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Chaos synchronization of the Sprott N System with parameter.

[section]1. Introduction

Since synchronization of chaotic systems was first introduced by Fujisaka and Yamada [1] and Pecora and Carroll [2]. Due to the importance and applications of coupled systems, ranging from chemical oscillators, coupled neurons, coupled circuits to mechanical oscillators, various synchronization schemes have been proposed by many scientists from different research fields [3-10]. Recently, a new type of chaotic synchronization-full state hybrid projective synchronization(FSHPS) in continuous-time chaotic and hyper-chaotic systems based on the Lyapunov's direct method is presented and investigated by wen[11],many notable results and a series of important applications to security communication regarding FSHPS has been presented in Refs [12-14].

We organize this paper as follows. In Section 2, the chaotic characteristic of the Sprott N autonomous system with parameter is studied by theoretical analysis and numerical simulation. In section 3, the scheme of full state hybrid projective synchronization(FSHPS) is given.A proper Smarandache controller is designed and the synchronization of the system is achieved under it.

[FIGURE 1 OMITTED]

[section]2. Dynamical behavior

A series of three dimensional autonomous systems is presented by J.C. Sprott in 1994 [15], in this paper, the N system of those is taken as example. The governing equations of the Sprott N system are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The equilibrium of the system (1) is P(-0.25,0.0.5).The Lyapunov exponents of the system are LEs=(0.076,0,-2.076),which shows the system is chaotic. In order to get abundance dynamical behavior of the system, the parameter /3 is leaded to the system. The governing equations of the Sprott N system with parameter [beta] can be described as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where x = ([x.sub.1], [x.sub.2], [x.sub.3]) is the state variable. The initial conditions are ([x.sub.1](0), [x.sub.2](0), [x.sub.3](0))=(1,5,2), when the parameter [beta] = 1.9, the system exits a chaotic attractor. The chaotic attractor in three-dimensional phase space is illustrated in Fig 1.

For this system, bifurcation can easily be detected by examining graphs of abs(z)versus the control parameter [beta]. The dynamical behavior of the system (2) can be characterized with its Lyapunov exponents which are computed numerically. The bifurcation diagram and the Lyapunov exponents spectrum are showed in Fig 2.

[FIGURE 2 OMITTED]

[section]3. Chaos synchronization base on FSHPS

We recall a class of automomous chaotic flows in the form of

x(t) = F(x), (3)

x = [([x.sub.1], [x.sub.2],...,[x.sub.n]).sup.T] is the state vector, and F(x) = [([F.sub.1](x), [F.sub.2](x),...,[F.sub.n](x)).sup.T] is continuous nonlinear vector function.

We take (3) as the drive system and the response system is given by

y(t) = F(y) + u, (4)

y = [([y.sub.1],[y.sub.2],...,[y.sub.n]).sup.T] is the state vector, and F(y) = [([F.sub.1](y),[F.sub.2](y),...,[F.sub.n](y)).sup.T] is continuous nonlinear vector function. u = u(x,y) = [([u.sub.1](x,y),[u.sub.2](x,y),...,[u.sub.n](x,y)).sup.T] is the controller to be determined for the purpose of full state hybrid projective synchronization.Let the vector error state be e(t) = y(t) -[alpha]x(t).Thus, the error dynamical system between the drive system (3) and the response system (4) is

e(t) = y(t) - [alpha]x(t) = F(x,y) + u, (5)

where F(x,y) = F(y) - [alpha]F(x) = [([F.sub.1](y) - [[alpha].sub.1][F.sub.1]](x),[F.sub.2](y) - [[alpha].sub.2][F.sub.2](x),...,[F.sub.n](y) - [[alpha].sub.n][F.sub.n](x)).sup.T].

In the following, we will give a simple principle to select suitable feedback controller a such that the two chaotic or hyper-chaotic systems are FSHPS. If the Lyapunov function candidate V is take as:

V = 1/2[e.sup.T][P.sub.e], (6)

where P is a positive definite constant matrix, obviously, V is positive define. One way choose as the corresponding identity matrix in most case. The time derivative of V along the trajectory of the error dynnmical system is as follows

V = [e.sup.T]P(u + F), (7)

Suppose that we can select an appropriate controller u such that V is negative definite. Then, based on the Lyapunov's direct method, the FSHPS of chaotic or hyper-chaotic flows is synchronization under nonlinear controller u .

In order to observe the FSHPS of system (2), we define the response system of (2) as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where u = [([u.sub.1], [u.sub.2], [u.sub.3]).sup.T] is the nonlinear controller to be designed for FSHPS of two Sprott N chaotic systems with two significantly different initial conditions.

Define the FSHPS error signal as e(t) = g(t) - cxx(t),i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [alpha] = diag([[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3]),and [[alpha].sub.1], [[alpha].sub.2] and [[alpha].sub.3] are different desired in advance scaling factors for FSHPS. The error dynnmical system can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The goal of control is to find a controller u = [([u.sub.1], [u.sub.2], [u.sub.3]).sup.T] for system (10) such that system (2) and (8) are in FSHPS. We now choose the control functions [u.sub.1], [u.sub.2] and [u.sub.3] as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

If the Lyapunov function candidate is taken as:

V = 1/2([e.sup.2.sub.1] + [e.sup.2.sub.2] + [e.sup.2.sub.3]), (12)

The time derivative of V along the trajectory of the error dynnmical system (10) is as follows

V = - ([e.sup.2.sub.1] + [e.sup.2.sub.2] + [e.sup.2.sub.3]), (13)

Since V is a positive definite function and V is a negative definite function, according to the Lyapunov's direct method, the error variables become zero as time tends to infinity, i.e., [lim.sub.t[right arrow][infinity]] || [y.sub.i] - [[alpha].sub.i][x.sub.i] || = 0, i = 1, 2, 3. This means that the two Sprott A systems are in FSHPS under the controller (11).

[FIGURE 3 OMITTED]

For the numerical simulations, fourth-order Runge-Kutta method is used to solve the systems of differential equations (2) and (8). The initial states of the drive system and response system are ([x.sub.1](0), [x.sub.2](0), [x.sub.3](0) = (1, 5, 2) and ([y.sub.1](0), [y.sub.2](0), [y.sub.3](0) = (11,15,12) The state errors between two Sprott systems are shown in Fig.3. Obviously, the synchronization errors converge asymptotically to zero and two systems are indeed achieved chaos synchronization.

(section)4. Conclusion and discussion

In the paper, the problem synchronization of the Sprott N system with parameter is investigated. An effective full state hybrid projective synchronization (FSHPS) controller and analytic expression of the controller for the system are designed. Because of the complete synchronization, anti-synchronization, projective synchronization are all included in FSHPS, our results contain and extend most existing works. But there are exist many interesting and difficult problems left our for in-depth study about this new type of synchronization behavior, therefore, further research into FSHPS and its application is still important and insightful, although it is not in the category of generalized synchronization.

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Xiaojun Liu ([dagger]), Wansheng He ([dagger]), Xianfeng Li([double dagger]), Lixin Yang ([dagger])

([dagger]) Department of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu, 741001, P.R.China

([double dagger]) School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University Lanzhou, Gansu. 730070, P.R.China

(1) This work is supported by the Gansu Provincial Education Department Foundation 0808-04 and Scientific Research Foundations of Tianshui Normal University of China TSB0818.
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Author:Xiaojun, Liu; Wansheng, He; Xianfeng, Li; Lixin, Yang
Publication:Scientia Magna
Article Type:Report
Geographic Code:9CHIN
Date:Jun 1, 2009
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