\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 224, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/224\hfil Periodic orbits]
{Periodic orbits in hyperchaotic Chen systems}

\author[S. Maza \hfil EJDE-2015/224\hfilneg]
{Susanna Maza}

\address{Susanna Maza \newline
Departament de Matem\`atica,
Universitat de Lleida,
Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain}
\email{smaza@matematica.udl.cat}

\thanks{Submitted March 25, 2015. Published August 28, 2015.}
\subjclass[2010]{37G15, 37G10, 34C07}
\keywords{Periodic orbit; zero-Hopf bifurcation averaging theory;
\hfill\break\indent  hyperchaotic Chen system}

\begin{abstract}
 In this work, we show a zero-Hopf bifurcation  in a Hyperchaotic Chen
 system. Using averaging theory, we prove the existence of two periodic
 orbits bifurcating from the zero-Hopf equilibria located at the origin
 of the Hyperchaotic Chen system.
\end{abstract}

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

\section{Introduction and statement of the results}\label{s1}

Hyperchaos has been widely investigated since in 1979
when Otto R\"ossler \cite{Ross} proposed one of the first hyperchaotic attractors. 
A hyperchaotic system is a chaotic system in which two or more Lyapunov 
exponents are positive indicating that the chaotic dynamics are expanded 
in more than one direction. This feature gives rise to complex attractors 
harder to control than chaotic ones. Thus, hyperchaos has been found to 
be useful in many fields such as: encryption, secure communications \cite{chen2}, 
nonlinear circuits \cite{li}, liquid mixing, lasers \cite{hak}, and many more.

It is worth to say that the minimal dimension of the phase space that embeds 
a hyperchaotic attractor must be four, so the typical examples of hyperchaotic 
systems have been introduced usually as extensions of known autonomous 
three-dimensional chaotic systems.
 Besides the hyperchaotic attractor of R\"ossler, the  hyperchaotic Lorenz-Haken 
system \cite{ning}, the Chua's circuit \cite{chua} and the hyperchaotic L\"u 
system \cite{lu} are well-known examples of hyperchaotic models as extensions 
of three-dimensional chaotic systems. 
The book \cite{Sprott} presents a catalog of systems exhibiting chaos.


In this work we study the hyperchaotic Chen system
\begin{equation}\label{chen-1}
\begin{gathered}
\dot{x} =  a (y - x) + w, \\
\dot{y} = d x + c y - x z,  \\
\dot{z} = x y - b z,\\
\dot{w} =y z + r w.
\end{gathered}
\end{equation}
This system was introduced in \cite{chen}, as an extension of 
a three-dimensional chaotic Chen system. Its study has generated considerable 
research, techniques of chaos synchronization, studies of controlling chaos, 
secure communications, power system protection and so on, see 
\cite{he,effa,park,smaoui} and the references therein.

The dynamics of Hyperchaotic systems has been studied mainly numerically 
and by computer simulations. In this work we perform an analytic analysis 
on the existence of periodic orbits of the differential system \eqref{chen-1} 
by applying {\it averaging theory of first order}.
 More precisely, we prove that a {\it zero-Hopf} bifurcation occurs in system 
\eqref{chen-1} bifurcating two limit cycles from the {\it zero-Hopf} equilibria 
as parameters vary. As far as I know, this is the first time that an analytic 
analysis of a zero-Hopf bifurcation in hyperchaotic Chen system is performed.

We recall that an equilibrium of a differential system 
$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$  with 
$\mathbf{f}:A \to \mathbb{R}^n$ and $A $ an open subset of $\mathbb{R}^n$  
is a zero-Hopf equilibrium if it has two pure imaginary conjugated eigenvalues 
and $n-2$ zero eigenvalues. The bifurcation of periodic orbits from 
zero-Hopf equilibria of three dimensional differentials systems
 has been studied via averaging theory in \cite{lli2}. 
See also \cite{glm2} for a study in systems with two slow and one fast variables. 
In \cite{glm} is performed an analysis of the periodic orbits bifurcating 
in the chaotic prototype-4 system of  R\"ossler. Recently,  was 
 published \cite{lli1}, a paper on a zero-Hopf bifurcation in the hyperchaotic 
lorenz system.
The main result of this work reads as follows.

\begin{theorem}\label{Teo-chen}
Consider the hyperchaotic  Chen system \eqref{chen-1} with $c=a$ for small
values of $b$ and $r$, and satisfying $b(a + d)$ $r>0$ and $a(a+d)<0$. 
Then a zero-Hopf bifurcation occurs emerging two limit cycles from the origin.
\end{theorem}

In section 2 we give a summary of the averaging theory of first order for 
finding limit cycles and we proof Theorem \ref{Teo-chen} in section 3.

\section{Averaging theory: perturbing an isochronous system}

We consider the problem of bifurcation of $T$-periodic solutions for a differential
 system of the form
\begin{equation}\label{pert-A1}
\dot{\mathbf{x}}= \mathbf{F}_0(t,\mathbf{x}) 
+ \varepsilon \mathbf{F}_1(t,\mathbf{x}, \varepsilon) \,,
\end{equation}
where $\varepsilon$ is a small positive parameter, $\mathbf{x} \in A$, 
where $A$ is an open subset of $\mathbb{R}^n$, $t\geq 0$. 
Moreover we assume that both $\mathbf{F}_0(t,\mathbf{x})$ and 
$\mathbf{F}_1(t,\mathbf{x}, \varepsilon)$ are
$\mathcal{C}^2$ functions and $T$-periodic in $t$.

The classical theory of averaging reduce the problem of finding $T$-periodic 
solutions of \eqref{pert-A1} for $\varepsilon>0$  small to the problem of 
finding simple zeros of the so-called {\it bifurcation functions}. 
Many methods encountered in the literature are based on this idea, 
see \cite{buica,ma,chicone3}.

We study here the particular case in which all the solutions of the unperturbed 
system
\begin{equation}\label{unper}
 \dot{\mathbf{x}}= \mathbf{F}_0(t,\mathbf{x})
\end{equation}
are $T$-periodic. Then we assume that the unperturbed system is 
isochronous and the problem in what we are interested in is the problem 
of the persistence of the periodic orbits under some perturbation.

We denote the linearization of \eqref{unper} along a periodic solution 
$\mathbf{x}(t,\mathbf{u})$  of \eqref{unper} such that
$\mathbf{x}(0,\mathbf{u})=\mathbf{u}$ by
$$
\dot{\mathbf{y}}=D_\mathbf{x}\mathbf{F}_0 (t,\mathbf{x}(t,\mathbf{u}))
$$
where $D_\mathbf{x}\mathbf{F}_0 $ is the Jacobian matrix of $F_0$ with 
respect to $\mathbf{x}$ and let $\Phi_\mathbf{u}(t)$ be some fundamental 
matrix of \eqref{unper}.
Assume that there exists an open set $V$ with closure $\bar{V} \subset A$ 
such that for each $\mathbf{u}\in \bar{V}$, $\mathbf{x}(t,\mathbf{u})$ 
is $T$-periodic. The following result gives an answer to the question of 
bifurcating periodic solutions
from the $T$-periodic solutions $\mathbf{x}(t,\mathbf{u})$.

\begin{theorem}\label{ave}
We assume that there exists an open set $V$ with $\bar{V} \subset A$ such 
that for each $\mathbf{u} \in \bar{V}$, $\mathbf{x}(t,\mathbf{u})$ is 
$T$-periodic. Consider the function $\mathbf{f}: \bar{V}\mapsto \mathbb{R}^n $ 
given by
\begin{equation}\label{bifur}
 \mathbf{f}(\mathbf{u})= \frac{1}{T} \int_0^{T}\Phi_\mathbf{u}^{-1} 
\mathbf{F}_1(t,\mathbf{x}(t,\mathbf{u})) \, dt \,. 
\end{equation}
If there exists $\mathbf{p} \in V$ with $\mathbf{f}(\mathbf{p})=0$ and 
$\det(D_\mathbf{u}\mathbf{f}(\mathbf{p})))\neq 0$ then there exists 
a $T$-periodic solution  $\gamma(t,\varepsilon)$ of system \eqref{pert-A1}
 such that  $\gamma(0,\varepsilon)\to \mathbf{p}$ as $\varepsilon \to 0$. 
Moreover, if all the eigenvalues of $\det(D_\mathbf{u}\mathbf{f}(\mathbf{p})))$  
have negative real part, the corresponding periodic orbit $\gamma(t,\varepsilon)$ 
is asymptotically stable for $\varepsilon$ sufficiently small.
\end{theorem}

For a proof of Theorem \ref{ave} see \cite{ma,Rose,buica2}.

\section{Proof of Theorem \ref{Teo-chen}}

The origin of system \eqref{chen-1} is always an equilibrium point of 
it under under any choice of parameters. The characteristic polynomial 
$p(\lambda)$ of the linearization of system \eqref{chen-1} at the 
equilibrium point located at the origin is given by
$$
p(\lambda) = (r - \lambda)(b + \lambda)(a (c + d - \lambda) + (c - \lambda)\lambda)
$$
The eigenvalues associated at the origin are
$$ 
\lambda_1=r, \quad \lambda_2=-b, \quad
 \lambda_{3,4}= \frac{1}{2} \big(-a + c \pm\sqrt{a^2 + 2ac + c^2 + 4ad} \big).
$$
This suggest to consider small values of the parameters $r$ and $b$ introducing 
the small parameter $\varepsilon$ in the following way 
$(r,b)\mapsto (\varepsilon r,\varepsilon b)$. 
Thus, taking $c=a$ the eigenvalues of system \eqref{chen-1} at the origin 
becomes $\lambda_{1,2}=0$ and $\lambda_{3,4}=\pm \sqrt{a(a+c)}$ when 
$\varepsilon\mapsto 0$. It follows that the origin is a zero-Hopf 
equilibrium when $\varepsilon\mapsto 0$ if $a(a+c)<0$.
 So, for this choice of parameters we have system \eqref{chen-1} written as
\begin{equation}\label{chen-2}
\begin{gathered}
\dot{x} =  a (y - x) + w, \\
\dot{y} = d x + a y - x z,  \\
\dot{z} = x y - b\varepsilon  z,\\
\dot{w} =y z + r\varepsilon w.
\end{gathered}
\end{equation}
Re-scaling the variables 
$(x,y,z,w)\mapsto (\varepsilon x,\varepsilon y,\varepsilon z,\varepsilon w)$, 
system \eqref{chen-2} becomes
\begin{equation} \label{chen-3}
 \begin{pmatrix} \dot{x} \\ \dot{y}\\ \dot{z} \\ \dot{w}  \\
\end{pmatrix}
=\begin{pmatrix}  a (y - x) + w \\ d x + a y \\0 \\0 \\
\end{pmatrix}
 + \varepsilon  \begin{pmatrix} 
0 \\- x z \\ x y - b z \\ y z + r w \end{pmatrix}.
\end{equation}
Now we have the Chen system \eqref{chen-1} written as differential system 
of the form \eqref{pert-A1}, and we can consider the problem of bifurcating 
periodic solutions of it by using averaging theory.
We have to solve first the unperturbed system of \eqref{chen-3}. 
The solution $\mathbf{x}(t,\mathbf{u})= (x(t),y(t),z(t),w(t))$ of
\begin{equation}\label{chen-4}
\begin{gathered}
\dot{x} =  a (y - x) + w, \\
\dot{y} = d x + a y,   \\
\dot{z} = 0, \\
\dot{w} =0, 
\end{gathered}
\end{equation}
satisfying the initial condition
 $ \mathbf{u}=(x(0),y(0),z(0),w(0))=(x_0,y_0,z_0,w_0) \in \mathbb{R}^4$ is
\begin{equation} \label{chen-5}
\begin{gathered}
\begin{aligned}
x(t) &=  \frac{1}{a + d} (w_0 +((a + d)x_0-w_0)\cosh{(\sqrt{a(a + d)} \ t)} \\
 &\quad + \frac{\sqrt{a + d}}{\sqrt{a}}(w_0 + a(y_0-x_0))
 \sinh{(\sqrt{a(a + d)}\ t})),
\end{aligned}  \\
\begin{aligned}
y(t) &=  \frac{1}{a(a + d)}((d w_0 + a(a + d)y_0)\cosh{(\sqrt{a (a + d)} \ t)}-d w_0\\
 &\quad + \sqrt{a}\sqrt{a + d}(dx_0 + ay_0)\sinh{(\sqrt{a(a + d) }\ t)}),
\end{aligned} \\
w(t) =  w_0,  \\
z(t) =  z_0.
\end{gathered}
\end{equation}
Notice that if $a (a + d)<0$ then
$\cosh{(\sqrt{a(a + d)}  t)}=\cos{(\sqrt{-a(a + d)}  t)}$ and
$\sinh{(\sqrt{a(a + d)}  t)}= i \sin{(\sqrt{-a(a + d)} t)}$
being $i^2=-1$. Hence, when  $a(a + d)<0$ the
solution \eqref{chen-5} of the unperturbed system \eqref{chen-4} is
\begin{equation} \label{chen-6}
\begin{gathered}
 x(t) =  \frac{1}{a + d} (w_0 +((a + d)x_0-w_0)\cos{(\Omega t)}
 - \frac{\Omega}{a}(w_0 + a(y_0-x_0))\sin{(\Omega t})),
\\
y(t) =  \frac{1}{a(a + d)}((d w_0 + a(a + d)y_0)\cos{(\Omega t)}
- d w_0- \Omega(dx_0 + ay_0)\sin{(\Omega t)}),
 \\
w(t) =  w_0,  \\
z(t) =  z_0.
\end{gathered}
\end{equation}
where $\Omega=\sqrt{-a(a + d)}$. We have that any solution \eqref{chen-6} of
the unperturbed system \eqref{chen-4} is periodic of period
$T=\frac{2 \pi}{\Omega}$. So system \eqref{chen-3} is, in fact, a perturbation of
an isochronous system when $a(a+c)<0$ and we can apply Theorem \ref{ave}.
The first variational system of \eqref{chen-4} along the solution
\eqref{chen-6} coincides with the unperturbed system \eqref{chen-4},
so the inverse of fundamental matrix solution $\Phi_\mathbf{u}(t)$
of \eqref{chen-4} is
\[
\begin{pmatrix}
\cos{(\Omega t)}+ \frac{a}{\Omega}\sin{(\Omega t)}
 & -\frac{a}{\Omega}\sin{(\Omega t)} & 0
 & \frac{1}{(a+d)} (1-\cos{(\Omega t)}+ \frac{\Omega}{a} \sin{(\Omega t)})  \\
-\frac{d}{\Omega}\sin{(\Omega t)}
  & \cos{(\Omega t)}- \frac{a}{\Omega}\sin{(\Omega t)} & 0
 & \frac{d}{a (a+d)}(\cos{(\Omega t)}-1) \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
\]
The bifurcation function \eqref{bifur} is given by
$$
 f(\mathbf{u})= \frac{\Omega}{2 \pi} \int_0^{\frac{2 \pi}{\Omega}}
\Phi_\mathbf{u}^{-1} F_1(t,\mathbf{x}_\mathbf{u}) \, dt
= (f_1,f_2,f_3,f_4) \,,
$$
where
\begin{gather*}
f_1(\mathbf{u})
 =  \frac{r w_0}{a+d} + \frac{dz_0((a + d)x_0-3w_0)}{2a(a + d)^2}
 - \frac{(a(y_0 - x_0)+w_0)z_0}{2(a + d)}\,,
\\
f_2(\mathbf{u})
 =  \frac{d(3dw_0 + a(a + d)y_0)z_0} {2 a^2 (a + d)^2}
 - \frac{d r w_0}{ a (a + d)}-\frac{(d x_0+ a y_0)z_0 } {2 (a + d)} ,
\\
\begin{aligned}
f_3(\mathbf{u})
&=   \frac{ d^2 (x_0^2 + 2 b z_0)}{2 (a + d)^2}
 + \frac{a(2abz_0-y_0(2w_0 + a(y_0-2x_0))}{2 (a + d)^2}\\
&\quad - \frac{(3w_0^2+2aw_0y_0) d}{2 a(a + d)^2}
 +\frac{a d(x_0^2 + 2x_0y_0 - y_0^2 + 4bz_0)) }{2 (a + d)^2}  \,,
\end{aligned}\\
f_4(\mathbf{u}) =  w_0 \Big( r - \frac{d z_0}{a (a + d)}\Big) \,.
\end{gather*}

The isolated zeros of the map 
$ \mathbf{u} \mapsto f(\mathbf{u}) 
= (f_1(\mathbf{u}),f_2(\mathbf{u}),f_3(\mathbf{u}),f_4(\mathbf{u}))$  are
\[
 \mathbf{p}_{1,2}
= \Big(\pm\frac{a \sqrt{b(a + d)  r}}{d} ,\mp \sqrt{ b(a + d)  r} ,
\frac{a (a + d)  r}{d},\pm \frac{a \sqrt{b (a + d) r} (a + d)}{d} \Big) 
 \]
Notice that $\mathbf{p}_{1,2} \in \mathbb{R}$ because of conditions of 
Theorem \ref{Teo-chen}.
The determinant of the Jacobian matrix of $\mathbf{f}$ at the points 
$\mathbf{p}_{1,2}$ is
$$
\det(D \mathbf{f}(\mathbf{p}_{1,2})) = \frac{b (a^4 + a^3 d - d^2) r^3}{2 d^2}.
$$
From condition $a(a+d)<0$ we have that $d\neq0$ and  
$a^4 + a^3 d - d^2= a^2(a(a+d)-d^2)\neq0$. Moreover, from  $b(a+d)r<0$ we 
have $b r\neq0$. Then we get $\det(D \mathbf{f}(\mathbf{p}_{1,2}))\neq0$ and
$\mathbf{p}_{1,2} $ are simples zeroes of $\mathbf{f}$. Hence, the averaging 
theory stated in Theorem \ref{ave} predicts the existence of two $T$-periodic orbits 
$\gamma_{1,2}(t,\varepsilon)$ of system \eqref{chen-2} with period 
$\frac{2 \pi}{\Omega}$  such that $\gamma_{1,2}(0,\varepsilon)\to (p_{1,2})$ as 
$\varepsilon \to 0$.


Since we have performed the re-scaling 
$(x,y,z,w)\mapsto (\varepsilon x,\varepsilon y,\varepsilon z,\varepsilon w)$ 
for bringing system \eqref{chen-1} to system \eqref{chen-2}, the solutions 
$\gamma_{1,2}(t,\varepsilon)$
of system \eqref{chen-2} provides the periodic orbits 
$\varepsilon \gamma_{1,2}(t,\varepsilon)$ of system \eqref{chen-1} 
tending to the zero-Hopf equilibrium as $\varepsilon \to 0$. 
This completes the proof.

\begin{remark} \label{rmk3} \rm
 Regarding the stability of the bifurcated periodic orbits, we know that 
the eigenvalues of $D \mathbf{f}(\mathbf{p}_{1,2})$ are
$$
\lambda_{1,2} =\frac{1}{2} \Big(b \pm \sqrt{b (b + 8 r)} \Big), \quad
\lambda_{3,4}=\frac{r}{2} \Big(1 \pm i \frac{a \Omega}{d}\Big) \,.
$$
Therefore, under our parameter restrictions we cannot have all the eigenvalues
 with negative real part. In consequence we cannot use Theorem \ref{ave} 
for the analysis of the periodic orbits $\gamma_{1,2}(t,\varepsilon)$.
\end{remark}

\subsection*{Acknowledgments}
 This research was supported by grant MTM2011-22877 from MICINN,
 and by grant 2014 SGR 1204 from  CIRIT.

\begin{thebibliography}{99}

\bibitem{buica2}  A. Buic\u{a}, J. P. Fran\c{c}oise, J. Llibre; 
\emph{Periodic solutions of nonlinear periodic differential
systems with a small parameter}, 
Comm. on Pure and Appl. Anal., \textbf{6}, 103--111, (2007).

\bibitem{buica}  A. Buic\u{a}, J. Llibre, O. Makarenkov; 
\emph{Asymptotic stability of periodic solutions for nonsmooth differential
equations with application to the nonsmooth van der Pol oscillator}, 
SIAM J.  Math.  Anal.  \textbf{40} (2009), 2478--2495.

\bibitem{lli1}   L. Cid-Montiel, J. Llibre, C. Stoica; 
\emph{Zero-Hopf bifurcation in a hyperchaotic Lorenz system}, 
Nonlinear Dynam. \textbf{75} (2014), no. 3, 561-–566.

\bibitem{chen2}  G. Chen, X. Dong; 
\emph{From chaos to order, Methodologies, Perspectives and applications}, 
World Scientific, Singapure, 1998, 12--27

\bibitem{chen}  A. M. Chen, J. A. Lu, J.H. L\"u,  S. M. Yu; 
\emph{Generating hyperchaos via state feedback control}, 
International Journal of Bifurcation and Chaos,  \textbf{15} (10)(2005), 3367--3375.

\bibitem{effa}  S. Effati, H. Saberi Nik, A. Jajarmi; 
\emph{Hyperchaos control of the hyperchaotic Chen system by optimal control design}. 
Nonlinear Dynam, \textbf{73} (2013), no. 1-2, 499-–508.

\bibitem{glm}  I. A. Garc\'ia, J. Llibre, S. Maza; 
\emph{Periodic orbits and their stability in the R\"ossler prototype-4 system}, 
Phys. Lett. A, \textbf{376} (2012), 2234–-2237.

\bibitem{glm2}  I. A. Garc\'ia, J. Llibre, S. Maza; 
\emph{On the periodic orbit bifurcating from a zero Hopf bifurcation in systems 
with two slow and one fast variables}, Appl. Math. Comput., \textbf{232} (2014), 84–90.

\bibitem{hak}  H. Haken; 
\emph{Analogy between higher instabilities in fluids and lasers},
 Physiscs Letters A, \textbf{53} (1975), 77--78.

\bibitem{he} A. S. Hegazi, A. E. Matouk; 
\emph{Dynamical behaviors and synchronization in the fractional order 
hyperchaotic Chen system}, Appl. Math. Lett., \textbf{24} (2011), no. 11, 1938–-1944.

\bibitem{chua}  T. Kapitaniak, L. Chua; 
\emph{Hyperchaotic attractors of unidirectionally-coupled Chu'as circuits}, 
Int. J. Bifurcat. chaos Appl. Sci. Eng.,  \textbf{364} (2006), 103--110.

\bibitem{li}  S. Li, G. Chen and X. Mou; 
\emph{On the Security of the Yi-Tan-Siew Chaotic Cipher}, 
IEEE Transactions on Circuits and Systems-II,  \textbf{51}, no 12, (2004), 665--669

 \bibitem{lu}  Y. X. Li, W. K. S. Tang, G. R. Chen; 
\emph{Generating hyperchaotic L\"u attractor via state feedback control}, Physica A,
 \textbf{53} (1975), 77--78.

\bibitem{lli2}  J. Llibre, C. Valls; 
\emph{Hopf bifurcation for some analytic differential systems in $\mathbb{R}^3$ 
via averaging theory}, Discrete Contin. Dyn. Syst.,
 \textbf{30} (2011), no. 3, 779–790.

\bibitem{ma}  I. G. Malkin; 
\emph{Some problems of the theory of nonlinear oscillations}, 
(Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. 492 pp.

\bibitem{ning}  S. C. Ning, H. Haken; 
\emph{Detuned lasers and the complex lorenz equation: 
subcritical and supercritical Hopf bifurcations}, Phys. Rev. A,
 \textbf{41} (1990), 3826--3837.


 \bibitem{park}  J. H. Park; 
\emph{Adaptive synchronization of hyperchaotic Chen system with uncertain parameters},
 Chaos Solitons Fractals, \textbf{26} (2005), no. 3, 959–-964.


\bibitem{chicone3}  M. B. H. Rhouma,  C. Chicone; 
\emph{On the continuation of periodic orbits}, Methods Appl. Anal.,
\textbf{7} (2000), 85–104.

\bibitem{Rose} M. Roseau; 
\emph{Vibrations non lin\'eaires et th\'eorie de la stabilit\'e},
 Springer Tracts in Natural Philosophy, Vol. 8 Springer-Verlag, Berlin-New York 1966.

\bibitem{Ross}  O. E. R\"ossler; 
\emph{An equation for hyperchaos}, Physiscs Letters A,
 \textbf{71} (2-3)(1979), 155--156.

\bibitem{smaoui}  N. Smaoui, A. Karouma, M. Zribi; 
\emph{Secure communications based on the synchronization of the hyperchaotic 
Chen and the unified chaotic systems}, 
Commun. Nonlinear Sci. Numer. Simul. \textbf{41} (2011), no. 8, 3279–3293.

\bibitem{Sprott}  J. C. Sprott; 
\emph{Elegant chaos. Algebraically simple chaotic flows}. 
World Scientific Publishing, 2010.

\end{thebibliography}

\end{document}