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

Abstract

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.

1. Introduction

Study of chaotic control and generalized synchronization has received great attention in the past several decades [1–10]; 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 [9–12]. Therefore, hyperchaotic system generates more complex dynamics than the low-dimensional chaotic system, which has much wider application than the low-dimensional chaotic system.

Until now, a variety of approaches have been proposed for the synchronization of low-dimensional chaotic systems, including Q-S method [13, 14], active control [15, 16], adaptive control [17–24], and time-delay feedback control [25]. Recently, Stenflo [26] presented a new hyperchaotic Lorenz-Stenflo (LS) system: ̇̇()̇̇𝑥=𝑎𝑦−𝑥+𝑑𝑤,𝑦=𝑥𝑐−𝑧−𝑦,𝑧=𝑦𝑥−𝑏𝑧,𝑤=−𝑥−𝑎𝑤. (1) In (1), 𝑥, 𝑦, 𝑧, and 𝑤 are the state variables of the system and 𝑎, 𝑏, 𝑐, and 𝑑 are real constant parameters. System (1) is generated from the originally three-dimensional Lorenz chaotic system by introducing a new control parameter 𝑏 and a state variable 𝑤.

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.

2. The Hyperchaotic Lorenz-Stenflo System

In the following we would like to consider the hyperchaotic cases of system (1). When 𝑎=1.0, 𝑏=0.7, 𝑐=26, and 𝑑=1.5, system (1) exhibits hyperchaotic behavior. Simulated results are depicted in Figures 1 and 2. Figures 1(a)–1(d) depict the projection of the chaotic attractor in different spaces; Figures 2(a)–2(d) depict the states of system (1) before being stabilized, respectively. The volume of the elements in the phase space 𝛿𝑋(𝑡)=𝛿𝑥𝛿𝑦𝛿𝑧𝛿𝑤 and the divergence of flow (1) are defined by ∇𝑋=𝜕𝑋+𝜕𝑥𝜕𝑋+𝜕𝑦𝜕𝑋+𝜕𝑧𝜕𝑋(),𝜕𝑤=−2𝑎+𝑏+1 (2) where 𝑋=(𝑥̇,𝑦,̇𝑧,̇𝑤̇)=[𝑎(𝑦−𝑥)+𝑑𝑤,𝑥(𝑐−𝑧)−𝑦,𝑦𝑥−𝑏𝑧,−𝑥−𝑎𝑤].

(a)

(b)

(c)

(d)

Figure 1 

Chaotic attractor in different spaces: (a) (𝑥,𝑦,𝑧), (b) (𝑥,𝑦,𝑤), (c) (𝑥,𝑤,𝑧), and (d) (𝑦,𝑤,𝑧).

(a)

(b)

(c)

(d)

Figure 2 

The states of system (1) before being stabilized: (a) 𝑥-𝑡, (b) 𝑦-𝑡, (c) 𝑧-𝑡, and (d) 𝑤-𝑡.

System (1) is dissipative when 2𝑎+𝑏+1>0. Moreover, an exponential contraction rate is given by 𝑑𝑋(𝑡)()𝑋(𝑡).𝑑𝑡=−2𝑎+𝑏+1 (3) It is clear that 𝑋(𝑡)=𝑋₀𝑒^{−(2𝑎+𝑏+1)𝑡}, which implies that the solutions of system (1) are bounded as 𝑡−>+∞. It is easy to find the three equilibria 𝐸₁(0,0,0,0), 𝐸₂(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑑+𝑎²,(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑎²,(𝑐−1)𝑎²−𝑑𝑎²,−𝑑+𝑎²𝑏𝑎²𝑐−𝑎²−𝑑𝑑+𝑎²𝑎,𝐸₃−(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑑+𝑎²,−(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑎²,(𝑐−1)𝑎²−𝑑𝑎²,𝑑+𝑎²𝑏𝑎²𝑐−𝑎²−𝑑𝑑+𝑎²𝑎. (4)

To determine the stability of the equilibria point 𝐸₁(0,0,0,0), evaluating the Jacobian matrix of system (1) at 𝐸₁ yields 𝐽|_𝐸1=−𝑎𝑎0𝑑𝑐−10000−𝑏0−100−𝑎. (5) If 𝑎=1.0, 𝑏=0.7, 𝑐=26, and 𝑑=1.5, the four eigenvalues of the characteristic polynomial of Jacobian matrix (5) are 𝜆₁=3.94975,𝜆₂=−5.9497,𝜆₃=−1,𝜆₄=−0.7. (6) Thus, the equilibria 𝐸₁ is a saddle point of the hyperchaotic system (1). The Jacobian matrix of system (1) at 𝐸₂ yields𝐽|_𝐸2=−𝑎𝑎0𝑑𝑐−1(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑑+𝑎²0(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑎²(𝑐−1)𝑎²−𝑑𝑏𝑑+𝑎²𝑑+𝑎².−𝑏0−100−𝑎 (7) If 𝑎=1.0, 𝑏=0.7, 𝑐=26, and 𝑑=1.5, the four eigenvalues of the characteristic polynomial of Jacobian matrix (7) are 𝜆₁=4.8887,𝜆₂𝜆=−6.3032,₃=−1.1427+0.5438𝑖,𝜆₄=−1.1427−0.5438𝑖. (8) Thus, the equilibria 𝐸₂ are unstable, and 𝐸₃ is similar.

3. The Hyperchaotic Control for Lorenz-Stenflo System

In this section, we control the hyperchaotic system (1) such that all trajections converge to the equilibrium point (0,0,0,0). The controlled hyperchaotic Lorenz-Stenflo system is given by 𝑥̇=𝑎𝑦−𝑥+𝑑𝑤+𝑢₁,𝑦̇=𝑥(𝑐−𝑧)−𝑦+𝑢₂,𝑧̇=𝑦𝑥−𝑏𝑧+𝑢₃,𝑤̇=−𝑥−𝑎𝑤+𝑢₄, (9) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are external control inputs which will be suitably derived from the trajectory of the chaotic system (1), specified by (𝑥,𝑦,𝑧,𝑤) to the equilibrium (0,0,0) of uncontrolled system 𝑢_𝑖=0(𝑖=1,2,3,4).

3.1. Linear Function Feedback Control

For the modified hyperchaotic Lorenz-Stenflo system (9), if one of the following feedback controllers 𝑢_𝑖(𝑖=1,2,3,4) is chosen for the system (9), then 𝑢₁=−𝑘₁𝑥₁, 𝑢₂=−𝑘₂𝑦, 𝑢₃=−𝑘₃𝑧, 𝑢₄=−𝑘₄𝑤, and where 𝑘_𝑖’s are feedback coefficients. Therefore controlled system (9) is rewritten as 𝑥̇=𝑎𝑦−𝑥+𝑑𝑤−𝑘₁̇()𝑥,𝑦=𝑥𝑐−𝑧−𝑦−𝑘₂̇𝑦,𝑧=𝑦𝑥−𝑏𝑧−𝑘₃̇𝑧,𝑤=−𝑥−𝑎𝑤−𝑘₄𝑤 (10) whose Jacobian matrix is 𝐽=−𝑎−𝑘₁𝑎0𝑑𝑐−1−𝑘₂0000−𝑏−𝑘₃0−100−𝑎−𝑘₄. (11) The characteristic equation of 𝐽 is 𝜆⁴+(−𝐵−𝐴−𝐷−𝐶)𝜆³+(𝐶𝐷+𝐴𝐵+𝐵𝐷+𝐵𝐶−𝑐𝑎+𝐴𝐷+𝐴𝐶+𝑑)𝜆²+()−𝐴𝐵𝐷−𝐴𝐵𝐶−𝐵𝐶𝐷+𝑐𝑎𝐷+𝑐𝑎𝐶−𝐴𝐶𝐷−𝑑𝐶−𝑑𝐵𝜆+𝐸=0, (12) where 𝐴=−𝑎−𝑘₁, 𝐵=−1−𝑘₂, 𝐶=−𝑏−𝑘₃, 𝐷=−𝑎−𝑘₄, and 𝐸=𝐴𝐵𝐶𝐷−𝑐𝑎𝐶𝐷+𝑑𝐵𝐶.

The abbreviated characteristic equation is 𝜆⁴+𝑅₁𝜆³+𝑅₂𝜆²+𝑅₃𝜆+𝐸=0, (13) where 𝑅₁=−𝐵−𝐴−𝐷−𝐶, 𝑅₂=𝐶𝐷+𝐴𝐵+𝐵𝐷+𝐵𝐶−𝑐𝑎+𝐴𝐷+𝐴𝐶+𝑑, and 𝑅₃=−𝐴𝐵𝐷−𝐴𝐵𝐶−𝐵𝐶𝐷+𝑐𝑎𝐷+𝑐𝑎𝐶−𝐴𝐶𝐷−𝑑𝐶−𝑑𝐵, 𝑅₄=𝐸.

According to the Routh-Hurwitz criterion, constraints are imposed as follows: 𝐻₁=𝑅₁𝐻=−𝐵−𝐴−𝐷−𝐶>0,₂=𝑅₁𝑅₃1𝑅₂=𝑅₁𝑅₂−𝑅₃=(−𝐴−𝐶−𝐵)𝐷²+(−2𝐴−2𝐶)𝐵−𝐵²−𝐴²−𝐶²𝐷+()𝐵−𝑑−2𝐴𝐶−𝐴−𝐶²−𝐴²𝐶+−2𝐴𝐶+𝑐𝑎−𝐴²−𝐶²𝐵+−𝑑−𝐶²𝐻+𝑐𝑎𝐴>0,₃=𝑅₁𝑅₃01𝑅₂𝐸0𝑅₁𝑅₃=𝑅₁𝑅₂𝑅₃−𝐸𝑅²₁−𝑅²₃𝐻>0,₄=𝑅₁𝑅₃001𝑅₂𝑅₄00𝑅₁𝑅₃001𝑅₂𝑅₄=𝑅₁𝑅₂𝑅₃𝑅₄−𝑅²₁𝑅²₄−𝑅²₃𝑅₄>0. (14)

This characteristic polynomial has four roots, all with negative real roots, under the condition of 𝐻₁>0, 𝐻₂>0, 𝐻₃>0, and 𝐻₄>0. Therefore, the equilibria (0,0,0,0) are the stable manifold 𝑊^𝑠 and the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable. The concrete dynamics of (9) can be demonstrated by Proposition 1, while the Maple program is demonstrated in Appendix A.

Proposition 1. If one chooses the control coefficients 𝑘₁=8, 𝑘₂=4, 𝑘₃=3, and 𝑘₄=2 and the parameters 𝑎=1.0, 𝑏=0.7, 𝑐=26, and 𝑑=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).

Proof. When the parameters were selected by the above value, we obtain the Jacobian matrix 𝐽=−9.01.001.526−50000−3.70−100−3.0. (15) The characteristic equation of 𝐽 is given by 𝜆⁴7+20𝜆10³2+1255𝜆²3+2954𝜆+2381320=0. (16) According to Appendix A, we easily obtain 𝐻₁7=2010>0,𝐻₂𝐻=2300>0,₃=578>0,𝐻₄=69500>0, (17) which yields the eigenvalues via the compute simulation 𝜆₁=−12.36844706,𝜆₂𝜆=−1.931149367,₃=−2.700403573,𝜆₄=−3.70000. (18) 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) using the fourth-order Runge-Kutta scheme with time step 0.01. The parameters and the corresponding feedback coefficients are given by the above value. The initial values are taken as [𝑥(0)=1,𝑦(0)=0.7,𝑧(0)=20,𝑤(0)=0.1]. The behaviors of the states (𝑥,𝑦,𝑧,𝑤) of the controlled chaotic Lorenz-Stenflo system (1) with time are displayed in Figures 3(a)–3(d).

(a)

(b)

(c)

(d)

Figure 3 

The states of system (9): (a) 𝑥-𝑡, (b) 𝑦-𝑡, (c) 𝑧-𝑡, (d) 𝑤-𝑡.

3.2. Speed Function Feedback Control

Suppose that 𝑢₁=𝑢₃=0; 𝑢₂ and 𝑢₄ are of the speed forms 𝑢₂=𝑘₂(−𝑏𝑧+𝑥𝑦) and 𝑢₄=−𝑘₄[𝑎(𝑦−𝑥)+𝑑𝑤], where 𝑘₂ and 𝑘₄ are speed feedback coefficients. Therefore the controlled chaotic system (9) is rewritten as ̇̇()𝑥=𝑎𝑦−𝑥+𝑑𝑤,𝑦=𝑥𝑐−𝑧−𝑦+𝑘₂,̇̇−𝑏𝑧+𝑥𝑦𝑧=𝑦𝑥−𝑏𝑧,𝑤=−𝑥−𝑎𝑤−𝑘₄𝑎𝑦−𝑥+𝑑𝑤. (19) The Jacobian matrix is 𝐽=−𝑎𝑎0𝑑𝑐−1−𝑘₂𝑏000−𝑏0−1+𝑘₄𝑎−𝑘₄𝑎0−𝑎−𝑘₄𝑑. (20) The characteristic equation of 𝐽 is 𝜆⁴+1−𝑘₁+2𝑎+𝑏𝜆³+2𝑎𝑏−𝑘₁𝑏−𝑘₁𝑎+2𝑎−𝑘₁+𝑏+𝑑+𝑎²−𝑐𝑎𝜆²+−𝑐𝑎²−𝑐𝑎𝑏+2𝑎𝑏+𝑑𝑏+𝑎²𝑏−𝑘₁𝑎+𝑎²−𝑘₁𝑏−𝑘₁𝑏𝑎+𝑑𝜆−𝑐𝑎²𝑏−𝑘₁𝑏𝑎+𝑎²𝑏+𝑑𝑏=0. (21)

Proposition 2. If one chooses the control coefficients: 𝑘₂=9, 𝑘₄=−1 and the parameters 𝑎=1.0, 𝑏=0.7, 𝑐=26, and 𝑑=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).

Similar to Proposition 1, the proof of Proposition 2 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 𝐽=−1.01.001.526−1−6.3000−0.70−2.01.000.5. (22) The characteristic equation of 𝐽 changes the following: 𝜆⁴+2.2𝜆³−21.95𝜆²−39.6𝜆−16.45=0. (23) The eigenvalues of the above equation are 𝜆₁=−5.10412,𝜆₂𝜆=−1,₃=−0.70000,𝜆₄=−4.60412. (24) Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (20) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [𝑥(0)=1,𝑦(0)=0.7,𝑧(0)=20,𝑤(0)=0.1]. The behaviors of the states (𝑥,𝑦,𝑧,𝑤) of the controlled chaotic Lorenz-Stenflo system (20) with time are displayed in Figures 4(a)–4(d).

(a)

(b)

(c)

(d)

Figure 4 

The states of system (19): (a) 𝑥-𝑡, (b) 𝑦-𝑡, (c) 𝑧-𝑡, and (d) 𝑤-𝑡.

3.3. The Doubly Periodic Function Feedback Control

Suppose that 𝑢₂=0, 𝑢₃=0, and 𝑢₄=0; 𝑢₁ is of the doubly periodic function 𝑢₁=𝑘₁𝑐𝑛(𝑥,𝑚), where 𝑘₁ is speed feedback coefficients and 0<𝑚<1 is the modulus of Jacobi elliptic function. Therefore the controlled chaotic system (9) is rewritten as 𝑥̇=𝑎𝑦−𝑥+𝑑𝑤+𝑘₁(),̇()̇̇𝑐𝑛𝑥,𝑚𝑦=𝑥𝑐−𝑧−𝑦,𝑧=𝑦𝑥−𝑏𝑧,𝑤=−𝑥−𝑎𝑤. (25) The Jacobian matrix is 𝐽=−𝑎+𝑘₁𝑎0𝑑𝑐−100−100−𝑎. (26) The characteristic equation of 𝐽 is 𝜆⁴+1−𝑘₁+2𝑎+𝑏𝜆³+2𝑎𝑏−𝑘₁𝑏−𝑘₁𝑎+2𝑎−𝑘₁+𝑏+𝑑+𝑎²−𝑐𝑎𝜆²+−𝑐𝑎²−𝑐𝑎𝑏+2𝑎𝑏+𝑑𝑏+𝑎²𝑏−𝑘₁𝑎+𝑎²−𝑘₁𝑏−𝑘₁𝑏𝑎+𝑑𝜆−𝑐𝑎²𝑏−𝑘₁𝑏𝑎+𝑎²𝑏+𝑑𝑏=0. (27)

Proposition 3. If one chooses the control coefficients 𝑘₁=−30, 𝑚=0.3 and the parameters 𝑎=1.0, 𝑏=0.7, 𝑐=26, and 𝑑=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).

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 𝐽=−31.01.001.526−10000−0.70−1.000−1.0. (28) The characteristic equation of 𝐽 changes the following: 𝜆⁴+33.7𝜆³+61.60𝜆²+33.450𝜆+4.55=0. (29) The eigenvalues of the above equation are 𝜆₁=−31.795569,𝜆₂𝜆=−0.20443,₃=−1,𝜆₄=−0.70000. (30) 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) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [𝑥(0)=1,𝑦(0)=0.7,𝑧(0)=20,𝑤(0)=0.1]. The behaviors of the states (𝑥,𝑦,𝑧,𝑤) of the controlled chaotic Lorenz-Stenflo system (25) with time are displayed in Figures 5(a)–5(d).

(a)

(b)

(c)

(d)

Figure 5 

The states of system (25): (a) 𝑥-𝑡, (b) 𝑦-𝑡, (c) 𝑧-𝑡, and (d) 𝑤-𝑡.

4. Globally Exponential Hyperchaotic Projective Synchronization Control

Consider two chaotic systems given by 𝑥̇_𝑚=𝑓𝑥_𝑚,𝑦̇,𝑡_𝑠=𝑔𝑦_𝑠,𝑡+𝑢𝑥_𝑚,𝑦_𝑠,𝑡, (31)(32) where 𝑥_𝑚=(𝑥_1𝑚,𝑥_2𝑚,…𝑥_𝑛𝑚)^𝑇, 𝑦_𝑠=(𝑦_1𝑠,𝑦_2𝑠,…𝑦_𝑛𝑠)^𝑇, 𝑓,𝑔∈𝐶^𝑟[𝑅₊×𝑅^𝑛,𝑅^𝑛], 𝑢∈𝐶^𝑟[𝑅₊×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛], and 𝑟≥1. 𝑅₊ comprises the set of non negative real numbers. Assume that (31) is the master system, (32) is the slave system, and 𝑢(𝑥_𝑚,𝑦_𝑠,𝑡) is the control vector. Let the error state be 𝑒(𝑡)=𝑒₁(𝑡),𝑒₂(𝑡)…,𝑒_𝑛(𝑡)^𝑇=𝑥_1𝑚−𝑎₁₁𝑦_1𝑠+𝑎₁₂𝑦_1𝑠,…,𝑥_𝑛𝑚−𝑎_𝑛1𝑦_𝑛𝑠+𝑎_𝑛2𝑦_1𝑠. (33) Then the error dynamics of 𝑒(𝑡) are defined by 𝑒(̇𝑡)=𝑓𝑥_𝑚,𝑡−𝑔𝑦_𝑠,𝑡−𝑢𝑥_𝑚,𝑦_𝑠,𝑡. (34) The slave and master systems are said to be exponential, hyperchaotic, projective and synchronized if, for all𝑥_𝑚(𝑡₀), 𝑦_𝑠(𝑡₀)∈𝑅^𝑛, and 𝑖∈𝑅^𝑛, ‖𝑥_𝑚−(𝑎_𝑖1𝑦_𝑠+𝑎_𝑖2)𝑦_𝑠‖→0 as 𝑡→∞.

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 (32) are globally and exponentially projective, synchronized, if there exists a positive definite quadratic polynomial 𝑉=(𝑒₁,𝑒₂,…𝑒_𝑛)𝑃(𝑒₁,𝑒₂,…𝑒_𝑛)^𝑇 such that 𝑑𝑉/𝑑𝑡=(𝑒₁,𝑒₂,…𝑒_𝑛)𝑄(𝑒₁,𝑒₂,…𝑒_𝑛)^𝑇. Moreover, the following negative Lyapunov exponent estimation for the error dynamical system (34) holds: ^𝑛_𝑖=1𝑒²_𝑖(𝑡)≤𝜆_max(𝑃)𝜆_min(𝑃)^𝑛_𝑖=1𝑒²_𝑖(0)−𝜆exp^max(𝑄)𝜆_min(𝑃)(𝑡), (35) where 𝑃=𝑃^𝑇∈𝑅^𝑛×𝑛 and 𝑄=𝑄^𝑇∈𝑅^𝑛×𝑛 are both positive definite matrices, 𝜆_max(𝑃) and 𝜆_min(𝑃) stand for the minimum and maximum eigenvalues of the matrix 𝑃, respectively, and 𝜆_min(𝑄) denotes the minimum eigenvalue of the matrix 𝑄.

In the following, we consider the hyperchaotic system (1) as a master system: 𝑥̇_𝑚=𝑎𝑦_𝑚−𝑥_𝑚+𝑑𝑤_𝑚,𝑦̇_𝑚=𝑥_𝑚𝑐−𝑧_𝑚−𝑦_𝑚,𝑧̇_𝑚=𝑦_𝑚𝑥_𝑚−𝑏𝑧_𝑚,𝑤̇_𝑚=−𝑥_𝑚−𝑎𝑤_𝑚 (36) and the system related to (36), given by 𝑥̇_𝑠=𝑎𝑦_𝑠−𝑥_𝑠+𝑑𝑤_𝑠+𝑢₁,𝑦̇_𝑠=𝑥_𝑠𝑐−𝑧_𝑠−𝑦_𝑠+𝑢₂,𝑧̇_𝑠=𝑦_𝑠𝑥_𝑠−𝑏𝑧_𝑠+𝑢₃,𝑤̇_𝑠=−𝑥_𝑠−𝑎𝑤_𝑠+𝑢₄, (37) as a slave system, where the subscripts “𝑚” and “𝑠” stand for the master system and slave system, respectively. Let the error state be 𝑒(𝑡)=𝑒₁,𝑒₂,𝑒₃,𝑒₄^𝑇=𝑥_𝑚−𝑎₁₁𝑥_𝑠+𝑎₁₂𝑥_𝑠,𝑦_𝑚−𝑎₂₁𝑦_𝑠+𝑎₂₂𝑦_𝑠,𝑧_𝑚−𝑎₃₁𝑧_𝑠+𝑎₃₂𝑧_𝑠,𝑤_𝑚−𝑎₄₁𝑤_𝑠+𝑎₄₂𝑤_𝑠^𝑇. (38) Then the derivative of 𝑒(𝑡) along the trajectories of (36) and (37), we obtain the error system: 𝑒₁̇=𝑎𝑦_𝑚−𝑥_𝑚+𝑑𝑤_𝑚−2𝑎₁₁𝑥_𝑠×𝑎𝑦_𝑠−𝑥_𝑠+𝑏𝑤_𝑠+𝑢₁−𝑎𝑦_𝑠−𝑥_𝑠+𝑏𝑤_𝑠+𝑢₁𝑎₁₂,𝑒₂̇=𝑐−𝑧_𝑚𝑥_𝑚−𝑦_𝑚−2𝑎₂₁𝑦_𝑠×𝑐−𝑧_𝑠𝑥_𝑠−𝑦_𝑠+𝑢₂−𝑐−𝑧_𝑠𝑥_𝑠−𝑦_𝑠+𝑢₂𝑎₂₂,𝑒₃̇=−𝑏𝑧_𝑚+𝑥_𝑚𝑦_𝑚−2𝑎₃₁𝑧_𝑠×−𝑑𝑧_𝑠+𝑥_𝑠𝑦_𝑠+𝑢₃−−𝑑𝑧_𝑠+𝑥_𝑠𝑦_𝑠+𝑢₃𝑎₃₂,𝑒₄̇=−𝑥_𝑚−𝑎𝑤_𝑚−2𝑎₄₁𝑤_𝑠−𝑥_𝑠−𝑎𝑤_𝑠+𝑢₄−−𝑥_𝑠−𝑎𝑤_𝑠+𝑢₄𝑎₄₂. (39) To demonstrate the synchronization control between systems (36) and (37), we have the following cases based on [29–44].

Case  1 (modified projective synchronization control).

Theorem I. When [𝑎₁₁,𝑎₁₂,𝑎₂₁,𝑎₂₂,𝑎₃₁,𝑎₃₂,𝑎₄₁,𝑎₄₂]=[0,𝑚,0,𝑚,0,𝑚,0,𝑚], 𝑎>0 and 𝑏>0. For the hyperchaotic system (1), if one of the following families of feedback controllers 𝑢_𝑖(𝑖=1,2,3,4) is given for the slave system (36): 𝑢₁=(𝑑−𝑏)𝑚𝑤₂−𝑣₁𝑚,𝑢₂=(1−𝑚)𝑚𝑥₂𝑧₂−𝑣₂𝑚,𝑢₃=𝑚𝑥₂𝑦₂−𝑏𝑧₂+𝑑𝑧₂−𝑥₂𝑦₂𝑚−𝑣₃𝑚,𝑢₄=−𝑥₁+𝑚𝑥₂−𝑣₄𝑚, (40) there exist many possible choices for 𝑣₁, 𝑣₂, 𝑣₃, and 𝑣₄. Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential modified projective synchronization can be achieved.

The concrete proof of Theorem I can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (42) with Runge-Kutta integration method. The initial values of the drive system and response system are 𝑥₁(0)=0.1, 𝑦₁(0)=0.1, 𝑧₁(0)=20, and 𝑤₁(0)=0.1 and 𝑥₂(0)=0.2, 𝑦₂(0)=0.2, 𝑧₂(0)=21, and 𝑤₂(0)=0.2 respectively. The parameters are chosen to be 𝑚=−2, 𝑎=1.0, 𝑏=0.7, 𝑑=1.5, and 𝑐=26 so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The numerical simulation of the master and the slave systems without active synchronization control law is shown in Figures 6(a)–6(d). The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 7(a)–7(d). The synchronization errors are shown in Figures 8(a)–8(d). Figures 9(a)–9(d) depicts the projection of the synchronized attractors. Figures 10(a)–10(f) depicts the phase portraits of the synchronized attractors.

(a)

(b)

(c)

(d)

Figure 6 

The solutions of the master and slave systems without active control law. (a) Signals 𝑥1 (the dashed line) and 𝑥2 (the solid line). (b) Signals 𝑦1 (the dashed line) and 𝑦2 (the solid line). (c) Signals 𝑧1 (the dashed line) and 𝑧2 (the solid line). (d) Signals 𝑤1 (the dashed line) and 𝑤2 (the solid line).

(a)

(b)

(c)

(d)

Figure 7 

The solutions of the drive and response systems with control law. (a) Signals 𝑥1 (the dashed line) and 𝑥2 (the solid line). (b) Signals 𝑦1 (the dashed line) and 𝑦2 (the solid line). (c) Signals 𝑧1 (the dashed line) and 𝑧2 (the solid line). (d) Signals 𝑤1 (the dashed line) and 𝑤2 (the solid line).

(a)

(b)

(c)

(d)

Figure 8 

The dynamics of synchronization errors. (a) Signal 𝑒1, (b) signal 𝑒2, (c) signal 𝑒3 and (d) signal 𝑒4.

(a)

(b)

(c)

(d)

Figure 9 

The projection of the synchronized attractors in different spaces: (a) (𝑥,𝑦,𝑧), (b) (𝑥,𝑦,𝑤), (c) (𝑥,𝑤,𝑧), and (d) (𝑦,𝑤,𝑧). The solid line denotes and the drive system; the dashed line denotes respond system synchronized.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10 

Simulated phase portraits of the chaotic system (1), projected on (a) the 𝑥-𝑦 plane, (b) the 𝑥-𝑧 plane, (c) the 𝑥-𝑤 plane, (d) the 𝑦-𝑧 plane, (e) the 𝑦-𝑤 plane, and (f) the 𝑧-𝑤 plane.

Case  2 (generalized projective synchronization control).

Theorem II. When [𝑎₁₁,𝑎₁₂,𝑎₂₁,𝑎₂₂,𝑎₃₁,𝑎₃₂,𝑎₄₁,𝑎₄₂]=[0,𝑚₁,0,𝑚₂,0,𝑚₃,0,𝑚₄], 𝑚_𝑖’s are different, 𝑎>0 and 𝑏>0. For the hyperchaotic system (1), if one of the following families of feedback controllers 𝑢_𝑖(𝑖=1,2,3,4) is given for the slave system (36): 𝑢₁=𝑚₂−𝑚₁𝑎𝑦₂+𝑑𝑚₄−𝑚₁𝑏𝑤₂−𝑣₁𝑚₁,𝑢₂=𝑚₁−𝑚₂𝑐𝑥₂+𝑚₂−𝑚₁𝑚₃𝑐𝑥₂−𝑣₂𝑚₂,𝑢₃=(𝑑−𝑏)𝑚₃𝑧₂+𝑚₁𝑚₂−𝑚₃𝑐𝑥₂−𝑣₃𝑚₃,𝑢₄=−𝑥₁+𝑚₄𝑥₂−𝑣₄𝑚₄, (41) there exist many possible choices for 𝑣₁, 𝑣₂, 𝑣₃, and 𝑣₄. 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 𝑚₁=2, 𝑚₂=3, 𝑚₃=3, and 𝑚₄=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).

(a)

(b)

(c)

(d)

Figure 11 

The solutions of the master and slave systems with control law. (a) Signals 𝑥1 (the dashed line) and 𝑥2 (the solid line). (b) Signals 𝑦1 (the dashed line) and 𝑦2 (the solid line). (c) Signals 𝑧1 (the dashed line) and 𝑧2 (the solid line). (d) Signals 𝑤1 (the dashed line) and 𝑤2 (the solid line).

(a)

(b)

(c)

(d)

Figure 12 

The dynamics of synchronization errors. (a) Signal 𝑒1, (b) signal 𝑒2, (c) signal 𝑒3, and (d) signal 𝑒4.

Case  3 (function synchronization control [45–58]).

Theorem III. When 𝑓₁=𝑎₁₁𝑥₂+𝑎₁₂, 𝑓₂=𝑎₂₁𝑦₂+𝑎₂₂, 𝑓₃=𝑎₃₁𝑧₂+𝑎₃₂, 𝑓₄=𝑎₄₁𝑤₂+𝑎₄₂, 𝑎_𝑖1≠0, 𝑎>0, 𝑏>0. For the hyperchaotic system (1), if one of the following families of feedback controllers 𝑢_𝑖(𝑖=1,2,3,4) is given for the slave system (36): 𝑢₁=𝑓₂−𝑓₁𝑎𝑦₂+𝑑𝑓₄−𝑓₁𝑏𝑤₂−𝑘₁𝑒₁𝑓₁,𝑢₂=𝑓₁−𝑓₂𝑐𝑥₂+𝑓₂−𝑓₁𝑓₃𝑐𝑥₂−𝑘₂𝑒₂𝑓₂,𝑢₃=(𝑑−𝑏)𝑓₃𝑧₂+𝑓₁𝑓₂−𝑓₃𝑐𝑥₂−𝑘₃𝑒₃𝑓₃,𝑢₄=−𝑥₁+𝑓₄𝑥₂−𝑘₄𝑒₄𝑓₄, (42) where 𝑘₁>0, 𝑘₂>0, 𝑘₃>0, and 𝑘₄>0, then the zero solution of the error dynamical system (39) is globally stable, and thus global function, projective synchronization occurs between the master systems (36) and (37).

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 of the master and the slave systems with active control law is presented in Figures 13(a)–13(d). The synchronization errors are shown in Figures 14(a)–14(d).

(a)

(b)

(c)

(d)

Figure 13 

The function projection of the synchronized attractors in different spaces: (a) (𝑥,𝑦,𝑧), (b) (𝑥,𝑦,𝑤), (c) (𝑥,𝑤,𝑧), and (d) (𝑦,𝑤,𝑧). The solid line denotes the drive system; the dashed line denotes respond system synchronized.

(a)

(b)

(c)

(d)

Figure 14 

The dynamics of synchronization errors. (a) Signal 𝑒1, (b) signal 𝑒2, (c) signal 𝑒3 and (d) signal 𝑒4.

5. 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.

Appendices

A. The Maple Program

 >restart: with(student): with(PDEtools): with(linalg): with (LinearAlgebra): >𝐽∶=𝑚𝑎𝑡𝑟𝑖𝑥(4,4,[[𝐴,𝑎,0,𝑑],[𝑐,𝐵,0,0],[0,0,𝐶,0],[−1,0,0,𝐷]]); >𝑈∶=𝑐ℎ𝑎𝑟𝑚𝑎𝑡(𝐽,𝜆); >𝑝𝑜𝑙𝑦1∶=𝑐𝑜𝑙𝑙𝑒𝑐𝑡(𝑑𝑒𝑡(𝑈),𝜆); >𝑎1∶=𝑐𝑜𝑒𝑓𝑓(𝑝𝑜𝑙𝑦1,𝜆,3); >𝑎2∶=𝑐𝑜𝑒𝑓𝑓(𝑝𝑜𝑙𝑦1,𝜆,2); >𝑎3∶=𝑐𝑜𝑒𝑓𝑓(𝑝𝑜𝑙𝑦1,𝜆,1); >𝑎4∶=𝑐𝑜𝑒𝑓𝑓(𝑝𝑜𝑙𝑦1,𝜆,0); >𝐻∶=𝑚𝑎𝑡𝑟𝑖𝑥(4,4,[𝑎1,𝑎3,0,0,1,𝑎2,𝑎4,0,0,𝑎1,𝑎3,0,0,1,𝑎2,𝑎4]); >𝐻11∶=𝑠𝑢𝑏𝑚𝑎𝑡𝑟𝑖𝑥(𝐻,1..1,1..1); >𝐻22∶=𝑠𝑢𝑏𝑚𝑎𝑡𝑟𝑖𝑥(𝐻,1..2,1..2); >𝐻33∶=𝑠𝑢𝑏𝑚𝑎𝑡𝑟𝑖𝑥(𝐻,1..3,1..3); >𝐻44∶=𝑠𝑢𝑏𝑚𝑎𝑡𝑟𝑖𝑥(𝐻,1..4,1..4); >𝐻1∶=𝑑𝑒𝑡(𝐻11); >𝐻2∶=𝑑𝑒𝑡(𝐻22); >𝐻3∶=𝑑𝑒𝑡(𝐻33);  >𝐻4∶=𝑑𝑒𝑡(𝐻44).

B. The Proof of Theorem

Proof of Theorem I. Consider the controller (40) and choose the following 𝑣_𝑖: 𝑣₁=−𝑎𝑒₂−𝑑𝑒₄,𝑣₂=−𝑐𝑒₁+𝑒₁𝑒₃+𝑒₁𝑚𝑧₂+𝑚𝑥₂𝑒₃,𝑣₃=−𝑒₂𝑒₁−𝑒₂𝑚𝑥₂−𝑚𝑦₂𝑒₁,𝑣₄=(𝑎−1)𝑒₄. (B.1) Then the system (39) is reduced into 𝑒₁̇=−𝑎𝑒₁,𝑒₂̇=−𝑒₂,𝑒₃̇=−𝑏𝑒₃,𝑒₄̇=−𝑒₄. (B.2) Let us consider the Lyapunov function for the system (B.2) as follows: 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄=12𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄. (B.3) In addition, the derivative of 𝑉 has the form 𝑑𝑉(𝑡)𝑑𝑡=−𝑎𝑒²₁+𝑒²₂+𝑏𝑒²₃+𝑒²₄, (B.4) which is negatively defined. So (B.2) is asymptotically stable. This implies that the two Lorenz-Stenflo hyperchaotic systems are projective and synchronized.

Proof of Theorem II. Consider the controller (41) and choose 𝑣_𝑖 as follows: 𝑣₁=−𝑎𝑒₂−𝑑𝑒₄,𝑣₂=−𝑐𝑒₁+𝑒₁𝑒₃+𝑒₁𝑚₃𝑧₂+𝑚₁𝑥₂𝑒₃,𝑣₃=−𝑒₂𝑒₁−𝑒₂𝑚₁𝑥₂−𝑚₂𝑦₂𝑒₁,𝑣₄=(𝑎−1)𝑒₄. (B.5) Then the following steps are the same (B.2), (B.3), and (B.4). So we know that the two Lorenz-Stenflo hyperchaotic systems are modified, projective, and synchronized.

Proof of Theorem III. Consider the controller (42) and choose the following positive definite, quadratic form of Lyapunov function: 𝑉(𝑡)=12𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄. (B.6) We differentiate 𝑉(𝑡) and substitute the trajectory of system (39) which yields 𝑑𝑉(𝑡)𝑑𝑡₍₁₈₎=𝑒₁𝑒₁̇+𝑒₂𝑒₂̇+𝑒₃𝑒₃̇=−𝑘₁𝑒²₁−𝑘₂𝑒²₂−𝑘₃𝑒²₃−𝑘₄𝑒²₄=−𝑒₁,𝑒₂,𝑒₃,𝑒₄𝑃𝑒₁,𝑒₂,𝑒₃,𝑒₄^𝑇, (B.7) where 𝑃=diag(𝑘₁,𝑘₂,𝑘₃,𝑘₄), 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.

Acknowledgments

This work is supported by the NSF of China (no. 11171355), the Ph.D. Programs Foundation of Ministry of Education of China (no. 20100171110040), and Guangdong Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University. The authors thank the handling editor and the reviewers for their valuable comments and suggestions, which improved the completeness of the paper.