 = 0, \quad A q_1 = q_0, \quad A^T p_1 = 0, \quad A^T p_0 = p_1, \\ \langle q_1, p_1 \rangle = \langle q_0, p_0 \rangle = 1, \quad \langle q_1, p_0 \rangle = \langle q_0, p_1 \rangle = 0. \end{gather*} The bilinear symmetric function is written as \[ B(x,y) = \left( 0,0, - a_1 (x_1 y_3 + x_3 y_1) -b_1 (x_1 y_2 + x_2 y_1) - 2 x_1 y_1 \right). \] From \eqref{bt2} and \eqref{bt3} and the previous calculations we have \begin{gather*} a = \frac{1}{2} \langle p_1,B(q_0,q_0) \rangle = \frac{-1}{a_0^2} \neq 0, \\ b = \langle p_0,B(q_0,q_0) \rangle + \langle p_1,B(q_0,q_1) \rangle = \frac{2-a_0 b_1}{a_0^3} \neq 0, \end{gather*} since $b_1 \neq 2/a_0$. Therefore, conditions (BT1), (BT2) and (BT3) are satisfied. See subsection \ref{S:32}. It remains to prove the transversality condition (BT4). Define the map \[ G: (x,y,z,b_0,c_0) \mapsto (f_1, f_2, f_3, T, D)(x,y,z,b_0,c_0). \] The transversality condition (BT4) is satisfied if the map $G$ is regular at $(0,0,0,0,0)$. Now, the determinant of the derivative of $G$ at $(0,0,0,0,0)$ is \[ \det DG (0,0,0,0,0) = 2 \neq 0, \] proving the regularity of $G$ at $(0,0,0,0,0)$. The theorem is proved. \end{proof} The number $a$ is negative and, from the assumption $b_1 \neq 2/a_0$, it follows that $b \neq 0$. Therefore, the sign $s$ of the product $a b$ is determined by the sign of $b_1 - 2/a_0$. Therefore it is possible to choose parameters for which $s=1$ or $s=-1$. Recall that the sign $s$ determines the stability of the limit cycle that bifurcates from the Hopf point or from the homoclinic loop. See subsection \ref{S:32}. \subsection{Fold-Hopf bifurcation analysis at $E_{\ast}$}\label{S:45} In this subsection we analyze the fold-Hopf bifurcation at the equilibrium point $E_{\ast}=(0,0,0)$ of system \eqref{eq:03} when the quadratic function $h$ has only one real zero. Without loss of generality, we consider $h(x) = x^2 + c_0$ at $c_0 = 0$. Thus system \eqref{eq:03} has the form presented in \eqref{bt:01}. We have the following theorem. \begin{theorem}\label{theo:fh} System \eqref{bt:01} undergoes a fold-Hopf bifurcation at the equilibrium point $E_{\ast} = (0,0,0)$ for parameter values \[ a_0 = c_0 = 0, \quad b_0 > 0, \quad b_1 \neq 0, \quad a_1 \notin \big\{ \frac{2}{b_0}, \frac{1}{b_0}, \frac{9}{10 b_0}, 0 \big\}. \] \end{theorem} \begin{proof} It is easy to see that $E_{\ast}=(0,0,0)$ is the only equilibrium point of system \eqref{bt:01} when $c_0 = 0$. Take the parameter values $a_0 = c_0 = 0$, $b_0 > 0$. The Jacobian matrix of system \eqref{bt:01} at $E_{\ast}$ is written as \[ A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -b_0 & 0 \end{pmatrix} \] and its characteristic polynomial is $p(\lambda) = \lambda (\lambda^2 + b_0)$. Thus we have the following eigenvalues $\lambda_1 = 0$ and $\lambda_{2,3} = \pm i \omega_0$, $\omega_0 = \sqrt{b_0}$. Consider the vectors \[ q_0 = \big( \frac{1}{b_0}, 0, 0 \big), \quad q_1 = \big( 1, i \sqrt{b_0}, -b_0 \big), \quad p_0 = \big( b_0, 0, 1 \big), \quad p_1 = \big( 0, \frac{i}{2 \sqrt{b_0}}, \frac{-1}{2 b_0} \big). \] It follows that \begin{gather*} A q_0 = 0, \quad A q_1 = i \omega_0 q_1, \quad A^T p_0 = 0, \quad A^T p_1 = - i \omega_0 p_1, \\ \langle p_0, q_0 \rangle = \langle p_1, q_1 \rangle = 1, \quad \langle p_1, q_0 \rangle = \langle p_0, q_1 \rangle = 0. \end{gather*} The multilinear symmetric functions $B$ and $C$ are written as \begin{gather*} B(x,y) = \left( 0,0, - a_1 (x_1 y_3 + x_3 y_1) -b_1 (x_1 y_2 + x_2 y_1) - 2 x_1 y_1 \right), \\ C(x,y,z) = (0,0,0). \end{gather*} Performing the calculations, the numbers $G_{200}$, $G_{110}$ and $G_{011}$ defined in \eqref{G200}, \eqref{G110}, \eqref{G011}, respectively, are \[ G_{200} = \frac{-2}{b_0^2}, \quad G_{110} = \frac{2-a_1 b_0+i b_1 \sqrt{b_0}}{2 b_0^2}, \quad G_{011} = 2 a_1 b_0-2. \] From \eqref{h020}, \eqref{h200}, \eqref{h011} and \eqref{h110}, the complex vectors $h_{200}$, $h_{020}$, $h_{110}$ and $h_{011}$ can be written as \begin{gather*} h_{200} = \Big( 0,-\frac{2}{b_0^3},0 \Big),\\ h_{020} = \Big( \frac{i a_1 b_0+b_1 \sqrt{b_0}-i}{3 b_0^{3/2}},\frac{-2 a_1 b_0+2 i b_1 \sqrt{b_0}+2}{3 b_0},-\frac{4 i \left(a_1 b_0-i b_1 \sqrt{b_0}-1\right)}{3 \sqrt{b_0}} \Big), \\ h_{110} = \Big( -\frac{3 i \left(a_1 b_0-i b_1 \sqrt{b_0}-2\right)}{4 b_0^{5/2}},\frac{a_1 b_0-i b_1 \sqrt{b_0}-2}{4 b_0^2},-\frac{i \left(a_1 b_0-i b_1 \sqrt{b_0}-2\right)}{4 b_0^{3/2}} \Big), \\ h_{011} = \Big( 0,2 a_1-\frac{2}{b_0},0 \Big). \end{gather*} Performing the calculations of the numbers $G_{300}$ \eqref{G300}, $G_{111}$ \eqref{G111}, $G_{210}$ \eqref{G210} and $G_{021}$ \eqref{G021}, respectively, we have \begin{gather*} G_{300} = \frac{6 b_1}{b_0^4}, \quad G_{111} = \frac{(3-2 a_1 b_0) b_1}{b_0^2}, \\ G_{210} = -\frac{i (a_1^2 b_0^2+b_1^2 b_0+4 a_1 b_0-12 i b_1 \sqrt{b_0}-12)}{4 b_0^{9/2}},\\ G_{021} = -\frac{i \big(5 a_1^2 b_0^2-b_1^2 b_0-9 i b_1 \sqrt{b_0}+a_1 (6 i b_0^{3/2} b_1-7 b_0)+2\big)}{6 b_0^{5/2}}. \end{gather*} Therefore, the numbers $b(0)$, $c(0)$, $d(0)$ defined in \eqref{b0} are \[ b(0) = -\frac{1}{b_0^2}, \quad c(0) = 2 (a_1 b_0-1), \quad d(0) = \frac{-a_1 b_0+3 i b_1 \sqrt{b_0}+2}{2 b_0^2}, \] while the number $e(0)$ defined in \eqref{e0} can be written as \[ e(0) = \frac{a_1 (9-10 a_1 b_0) b_1}{16 b_0^3 (a_1 b_0-1)}. \] The number $b(0)$ is negative and, from the assumption $a_1 \neq 1/b_0$, it follows that $c(0) \neq 0$. Therefore, the sign $s$ of the product $b(0) c(0)$ is determined by the sign of $a_1 b_0-1$. On the other hand, from our assumptions it follows that $e(0) \neq 0$ and its sign can be determined easily if we fix some parameters. So (FH1) is satisfied. It remains to prove the transversality condition (FH2) which is equivalent to the nonvanishing of $\det DG(x,y,z,a_0,c_0)$ evaluated at $(x,y,z,a_0,c_0) = (0,0,0,0,0)$, where the map $G$ is defined by \[ G(x,y,z,a_0,c_0) = (f(x,y,z,a_0,c_0), \operatorname{Tr} (f_x(x,y,z,a_0,c_0)), \det (f_x(x,y,z,a_0,c_0))). \] By simple calculations it follows that $\det DG(0,0,0,0,0) = 2 \neq 0$. Finally, from \eqref{stheta} we have \[ \theta (0) = \frac{1}{4} \left(a_1 b_0-2\right) \neq 0. \] The proof is complete. \end{proof} It is possible to choose parameters so that $s=1$ and $\theta (0) < 0$. For example, taking $0 < a_1 < 1/b_0$, $b_0 > 0$, it follows that $0 < a_1 < 2/b_0$ and therefore $s = 1$ and $\theta (0) < 0$. Thus a nontrivial invariant set bifurcates from the equilibrium under variation of the parameters. See \cite[pp. 341--343]{kuznet}. \section{Concluding remarks}\label{S:5} This paper starts with the stability analysis which accounts for the characterization, in the space of parameters, of the structural as well as Lyapunov stability of the equilibria of system \eqref{eq:03}. It continues, after a suitable choice of parameters, with recounting the extension of the analysis to the first order, codimension one stable points, happening on the complement of the curves $\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_3$ and $\mathcal{L}_4$ (see Figures \ref{fig1} and \ref{fig2}) in the critical surfaces $\mathcal{H}_{00}$ and $\mathcal{H}_{10}$ where the criterium of Lyapunov holds based on the calculation of the first Lyapunov coefficient. Here the bifurcation analysis at the equilibrium points of system \eqref{syst2} is pushed forward to the calculation of the second and third Lyapunov coefficients which make possible the determination of the Lyapunov as well as higher order structural stability at the equilibrium points $E_0$ and $E_1$. See Theorems \ref{theo:02}, \ref{theo:03}, \ref{theo:05}, \ref{theo:06}, \ref{theo:07}, \ref{theo:08}. With the analytic data provided in the analysis performed here, the bifurcation diagrams can be established along the points of the curves $\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_3$ and $\mathcal{L}_4$ where the first Lyapunov coefficient vanishes. These bifurcation diagrams provide a qualitative synthesis of the dynamical conclusions achieved here at the parameter values where system \eqref{syst2} achieves most complex equilibrium points. Concerning with the vanishing of the Lyapunov coefficients in a quadratic system (see Theorems \ref{theo:03} and \ref{theo:05}) a question about the stability of the equilibria $E_0$ and $E_1$ is formulated. See Question \ref{q:01}. Another question (see Question \ref{q:02}) about the number of small limit cycles that can bifurcate from the equilibria $E_0$ and $E_1$, for a suitable perturbation of the parameters, is also presented. Two other codimension 2 bifurcations are also analyzed: Bogdanov-Takens and fold-Hopf bifurcations. See Theorems \ref{theo:bt} and \ref{theo:fh}. With the analytic data provided here, the bifurcation diagrams can be established leading to the existence of global bifurcations such as homoclinic ones. There is also the possibility of torus bifurcation. Finally, we would like to stress that although this work ultimately focuses a quadratic three dimensional system of differential equations \eqref{eq:03}, the method of analysis and calculations explained in Section \ref{S:4} can be adapted to the study of other polynomial systems. A cubic three dimensional system analogous to \eqref{eq:03} will be the subject of a future work. \subsection*{Acknowledgements} This work was done under the project APQ-01511-09 from FAPEMIG . \begin{thebibliography}{00} \bibitem{bautin} {N. N. Bautin}; \emph{On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type}, Amer. Math. Soc. Transl., \textbf{100} (1962), 396--413. Translated from Math. USSR-Sb, \textbf{30} (1952), 181–-196. \bibitem{chen} {G. Chen and T. Ueta}; \emph{Yet another chaotic attractor}, Int. J. Bifur. 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