Study of Dynamical Properties and Effective of a State u for Hyperchaotic Pan Systems

89 Study of Dynamical Properties and Effective of a State u for Hyperchaotic Pan Systems Saad Fawzi AL-Azzawi saad_fawzi78@yahoo.com Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul, Iraq Received on: 27/05/2012 Accepted on: 18/09/2012 ABSTRACT In this paper, we propose a new four-dimensional continuous autonomous hyperchaotic system which is built by adding a nonlinear controller to a threedimensional chaotic Pan system. The new system is analyzed both theoretically and numerically by studying dynamical behaviors for this system, including equilibrium points, Lyapunov exponents, stability and bifurcation, also we study the stability, bifurcation and symmetry for another four dimensional system which is generated by Pan in (2011) from the original system, we compared between two systems and found the difference between them. Finally, we show the effective of state u on the two systems.


1-Introduction:
In the recent years, hyperchaos generation and control have been extensively studied due to its theoretical and practical applications in the fields of communications, laser, neural work, nonlinear circuit, mathematics, and so on [6,7,10] historically, hyperchaos was firstly reported by ssler o R   . That is, the noted fourdimensional(4D) hyperchaoticsystem [7,10]. ssler o R   Generally, a hyperchaotic system is classified as a chaotic system with more than one positive Lyapunov exponents [2,3 ,6,7,9], and has more complex dynamical behaviors than chaotic system [3,9].Very recently, hyperchaos was found numerically and experimentally by adding a simple state feedback controller [6,7,10].
In [5] Pan and etc propose four dimensional hyperchaotic pan system(2011) via state feedback control then transformation to fractional-order hyperchaotic system (FOHS) and studied chaos synchronization for this system.
In this paper, we generated a new modified hyperchaotic Pan system based on a three-dimensional Pan system by introducing a nonlinear state feedback controller, and we study some basic properties and behaviors for this system, also you study the dynamical behaviors of hyperchaotic Pan system(2011) and compare between two systems. Finally, we show the effect of state u on two systems.

2-Hyperchaotic Pan System
The Pan system or L u  -like system(2010) [4,5] is described by  [5]. The Lyapunov exponents spectrum and attractor of the system (1) is shown in Fig.1 [5] ,and Fig.2 [5]. In order to obtain hyperchaotic , the important requirements as follows [2,5,7,8, 9,10 ,11] : (1) The minimal dimension of the phase space of an autonomous system is at least four. (2) The number of terms in the coupled equations giving rise to instability should be, at least, two, of which, at least, one should have a nonlinear function. Based on pan system and above two basic requirements, one can construct fourthorder hyperchaot system ,by introducing a state feedback controller, as follows: are constant parameters. this system is called hyperchaotic Pan system, When 28 , [5], the Lyapunov exponents , 0000 . 0 , 2492 . 0 , 7340 . 0 by Wolf Algorithm [5]. The Lyapunov exponents spectrum and attractor of the system (2) are shown in Fig.3 [5] , and Fig.4 [5]. 3-Helping Results:

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

Remark 2 (Lyapunov Exponents) [2,7]:
The dynamical behaviors of this system can be classified as follows: the system is a equilibrium point. In the context of ordinary differential equations ODEs, the word "Bifurcation" has come to mean any marked change in the structure of the orbits of a system (usually nonlinear) as a parameter passes through a critical value [1].

Remark 3(Hopf Bifurcation) [9]:
Any system has a Hopf bifurcation if the following condition is satisfied: 1-The Jacobian matrix has two purely imaginary roots and no other roots with zero real parts.

4-Main Results:
Based on Pan system and generating conditions, we can construct a new four dimension hyperchaotic system by introducing a state feedback controller, as follows: Add controller u to the second equation of system (1), , then we obtain a new hyperchaotic system are constant parameters. For simplification, system (3) is called a new modified hyperchaotic Pan system in this paper.
In the following we briefly describe some dynamical behaviors of the new hyperchaotic system (3).

4.1-Dissipative of System(3):
The divergence of the new four dimensional system (3) is Thus, when 0  + b a the new system (3) is dissipative, consequently, all trajectories of the new system ultimately arrive at an attractor .

4.2-Equilibrium Point of System(3) :
The equilibrium of system (3) satisfies the following equations:

.3-Lyapunov Exponents and Lyapunov Dimension :
We calculate the Lyapunov exponents for a new modified hyperchaotic system with the Wolf Algorithm by using MATLAB software, the numerical simulation is carried out with 10 , for the initial value has the following cases: consequently, the solution of system (3)  are the complex solutions and 2  the real solution of equation (7) then, from . This easily leads to In the following we will prove that the system (3) displays a Hopf bifurcation at the point ) Theorem 2: If 0 = c , equation (7) has negative solutions 0 together with a pair of purely imaginary roots , therefore, the system (3) displays a Hopf bifurcation at the point ) , the real part and imaginary part of the ) ( 0 c c  respectively are:

-Dynamical Behaviors of Hyperchaotic Pan System
In the following, we briefly describe some dynamical behaviors of system (2).

5.1-Symmetry:
Note that the invariance of the system (2) under the transformation ) , , , ,i.e. under reflection in the z-axis . the symmetry persists for all values of the system parameters.

5.2-Dissipative of System (2):
The divergence of the new four dimensional system (2) is Thus when 0 the new system (2) is dissipative, consequently all trajectories of the new system ultimately arrive at an attractor .

Proposition 2:
The zero solution of system (2) has not purely imaginary roots if and ( 0 c critical value). In this case, the solutions of , therefore not satisfied one condition of Hopf bifurcation. Due to the system is invariant under the transformation, so one only needs to consider the stability of anyone of the both. The stability of the system (2)at equilibrium point 1 p is analyzed in this paper. Theorem4 : The solution of system (2)at the equilibrium point 1 Proof: Now to find Jacobian matrix at 1 p we need the following transformation Under the linear transformation ) , , , The equilibrium point 1 p of the system (2) Using Routh-Hurwitz criterion, the equation (14) has all roots with negative real parts if and only if the conditions are satisfied as follows: .
Consequently, conditions of Routh-Hurwitz are satisfied in (1) therefore, the is system (2) is asymptotically stable under the conditions: We explain the difference between the two systems by using of the following table

6-Conclusions:
In this paper a new four dimensional hyperchaotic system called a new modified hyperchaotic Pan system was presented. and some properties and dynamic behaviors for this system have been investigated, we studied the effective location and sign for state u of the system. We conclued that the effect of state u was different from system to another, where the state u was effective of a new system while was not effective on hyperchaotic pan system.