Stability Analysis of Unified Chaotic System

د للاخ نم دحوملا برطضملا ماظنلا ةيرارقتسا داجيإ وه ثحبلا اذه نم فدهلا ةسار ةفلتخم ةمظنأ ، ميق ىلع دامتعلااب ةمظنأ ةثلاث ىلإ دحوملا ماظنلا مسقني ثيح θ ، امدنع ثيحب نوكت 8 . 0 0 < ≤ θ امدنعو سنرول ماظن حبصي دحوملا برطضملا ماظنلا ناف 8 . 0 = θ  يل ماظن حبصي دحوملا ماظنلا ناف امدنعو 1 8 . 0 ≤ < θ حبصي دحوملا ماظنلا ناف ماظن  نيشت . برطضملا ماظنلا ةيرارقتسا ىلإ اندوقي ةمظنلأا هذه ةيرارقتسا ليلحت ةساردبو دحوملا .

; the unified chaotic system becomes Chen system. Investigations the stability analysis of these systems leads to the stability of the unified chaotic system.

1-Introduction:
Over the last three decades, Chaos as very interesting nonlinear phenomena has been widely studied in various scientific fields. Recently, there has been increasing interest in controlling and utilizing chaos, particularly within the physicists and mathematicians and engineering and technological communities [5].
In 1963, Lorenz found the first chaotic system, which is a third order autonomous system with only two multiplication-type quadratic terms but displays very complex dynamical behaviors [10]. In 1999, Chen found another similar but topologically non-equivalent chaotic system , Chen system [3]. In 2002, L u  and Chen found a new chaotic system, bearing the name of the L u  system [1]. In the same year, L u  et al. unified above the three chaotic systems into one chaotic system which is called unified chaotic system [2]. The unified chaotic system was produced as a kind of unique and unified classification between the Lorenz and the Chen attractors, both in theory and in simulation. Of particularly interesting is that this unified system is likely the simplest chaotic system that bridges the gap between the Lorenz and the Chen system, and so it contributes to a better understanding of a family of 3dimensional autonomous quadratic chaotic systems [4].
The nonlinear differential equations that describe the unified chaotic system are following form [2,4,8]: , this system which describes a large family of chaotic systems containing the Lorenz and Chen system as two extremes and the L u  system as a transition in between [10]. Recently, there are some analytical results reported about these chaotic systems, which are called the Lorenz family [3]. ; it becomes the generalized L u  system; when system (1) is called the generalized Chen system [2,9].
The three critical points of system (1) can be found in [4] as follows: and studied the stability at these points. Finally, cleared that this system is unstable for , without depended on the Lorenz, Chen and L u  systems.
In this paper, we find the stability conditions of L u  system and obtain on helping results; finally, we find the same results which found in [4], but with another method, by using the three chaotic systems to investigate the stability of system (1).

2-Preliminaries:
The following notations will be used for the remaining of this paper. The mathematical model of Lorenz system is a system of nonlinear ordinary differential equations which has the following form: , and Lorenz studied this system when , [6]. Chen system it also has the form of nonlinear ordinary differential equations: The mathematical model of L u  system also is a system of nonlinear ordinary differential equations which has the form: One can see that the three dynamical systems, all are nonlinear ordinary differential equation systems of third degree. And all x derivative in those systems are multiplied by a number  . The only difference is the form of y derivative [1]. According to Vanecek and Celikovsky definition. They separated the system into linear and quadratic parts. In the linear part of system described by the matrix  [8,9].

3-Helping Results:
Regarding the basic dynamical behaviors of the Lorenz system, we have the following observations.

4-Main results:
It is easy to verify that the L u  system has three critical points: The characteristic equation is: , and three eigenvalues corresponding to the critical point 0 S are: In the following, we consider the stability of the system (4) at the critical points + S and − S , Because the system is invariant under the transformation [1], so one only needs to consider the stability of any one of the both . The stability of the system (4) at critical point + S is analyzed in this paper.
Under the linear transformation The critical point + S of the system (4) is swiched to the new critical point ) 0 , 0 , 0 ( 0 S  of the system (13) under the linear transformation, in the following, the stability of system (13) at the critical point 0 S  is considered. The Jacobian matrix of the system (13) at and the characteristic equation is : , consequently we must prove that r r We will use the following Corollary, which enables us to find the Jacobian matrix directly for Lorenz family at So, the Jacobian matrix at + S by using Corollary 1 is given by: and the characteristic equation is :

-Conclusion:
In this paper, we have investigated the stability of unified chaotic system by using three chaotic systems (Lorenz, Chen and L u  systems). By this method we justified the same results which have been found by previous methods. An illustrative example shows the effectiveness and feasibility of this method.