Chaos, Solitons and Fractals 26 (2005) 353–361 www.elsevier.com/locate/chaos
Controlling and tracking hyperchaotic Ro¨ssler system via active backstepping design q Hao Zhang *, Xi-kui Ma, Ming Li, Jian-long Zou Group of Theory and New Technology of Electrical Engineering, School of Electrical Engineering, Xi’an Jiaotong University, Xian 710049, PR China Accepted 20 December 2004
Abstract This paper presents a novel active backstepping control approach for controlling hyperchaotic Ro¨ssler system to a steady state as well as tracking of any desire trajectory to be achieved in a systematic way. The proposed method is a systematic design approach and consists in a recursive procedure that interlaces the choice of a Lyapunov function with the design of active control. Numerical results show that the controller is singularity free and the closed-loop system is stable globally. Especially, the main feature of this technique is that it gives the flexibility to construct a control law. Finally, numerical experiments verify the feasibility and effectiveness of the proposed control technique. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work by Ott, Grebogi and York (OGY) [1], chaos control has received increasing attention and has become a very active topic in nonlinear sciences. Various effective methods [2–8,17–26] have been proposed over the last decade to achieve the control and stabilization of chaotic systems. Furthermore, some of these methods have been successfully applied to experimental systems such as power electronics, laser, electromagnetic compatibility (EMC) and so on [5,6,8]. However, most of these aforementioned methods are only suitable for some low dimensional chaotic systems with only one positive Lyapunov exponent. Recently, specialists from nonlinear science turned their attention to the study of controlling hyperchaotic systems [9–14]. Colet et al. [9] used a single parameter feedback method to stabilize remnants of periodic orbits in a hyperchaotic laser system. Based on the OGY method, Yang et al. [10] proposed a remarkable YLM control method to regulate the hyperchaotic behavior of discrete-time systems to their inherent unstable fixed points. Bu [11] and Oliveira and Rosolen [12] proposed the YLM improved method to successfully control the hyperchaotic dynamics to fixed points. Hsieh et al. [13] presented a feedback controller for the stabilization of hyperchaotic Ro¨ssler system to periodic motion. More recently, Jiang et al. [14] also used sliding mode control method to suppress
q *
Project supported by the Doctorate Foundation of Xian Jiaotong University, PR China (Grant No. DFXJTU2003-7). Corresponding author. Tel.: +86 29 826 56905. E-mail address:
[email protected] (H. Zhang).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.12.032
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hyperchaos in Ro¨ssler system. Due to these developments, hyperchaos control as well as chaos control has attracted revived interests in the nonlinear control community, and has become a subject of much on-going research. In recent years, backstepping design [17–21] and active control [22–26] have been widely recognized as two powerful design methods to control chaos. Backstepping design can guarantee global stabilities, tracking, and transient performance for a broad class of strict-feedback nonlinear systems [15,16]. Active control technique gives the flexibility to construct a control law so that it can be used widely to control various nonlinear systems including chaotic systems. Consequently, the main aim of this paper is in an attempt to use a combination of the two control approaches, i.e., active backstepping method, to control hyperchaotic systems. In this paper, we propose a novel active controller to suppress hyperchaos in Ro¨ssler system, based on the backstepping design. The proposed approach is a systematic design approach and consists in a recursive procedure that skillfully interlaces the choice of a Lyapunov function with the design of active control. Besides, the control scheme can also be used to track any desired trajectories in a systematic way. Finally, the effectiveness and feasibility of the proposed control technique are numerically verified. The paper is organized as follows: In Section 2, a brief description of the hyperchaotic Ro¨ssler system is introduced. Sections 3 and 4 discuss the design of an active backstepping controller for suppressing hyperchaotic motion as well as tracking of any desire trajectory. Numerical simulation is given for illustration and verification in Section 5. Finally, some concluding remarks and comments are given.
2. Problem description In 1976, Ro¨ssler [27] first introduced Ro¨ssler system, which is a simple and prototypical equation with chaos similar to the Lorenz model of turbulence that contains just one (second order) nonlinearity in one variable. Due to its simplicity, the Ro¨ssler system has become a standard chaotic system to verify the effectiveness of the chaos control strategy [7]. In 1979, Ro¨ssler [28] proposed further a four-variable oscillator, i.e., hyperchaotic Ro¨ssler system, which contains only one nonlinear term of quadratic type and produce chaos with two directions of hyperbolic instability on the strange attractor. The hyperchaotic Ro¨ssler system [28] can be described as follows: 8 x_ 1 ¼ bx2 þ cx1 > > > > > < x_ 2 ¼ 3 þ x2 x3 ð1Þ > > > x_ 3 ¼ x2 x4 > > : x_ 4 ¼ x1 þ x3 þ ax4 where a = 0.25, b = 0.5 and c = 0.05. This system exhibits a hyperchaotic behavior starting from proper initial conditions [28]. (See Fig. 1.) Numerical evidence indicates that the hyperchaotic attractor has two positive Lyapunov exponents, i.e., k1 = 0.109 and k2 = 0.024. In this paper we will expect to design an appropriate active controller to drive the hyperchaotic attractor to a stable state, i.e., unstable periodic orbit or equilibrium point. In fact, it is very useful for actual engineering. According to the active control theory [22–26], the controlled hyperchaotic system can be written in the form 8 x_ 1 ¼ bx2 þ cx1 þ u1 > > > < x_ 2 ¼ 3 þ x2 x3 þ u2 ð2Þ > > > x_ 3 ¼ x2 x4 þ u3 : x_ 4 ¼ x1 þ x3 þ ax4 þ u4 where ui, i = 1, 2, 3, 4 is a control input. For brevity, x = (x1, x2, x3, x4)T denotes the state vector of the controlled system and u(t) = (u1, u2, u3, u4)T is the control input vector. In practical applications, the controller to be designed must be simple, efficient and easy to implement. Thus, let u1 = 0, then the controlled dynamics can be rewritten as 8 x_ 1 ¼ bx2 þ cx1 > > > < x_ ¼ 3 þ x x þ u 2 2 3 2 ð3Þ > _ x ¼ x x þ u 3 2 4 3 > > : x_ 4 ¼ x1 þ x3 þ ax4 þ u4
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Fig. 1. Typical phase portraits of hyperchaotic Ro¨ssler system.
3. Controlling hyperchaos in Ro¨ssler system In the following we will use backstepping design method to design an active controller for suppressing hyperchaos in Ro¨ssler system. Let the target (unstable) periodic orbit of the Ro¨ssler system be ~x ¼ ð~x1 ; ~x2 ; ~x3 ; ~x4 ÞT , which is itself a solution of the system (1), namely, it satisfies 8 > ~x_ 1 ¼ b~x2 þ c~x1 > > <_ ~x2 ¼ 3 þ ~x2~x3 ð4Þ > ~x_ 3 ¼ ~x2 ~x4 > > :_ ~x4 ¼ ~x1 þ ~x3 þ a~x4 Thus, an important problem of controlling chaos under investigation here is to select an appropriate control function u(t) such that the trajectory of the controlled system (3) asymptotically approaches the target periodic orbit ~x, in the sense that ð5Þ limit kx ~xk ¼ 0 t!1
where kÆk is the Euclidean norm.
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A subtraction of Eq. (4) from Eq. (3) now gives 8 e_ ¼ be2 þ ce1 > > > 1 < e_ 2 ¼ e2 e3 þ e2~x3 þ ~x2 e3 þ u2 > e_ 3 ¼ e2 e4 þ u3 > > : e_ 4 ¼ e1 þ e3 þ ae4 þ u4
ð6Þ
where e1 ¼ x1 ~x1 , e2 ¼ x2 ~x2 , e3 ¼ x3 ~x3 and e4 ¼ x4 ~x4 . Eq. (6) describes the error dynamics and can be considered in terms of a control problem where the system to be controlled is a nonlinear system with a control input u(t). As long as the control input can stabilize the system, the error vector e = (e1, e2, e3, e4)T converges to zero as time t goes to infinity. This implies that the trajectory of the controlled system (3) asymptotically approaches the target periodic orbit ~x. The backstepping design procedure is recursive. At the ith step, the ith-order subsystem is stabilized with respect to a Lyapunov function Vi by the design of a virtual control ai and a control input function ui. Now we begin to design the active controller based on the backstepping design method, as follows: Step 1. Let z1 = e1, then we can obtain its derivative z_ 1 ¼ e_ 1 ¼ be2 þ cz1
ð7Þ
where e2 = a1(z1) is regarded as an virtual controller. For the design of a1(z1) to stabilize z1-subsystem (7), we can choose the following Lyapunov function: 1 V 1 ¼ z21 2
ð8Þ
The derivative of V1 is V_ 1 ¼ z1 z_ 1 ¼ z1 ½ba1 ðz1 Þ þ cz1
ð9Þ
If we choose a1 ðz1 Þ ¼ V_ 1 is negative definite. This implies that the z1-subsystem (7) is asymptotically stable. Since the virtual control function a1(z1) is estimative, the error between e2 and a1(z1) is cþ1 z, b 1
z2 ¼ e2 a1 ðz1 Þ We can obtain the following (z1, z2)-subsystem: z_ 1 ¼ bz2 z1 ~x3 z1 þ ðc þ 1Þz2 cþ1 z_ 2 ¼ z2 cþ1 z þ ~x2 e3 þ z2~x3 cþ1 z þ u2 b 1 b b 1
ð10Þ
ð11Þ
where e3 = a2(z1, z2) is regarded as an virtual controller. Step 2. In order to stabilize the (z1, z2)-subsystem (11), we can choose a Lyapunov function V2 defined by 1 V 2 ¼ V 1 þ z22 2
ð12Þ
Its derivative is given by V_ 2 ¼ V_ 1 þ z2 z_ 2 ¼ z21 þ bz1 z2 þ z2 z_ 2
cþ1 cþ1 cþ1 ~x3 z1 þ u2 z1 þ ~x2 a2 ðz1 ; z2 Þ þ z2 bz1 þ ~x3 z2 þ ðc þ 1Þz2 z1 ¼ z21 þ z2 z1 b b b
ð13Þ
~x3 Þz1 ðc þ 2 þ ~x3 Þz2 , then V_ 2 ¼ z21 z22 < 0 makes (z1, z2)-subsystem (11) If a2(z1, z2) = 0 and u2 ¼ ðb cþ1 cþ1 b b asymptotically stable. Similarly, assume that z3 = e3 a2(z1, z2), then we can yield the following (z1, z2, z3)-subsystem: 8 < z_ 1 ¼ bz2 z1 z_ 2 ¼ ðz2 cþ1 z þ ~x2 Þz3 z2 bz1 ð14Þ b 1 : z_ 3 ¼ z2 þ cþ1 z e þ u 1 4 3 b where e4 = a3(z1, z2, z3) is regarded as an virtual controller. Step 3. We can choose a Lyapunov function V3 defined by 1 V 3 ¼ V 2 þ z23 2 in order to make the (z1, z2, z3)-subsystem (14) stable.
ð15Þ
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The derivative of V3 is
cþ1 cþ1 V_ 3 ¼ V_ 2 þ z3 z_ 3 ¼ z21 z22 z3 a3 ðz1 ; z2 ; z3 Þ þ z3 z1 z2 þ ~x2 z2 z2 þ z1 z1 z2 þ u3 b b
ð16Þ
If we choose a3(z1, z2, z3) = 0 and u3 ¼ ð1 cþ1 Þz1 z2 ð~x2 1Þz2 cþ1 z z3 , V_ 3 ¼ z21 z22 z23 < 0 makes b b 1 (z1, z2, z3)-subsystem (14) asymptotically stable. Let z4 = e4 a3(z1, z2, z3), then we can have the following (z1, z2, z3, z4)-system: 8 z_ 1 ¼ bz2 z1 > > > > > < z_ 2 ¼ z2 cþ1 z1 þ ~x2 z3 z2 bz1 b ð17Þ > z_ 3 ¼ ~x2 z2 1 cþ1z1 z2 z4 z3 > > b > > : z_ 4 ¼ z1 þ z3 þ az4 þ u4 Step 4. In order to stabilize the (z1, z2, z3, z4)-system (17), we can choose a Lyapunov function V4 defined by 1 V 4 ¼ V 3 þ z24 2
ð18Þ
The derivative of V4 is expressed as V_ 4 ¼ V_ 3 þ z4 z_ 4 ¼ z21 z22 z23 þ z4 ðaz4 þ z1 þ u4 Þ If u4 = z1 (a + 1)z4, then V_ 3 ¼ Finally, the fill dimension (z1, z2, z3, z4)-system is 8 z_ 1 ¼ bz2 z1 > > > > > < z_ 2 ¼ z2 cþ1 z1 þ ~x2 z3 z2 bz1 b > > z1 z2 z4 z3 z_ 3 ¼ ~x2 z2 1 cþ1 > b > > : z_ 4 ¼ z3 z4 z21
z22
z23
z24
ð19Þ < 0 makes (z1, z2, z3, z4)-system (17) asymptotically stable.
ð20Þ
Since V_ 4 is negative definite, it follows from LaSalle–Yoshizawa theorem [16] that in the (z1, z2, z3, z4) coordinates the equilibrium (0, 0, 0, 0) is global asymptotically stable. In view of z1 = e1, z2 ¼ e2 a1 ðz1 Þ ¼ e2 þ cþ1 z; b 1 z3 ¼ e3 a2 ðz1 ; z2 Þ ¼ e3 , and z4 = e4 a3(z1, z2, z3) = e4, this implies that e1, e2, e3 and e4 go to zeros asymptotically. In other words, the trajectory of the controlled system (3) asymptotically approaches the target periodic orbit ~x.
4. Tracking any desired trajectory In this section we will discuss how to choose a control law u(t) so that we can track any desired trajectory yr(t) with a scalar output y = x1. In the following, the active controller will be designed in terms of the backstepping design method, as follows: Step 1. Assume that z1 = y yr = x1 yr, then we can obtain its derivative z_ 1 ¼ bx2 þ cðz1 þ y r Þ y_ r
ð21Þ
where x2 = a1(z1) is regarded as an virtual controller. For the design of a1(z1) to stabilize z1-subsystem (21), we can choose the following Lyapunov function: 1 V 1 ¼ z21 2
ð22Þ
The derivative of V1 is V_ 1 ¼ z1 z_ 1 ¼ z1 ½ba1 ðz1 Þ þ cz1 þ cy r y_ r
ð23Þ
Here, we choose a1 ðz1 Þ ¼ 1b ½ðc þ 1Þz1 þ cy r y_ r such that V_ 1 is negative definite. This implies that the z1-subsystem (21) is asymptotically stable. Since the virtual control function a1(z1) is estimative, the error between x2 and a1(z1) is z2 ¼ x2 a1 ðz1 Þ
ð24Þ
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We can obtain the following (z1, z2)-subsystem: ( z_ 1 ¼ bz2 z1 z_ 2 ¼ 3 þ z2 x3 1b ½ðc þ 1Þz1 þ cy r y_ r x3 þ cþ1 ðbz2 z1 Þ þ 1b ðc_y r €y r Þ þ u2 b
ð25Þ
where x3 = a2(z1, z2) is regarded as an virtual controller. Step 2. In order to stabilize the (z1, z2)-subsystem (25), we can choose a Lyapunov function V2 defined by 1 V 2 ¼ V 1 þ z22 2
ð26Þ
Its derivative is given by V_ 2 ¼ V_ 1 þ z2 z_ 2
1 cþ1 c 1 ðbz2 z1 Þ þ y_ r €y r bz1 þ u2 ¼ z21 þ z2 z2 ½ðc þ 1Þz1 þ cy r y_ r a2 ðz1 ; z2 Þ þ z2 3 þ b b b b
ð27Þ
ðbz2 z1 Þ bc y_ r þ 1b €y r þ bz1 z2 , thenV_ 2 ¼ z21 z22 < 0 makes (z1, z2)-subsystem If a2(z1, z2) = 0 and u2 ¼ 3 cþ1 b (25) asymptotically stable. Similarly, assume that z3 = x3 a2(z1, z2), then we can obtain the following (z1, z2, z3)-subsystem 8 z_ ¼ bz2 z1 > < 1 z_ 2 ¼ z2 z3 1b ½ðc þ 1Þz1 þ cy r y_ r z3 cþ1 ðbz2 z1 Þ bc y_ r þ 1b €y r þ bz1 z2 ð28Þ b > : 1 z_ 3 ¼ z2 þ b ½ðc þ 1Þz1 þ cy r y_ r x4 þ u3 where x4 = a3(z1, z2, z3) is regarded as an virtual controller. Step 3. In order to stabilize the (z1, z2, z3)-subsystem (28), we can choose a Lyapunov function V3 defined by 1 V 3 ¼ V 2 þ z23 2
ð29Þ
Its derivative is given by
1 z2 2 2 2 _ _ V 3 ¼ V 2 þ z3 z_ 3 ¼ z1 z2 z3 a4 ðz1 ; z2 ; z3 Þ þ z3 z2 z2 þ ½ðc þ 1Þz1 þ cy r y_ r þ u3 b
ð30Þ
2 If a3(z1, z2, z3) = 0 and u3 ¼ z22 þ z2 1z ½ðc þ 1Þz1 þ cy r y_ r z3 , then V_ 3 ¼ z21 z22 z23 < 0 makes (z1, z2, z3)b subsystem (28) asymptotically stable. Similarly, assume that z4 = x4 a3(z1, z2, z3), then we can obtain the following (z1, z2, z3, z4)-system: 8 z_ 1 ¼ bz2 z1 > > > > < z_ 2 ¼ z2 z3 1 ½ðc þ 1Þz1 þ cy y_ z3 cþ1 ðbz2 z1 Þ c y_ þ 1 €y þ bz1 z2 r r b b b r b r ð31Þ z2 2 > _ z _ ¼ z z z z z þ ½ðc þ 1Þz þ cy y
> 3 1 2 3 4 1 r r 2 b > > : z_ 4 ¼ z3 þ az4 þ z1 þ u4
Step 4. In order to stabilize the (z1, z2, z3, z4)-system (31), we can choose a Lyapunov function V4 defined by 1 V 4 ¼ V 3 þ z24 2
ð32Þ
Its derivative is given by V_ 4 ¼ V_ 3 þ z4 z_ 4 ¼ z21 z22 z23 þ z4 ðaz4 þ z1 þ u4 Þ If u4 = z1 (a + 1)z4, then V_ 3 ¼ z21 z22 z23 z24 < 0 makes (z1, z2, z3, z4)-system (31) asymptotically stable. Finally, we can obtain the fill dimension (z1, z2, z3, z4)-system 8 z_ 1 ¼ bz2 z1 > > > > < z_ 2 ¼ z2 z3 1 ½ðc þ 1Þz1 þ cy y_ z3 cþ1 ðbz2 z1 Þ c y_ þ 1 €y þ bz1 z2 r r b b b r b r z2 2 > z_ 3 ¼ z1 z2 z3 z4 z2 þ b ½ðc þ 1Þz1 þ cy r y_ r
> > > : z_ 4 ¼ z3 z4
ð33Þ
ð34Þ
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Likewise, since V_ 4 is negative definite, it follows from LaSalle–Yoshizawa theorem [16] that in the (z1, z2, z3, z4) coordinates the equilibrium (0, 0, 0, 0) is global asymptotically stable. In view of z1 = y yr, z2 ¼ x2 a1 ðz1 Þ ¼ x2 þ 1b ½ðc þ 1Þz1 þ cy r y_ r , z3 = x3 a2(z1, z2) = x3, and z4 = x4 a3(z1, z2, z3) = x4, this implies that the output trajectory of the controlled system (3) asymptotically approaches the target trajectory yr.
5. Numerical experiments We have introduced the novel active backstepping control approach for the control and tracking problems of hyperchaotic Ro¨ssler system. In what follows, numerical experiments are given to verify the effectiveness of the control approach. Case 1 (Controlling hyperchaotic behavior to a stable state). It is well known that two equilibrium points of the pffiffiffiffi pffiffiffiffi ffi ; p2ffiffiffiffi ; 3 213 ; p2ffiffiffiffi and p20ffiffiffiffi ; p2ffiffiffiffi ; 3 213 ; p2ffiffiffiffi , respectively. Thus, our purpose hyperchaotic Ro¨ssler system are p20ffiffiffi 13 13 13 13 13 13 is to design an active backstepping controller to make trajectory of the controlled chaotic system asymptotically pffiffiffiffi ffi ; p2ffiffiffiffi ; 3 213 ; p2ffiffiffiffi . approach the desired equilibrium point ð~x1 ; ~x2 ; ~x3 ; ~x4 Þ ¼ p20ffiffiffi 13 13 13
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Fig. 2. Controlling hyperchaotic Ro¨ssler system using active backstepping design: (a) time waveform of the state x1, (b) time waveform of the state x2, (c) time waveform of the state x3 and (d) time waveform of the state x4.
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According to the foregoing design method, we can obtain the control functions 8 pffiffiffiffi pffiffiffiffi 13 > u2 ¼ c þ 2 þ 3 213 z2 b cþ1 3ðcþ1Þ z1 > 2b b > < ffi þ 1 z2 cþ1 u3 ¼ 1 cþ1 z z3 z1 z2 þ p2ffiffiffi > b b 1 13 > > : u4 ¼ z1 ða þ 1Þz4
ð35Þ
to make the controlled system (3) asymptotically stable. The numerical experiment is carried out with the initial condition (15, 10, 20, 10) and the four-order Runge–Kutta algorithm with step size of 0.001. After 50 s, the motion trajectories have entered into the hyperchaotic attractor. From then on we control the hyperchaotic system by applying the above active controller u(x). The simulation results are shown in Figs. 2 and 3. Fig. 2(a)–(d) displays the time waveforms of all the system state variables before and after the controller is activated. Fig. 3 depicts the time waveforms of the error e. From Figs. 2 and 3, one can seepffiffiffithat as ffi ffi ; p2ffiffiffiffi ; 3 213 ; p2ffiffiffiffiÞ. the control is activated, the system orbit eventually converges to the desired equilibrium point ð p20ffiffiffi 13 13 13
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Fig. 3. When controlling Ro¨ssler system via active backstepping design, time waveforms of the error variables, e1 (solid), e2 (dashed), e3 (dotted) and e4 (dashdot).
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Fig. 4. Tracking yr = sin t using active backstepping design: (a) time waveform of the output y = x1 of hyperchaotic Ro¨ssler system and (b) time waveform of the error y yr.
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Case 2 (Tracking a desired sine waveform). As s second example, we will design a control law so that a scalar output y = x1 can track a desired sine waveform yr = sin t. Using the above design technique, we can obtain the control functions 8 cþ1 c 1 > < u2 ¼ 3 b ðbz2 z1 Þ b cos t b sin t þ bz1 z2 2 ð36Þ u3 ¼ z22 þ z2 1z ½ðc þ 1Þz1 þ c sin t cos t z3 b > : u4 ¼ z1 ða þ 1Þz4 to make the output trajectory of the controlled system (3) asymptotically approach the target trajectory yr. Similarly, let the initial conditions be (15, 6, 20, 5). After 50 s, the motion trajectories have entered into the hyperchaotic attractor. From then on the active controller is activated. The numerical results are illustrated in Fig. 4. As expected, one can observe that the output trajectory y asymptotically approach the target trajectory yr in Fig. 4(a) and (b).
6. Conclusions In this paper, an effective control method for controlling hyperchaotic Ro¨ssler system has been proposed using active backstepping design. The proposed control approach enables stabilization of chaotic motion to a steady state as well as tracking of any desire trajectory to be achieved in a systematic way. A comparison with the aforementioned methods clearly highlights the advantages of the proposed approach, as follows: (i) The design method can overcome the controller singularity problem resulting from the nonlinear term of quadratic type [19–21]. (ii) The proposed approach provides a systematic design approach for chaos control and synchronization, and guarantees the global stabilities of the closed-loop system. (iii) Especially, the main feature of this approach is that it gives the flexibility to construct a control law so that the control strategy can be easily extended to other higher dimensional hyperchaotic systems.
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