New Study of Stability for New Lorenz-like System

In this paper, we studied differential system like of three dimensional Lorenz system. Nonlinear characteristic and basic dynamic properties of three dimensional autonomous system are studied by means of nonlinear dynamics theory, including the stability and we found that the value 4 1 − = b is effected by the form of the roots.

. It's notable that the Lorenz system has seven terms on the righthand side , two of which are nonlinear ) , ( xz xy . [5] In 1999 , Chen found a similar, but nonequivalent chaotic attractor [3], which is now known to the dual of the Lorenz system, in a sense defined [2]: when expressing the system in linear and nonlinear where the linear part has the constant matrix 3  . In 2002, Lü and Chen reported a new chaotic system called Lü system [8] , which satisfies the condition 0 21 12 = a a , and bridges the gap between the Lorenz and Chen systems [5].
At the same year, a unified chaotic system was created that connects Chen chaotic system to the Lorenz chaotic system through the Lü chaotic system [7]. Last year Li, et al [5] presented a new Lorenzlike system where the sign of the crucial condition 21 12 a a is only determined by parameter b [9].

2-Dynamical Behavior of the New Chaotic Attractor:
New Lorenz-like system, it's mathematical model that is a system of nonlinear ordinary differential equations which has the following form: is the state variables of the system. a, b and c are parameters . There are six terms on the right-hand side, but only relies on two quadratic nonlinearities xy and xz. [5] The system (1) would satisfy one of the following cases via the crucial condition set by Vanecek and Celikovsky [10] . This shows that the sign of the crucial condition 21 12 a a is determined only by parameter b [9]. The following briefly describe some basic properties of the system (1)

1-Symmetry and Invariance:
First, note that the system (1) has a symmetry S because the transformation: Which permits system invariant for all values of the system parameters a, b and c. Obviously, the zaxis itself is an orbit, that is if . Moreover, the orbit on axis z − tends to the origin as . And the transformation S indicates that the system is symmetrical on the axis z − , for instance, if  is the solution of the system, and  S is too. [5] 2-Dissipative: The system (1) can be a dissipative system, because the divergence of the vector field, also called the trace of the Jacobian matrix is negative if and only if the sum of the parameters a and c is positive, that is

So, the system will always be dissipative if and only if when
, with an exponential: , see [5].

3-Critical Points:
The critical points of the system (1) can be easily found by solving the three , which lead to It can be easily verified that there are three critical points ) , , , there is only one critical point but which dependent on the value of the parameter b, which is denoted as . [5] In [5], this system is studied when the parameter . 0  bc In [9], shows that this system at the origin is asymptotically stable when 0  b , and unstable when 0 In this paper we studied the same system when the parameter 0 is affected on the form of the roots.
The Chaotic Attractor of the New Lorenz-like System (1)

Remark 1 (Routh-Hurwitz Test) [1]:
All roots of the indicated polynomial have negative real parts precisely when the given conditions are met :

Remark 2 (Critical Cases) [1]:
Critical cases in the theory of stability for differential equation means, that cases when the real part of all roots of the characteristic equation has no positive with the real part of at least one root being zero , other express which is neither stable nor unstable.

Remark 3 [5]:
The characteristic polynomial of system (1) at the critical point 0 P is :

Theorem 1 [5]:
The solution of system (1) at the critical point ) Now, we only need to consider the stability of system (1) at ) , , After substituting (6) in (1) we get :

Remark 4 [5]:
The characteristic polynomial of system (7)  It's easy to verify that the equation (8)
Proof: Using Routh -Hurwitz criterion, the equation (5), has all roots with negative real parts if and only if the conditions are satisfied as follows: where c a, are positive parameters, , the proof of the first condition is complete. While if 0 = b , then satisfied remark 2 , hence the system (1) is critical case, the proof is complete. , has the following cases: , we get c − = The proof is complete.
, then the critical points  p are all unstable.
, then the system (7) is a critical case.
Proof : Using Routh -Hurwitz criterion, the equation (8)  , also we must proof that C AB  , therefore , the proof of the first condition is complete.
, then one of Routh-Hurwitz conditions is not satisfied. Consequently, the system (7) is unstable. Finally, if ) 1 ( , then satisfied remark 2 , hence the system (7) is critical case, the proof is complete. . Hence, the system is a critical case, and third part of theorem (5) is hold.

6-Conclusions:
In this paper, we studied the stability for the New Lorenz-like system by using Routh-Hurwitz method (this method depended on the previous methods on the estimate of the signs of the roots without finding the value of these roots), and we conclude that the stability of this system depended on the parameter b.