Chaotic Control and Generalized Synchronization for a Hyperchaotic Lorenz-Stenflo System

This paper is devoted to investigate the tracking control and generalized synchronization of the hyperchaotic Lorenz-Stenflo system using the tracking model and the feedback control scheme. We suppress the chaos to unstable equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls. Novel tracking controllers with corresponding parameter update laws are designed such that the Lorenz-Stenflo systems can be synchronized asymptotically. Moreover, numerical simulations are presented to demonstrate the effectiveness, through the contrast between the orbits before being stabilized and the ones after being stabilized.


Introduction
Study of chaotic control and generalized synchronization has received great attention in the past several decades [1][2][3][4][5][6][7][8][9][10]; many hyperchaotic systems have been proposed and studied in the last decade, for example, a new hyperchaotic Rössler system [4], the hyperchaotic L system [5], Chua's circuit [6], the hyperchaotic Chen system [7,8], and so forth. Hyperchaotic system has been proposed for secure communication and the presence of more than one positive Lyapunov exponent clearly improves the security of the communication scheme [9][10][11][12]. Therefore, hyperchaotic system generates more complex dynamics than the low-dimensional chaotic system, which has much wider application than the low-dimensional chaotic system.
In this paper, we will consider chaos control and generalized synchronization related to hyperchaotic Lorenz-Stenflo system. we found that the feedback control achieved in the low-dimensional system like many other studies of dynamics in low-dimensional systems. We suppress the hyperchaotic Lorenz-Stenflo system to unstabilize equilibrium via three control methods: linear feedback control, speed feedback control, and doubly-periodic function feedback control. By designing a nonlinear controller, we achieve the generalized synchronization of two Lorenz-Stenflo systems up to a scaling factor. Moreover, numerical simulations are applied to verify the effectiveness of the obtained controllers.

The Hyperchaotic Lorenz-Stenflo System
In the following we would like to consider the hyperchaotic cases of system (1). When = 1.0, = 0.7, = 26, and = 1. stabilized, respectively. The volume of the elements in the phase space ( ) = and the divergence of flow (1) are defined by System (1) is dissipative when 2 + + 1 > 0. Moreover, an exponential contraction rate is given by It is clear that ( ) = 0 −(2 + +1) , which implies that the solutions of system (1) are bounded as − > +∞. It is easy to find the three equilibria 1 (0, 0, 0, 0), To determine the stability of the equilibria point 1 (0, 0, 0, 0), evaluating the Jacobian matrix of system (1) at 1 yields Abstract and Applied Analysis Thus, the equilibria 1 is a saddle point of the hyperchaotic system (1). The Jacobian matrix of system (1) at 2 yields ) .
Thus, the equilibria 2 are unstable, and 3 is similar.

The abbreviated characteristic equation is
According to the Routh-Hurwitz criterion, constraints are imposed as follows: This characteristic polynomial has four roots, all with negative real roots, under the condition of 1 > 0, 2 > 0, The characteristic equation of is given by According to Appendix A, we easily obtain 1 = 20 which yields the eigenvalues via the compute simulation Thus the zero solution of system (9) is exponentially stable, Proposition 1 is proved.
Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (1)
In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix The Jacobian matrix is The characteristic equation of is Abstract and Applied Analysis  The proof of Proposition 3 is the same as Proposition 1, which is straightforward and thus is omitted.
In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix ) .
The characteristic equation of changes the following: Thus the zero solution of system (9) is exponentially stable; Proposition 3 is proved. Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (25)
Lemma 4 (see [27,28]). The zero solution of the error dynamical system (34) is globally and exponentially stable; the master-slave systems (31) and ( where = ∈ × and = ∈ × are both positive definite matrices, max ( ) and min ( ) stand for the minimum and maximum eigenvalues of the matrix , respectively, and min ( ) denotes the minimum eigenvalue of the matrix .
In the following, we consider the hyperchaotic system (1) as a master system:̇= as a slave system, where the subscripts " " and " " stand for the master system and slave system, respectively. Let the error state be
Abstract and Applied Analysis

14
Abstract and Applied Analysis there exist many possible choices for V 1 , V 2 , V 3 , and V 4 . Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential generalized projective synchronization.
The concrete proof of Theorem II can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The parameters are chosen to be 1 = 2, 2 = 3, 3 = 3, and 4 = 5. The initial values and others parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 11(a)-11(d). The synchronization errors are shown in Figures 12(a)-12(d).
The concrete Proof of Theorem III can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The initial values and the parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions Abstract and Applied Analysis

Summary and Conclusions
In this paper, we have introduced the tracking control and generalized synchronization of the hyperchaotic system which is different from the Lorenz-Stenflo attractor. We suppress the chaos to unstabilize equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls of two Lorenz-Stenflo systems. As a result, some powerful controllers are obtained. Then, we investigate the hyperchaotic system applying the complex system calculus technique. Moreover, numerical simulations are used to verify the effectiveness of our results, through the contrast between the orbits before being stabilized and the ones after being stabilized.

B. The Proof of Theorem
Proof of Theorem I. Consider the controller (40) and choose the following V : where = diag( 1 , 2 , 3 , 4 ), which implies that the conclusion of Theorem III is true. So (39) is asymptotically stable. This implies that the two Lorenz-Stenflo hyperchaotic systems are projective synchronized functions.