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Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system
Nonlinear Analysis 71 (2009) 2441–2446
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system Congxu Zhu School of Information Science and Engineering, Central South University, Changsha, 410083, China
article
info
Article history: Received 17 January 2008 Accepted 15 January 2009 Keywords: Chaotic system Stabilization of unstable equilibrium points Dislocated feedback control Enhancing feedback control Speed feedback control
abstract This paper studies the stabilization of unstable equilibrium points in a new chaotic system. Three different methods, the dislocated feedback control method, the enhancing feedback control method and the speed feedback control method, are used to stabilize unstable equilibrium points in the new chaotic system. On the basis of the Routh–Hurwitz theorem and the linearization model of a system at the adjacent equilibrium point, the conditions of stabilization are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the three different methods. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Dynamic chaos is a very interesting non-linear effect, which has been intensively studied since Lorenz [1] found the first canonical chaotic attractor in 1963. The effect is very common; it has been detected in a large number of dynamic systems of various physical natures. However, this effect is usually undesirable in practice, and it restricts the operating range of many electronic and mechanical devices. Recently, control of this kind of complex dynamical system has attracted a great deal of attention within the engineering community. Research efforts have investigated the chaos control in many physical chaotic systems [2–11]. Different control strategies for stabilizing chaos have been proposed, such as adaptive control, time delay control, and fuzzy control. Generally speaking, there are two main approaches for controlling chaos: feedback control and non-feedback control. The feedback control approach offers many advantages such as robustness and computational complexity over the non-feedback control method. Very recently, Cai et al. presented a new chaotic dynamical system [12]. It is a three-dimensional autonomous system described by
x˙ = a(y − x) y˙ = bx + cy − xz z˙ = x2 − hz
(1)
where a, b, c and h are system parameters. System (1) has a chaotic attractor as shown in Fig.√ 1 when a =√20, b = 14, c = 10.6, h = √ 2.8. It is obvious √ that system (1) has three unstable equilibrium points O1 (0, 0, 0), O2 ( h(b + c ), h(b + c ), b+c ) and O3 (− h(b + c ), − h(b + c ), b + c ). The aim of this article is to discuss three novel feedback control strategies for the new chaotic system (1). The rest of the paper is organized as follows. Section 2 presents the dislocated feedback control strategy; the enhancing feedback control strategy and the speed feedback control strategy are presented in Sections 3 and 4, respectively. Section 5 concludes the paper.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.127
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C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
Fig. 1. The new chaotic attractor.
Fig. 2. The state of system (2) converging to O1 (0, 0, 0) when k = 25, and x(0) = 20, y(0) = 20, z (0) = 20.
2. Dislocated feedback control strategies For the ordinary feedback control, the system variable is often multiplied by a coefficient as the feedback gain is added to the right-hand side of the corresponding equation. If such a variable is added to the right-hand side of another equation, then this method is called dislocated feedback control. Theorem 1. Let the controlled new chaotic system be
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − kx z˙ = x2 − 2.8z
(2)
where k is the feedback coefficient; when k > 24.6, system (2) will gradually converge to the unstable equilibrium point O1 (0, 0, 0). Proof. The Jacobi matrix of system (2) as regards the unstable equilibrium point O1 (0, 0, 0) is
−20 J = 14 − k 0
20 10.6 0
0 0 −2.8
!
and then the characteristic equation is
λ3 + c1 λ2 + c2 λ + c3 = 0 where c1 = 12.2,
c2 = 20k − 465.68,
c3 = 56k − 1377.6.
Obviously, when k > 24.6, c1 > 0, c2 > 0, c3 > 0 and c1 c2 − c3 > 0. According to the Routh–Hurwitz theorem, system (2) will gradually converge to the unstable equilibrium point O1 (0, 0, 0). The proof is thus completed (see Fig. 2).
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
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Fig. 3. The state of system (3) converging to O2 (8.2994, 8.2994, 24.6) when k = 16, and x(0) = 20, y(0) = 20, z (0) = 20.
Furthermore, if the controlled new chaotic system is
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − k(x − α) z˙ = x2 − 2.8z
(3)
when α = 8.2994, k > 13.3729, system (3) will gradually converge to another unstable equilibrium point O2 (8.2994, 8.2994, 24.6) (see Fig. 3). For demonstrating this conclusion, we do the following transformations: x1 = x − α , y1 = y − α , z1 = z − β . If α = 8.2994, β = 24.6, then system (3) has the following form:
x˙ 1 = 20(y1 − x1 ) y˙ 1 = (−10.6 − k)x1 + 10.6y1 − x1 z1 − 8.2994z1 z˙ = x2 + 16.5988x − 2.8z 1 1 1 1
(30 )
when k > 13.3729; one can easily prove that system (30 ) will gradually converge to unstable equilibrium point O1 (0, 0, 0), that is system (3) will gradually converge to the unstable equilibrium point O2 (8.2994, 8.2994, 24.6). Similarly, if α = −8.2994, β = 24.6, system (3) will gradually converge to another unstable equilibrium point O3 (−8.2994, −8.2994, 24.6). In addition, if the controlled new chaotic system is
x˙ = 20(y − x) − ky y˙ = 14x + 10.6y − xz z˙ = x2 − 2.8z
(4)
when k > 4.8571, then system (4) will gradually converge to unstable equilibrium point O1 (0, 0, 0). 3. Enhancing feedback control strategies It is difficult to control a complex system by means of only one feedback variable, and in such cases the feedback gain is always very large. So we consider using multiple variables multiplied by a proper coefficient as the feedback gain. This method is called enhancing feedback control. Theorem 2. For the particular structure of the new chaotic system, the control method is considered as follows:
x˙ = 20(y − x) − kx y˙ = 14x + 10.6y − xz − ky z˙ = x2 − 2.8z
(5)
when k > 17.9736; system (5) will converge to the unstable equilibrium point O1 (0, 0, 0). Proof. The Jacobi matrix of system (5) as regards the unstable equilibrium point O1 (0, 0, 0) is
−20 − k J =
14 0
20 10.6 − k 0
0 0 −2.8
and then the characteristic equation is
λ3 + c1 λ2 + c2 λ + c3 = 0
!
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C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
Fig. 4. The state of system (5) converging to O1 (0, 0, 0) when k = 18, and x(0) = 20, y(0) = 10, z (0) = 25.
where c1 = 2k + 12.2,
c2 = k2 + 15k − 465.68,
c3 = k2 + 9.4k − 492.
It can easily be seen that when k > 17.9736, we can obtain c1 > 0, c2 > 0, c3 > 0 and c1 c2 − c3 > 0. According to the Routh–Hurwitz theorem, system (5) will gradually converge to the unstable equilibrium point O1 (0, 0, 0) (see Fig. 4). In addition, if the controlled new chaotic system is
x˙ = 20(y − x) − k(x − α) y˙ = 14x + 10.6y − xz − k(y − α) z˙ = x2 − 2.8z
(6)
when k > 6.1, and α = 8.2994 or α = −8.2994, it can thus be demonstrated that the system is controlled to the points O2 (8.2994, 8.2994, 24.6) and O3 (−8.2994, −8.2994, 24.6) respectively by a similar method. Considering the case of the ordinary feedback control
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − ky z˙ = x2 − 2.8z
(7)
when k > 24.6, it can be demonstrated that system (7) will gradually converge to the unstable equilibrium point O1 (0, 0, 0). Obviously, the infimum of the enhancing feedback coefficient for the new chaotic system is smaller than that of the ordinary feedback coefficient. 4. Speed feedback control strategies For the feedback control, the independent variable of a system function is often multiplied by a coefficient as the feedback gain, so the method is called displacement feedback control. Similarly, if the derivative of an independent variable is multiplied by a coefficient as the feedback gain, it is called speed feedback control. Theorem 3. Let the controlled new chaotic system be
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − kx˙ z˙ = x2 − 2.8z
(8)
where k is the feedback coefficient; when k > 1.03, system (8) will gradually converge to the unstable equilibrium point O2 (8.2994, 8.2994, 24.6). Proof. Let x1 = x − α , y1 = y − α , z1 = z − β ; if α = 8.2994 and β = 24.6, then one can transform system (8) into the following form:
x˙ 1 = 20(y1 − x1 ) y˙ 1 = (20k − 10.6)x1 + (10.6 − 20k)y1 − α z1 − x1 z1 z˙ = x2 + 2α x − 2.8z . 1 1 1
(9)
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
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Fig. 5. The state of system (8) converging to O2 (8.2994, 8.2994, 24.6) when k = 1.5, and x(0) = 20, y(0) = 5, z (0) = 25.
The Jacobi matrix of (9) as regards the unstable equilibrium point O1 (0, 0, 0) is
−20 J = 20k − 10.6 2α
20 10.6 − 20k 0
0
!
−α −2.8
and then the characteristic equation is
λ3 + c1 λ2 + c2 λ + c3 = 0 where c1 = 20k + 12.2,
c2 = 56k + 26.32,
c3 = 40α 2 = 2755.2.
Obviously, when k > 1.03, then c1 > 0, c2 > 0, c3 > 0 and c1 c2 − c3 > 0. According to the Routh–Hurwitz theorem, system (9) will gradually converge to the point O1 (0, 0, 0), that is system (8) will gradually converge to the unstable equilibrium point O2 (8.2994, 8.2994, 24.6). The proof is thus completed (see Fig. 5). Similarly, if α = −8.2994 and β = 24.6, when k > 1.03, the system (8) can converge to the unstable equilibrium point O3 (−8.2994, −8.2994, 24.6). But the new chaotic system cannot converge to the point O1 (0, 0, 0) under this speed feedback control strategy. Obviously, the infimum of the speed feedback coefficient for the new chaotic system is also smaller than that of the ordinary feedback coefficient. 5. Conclusions This article adopts the dislocated feedback control method, the enhancing feedback control method and the speed feedback control method to research stabilization of unstable equilibrium points in a new chaotic system. On the basis of the Routh–Hurwitz theorem and the linearization model of a system at the adjacent equilibrium point, the author gets the sufficient conditions for achieving stabilization of unstable equilibrium points in the new chaotic system theoretically. It should be noted that, comparing enhancing feedback control and speed feedback control with ordinary feedback control, the coefficients of the former are smaller than the latter, so the complexity and cost are reduced. Numerical simulations show the effectiveness of the three different methods. Acknowledgement This work was supported by the Chinese Provincial Natural Science Foundation of Hunan Province (No: 06JJ5098). References [1] [2] [3] [4] [5]
E.N. Lorenz, Deterministic non-periodic flows, Journal of the Atmospheric Sciences 20 (1963) 130–141. E. Ott, C. Grebogi, Controlling chaos, Physical Review Letters 64 (1990) 1196–1199. A. Alasty, H. Salarieh, Controlling the chaos using fuzzy estimation of OGY and Pyragas controllers, Chaos, Solitons and Fractals 26 (2005) 379–392. X. Liao, P. Yu, Chaos control for the family of Rössler systems using feedback controllers, Chaos, Solitons and Fractals 29 (2006) 91–107. H.-T. Yau, C.-L. Chenc, Chaos control of Lorenz systems using adaptive controller with input saturation, Chaos, Solitons and Fractals 34 (2007) 1567–1574. [6] H.N. Agiza, Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons and Fractals 13 (2002) 341–352. [7] H.N. Agiza, Chaos synchronization of Lü dynamical system, Nonlinear Analysis 58 (2004) 11–20. [8] L. Shen, M. Wang, Adaptive control of chaotic systems based on a single layer neural network, Physics Letters A 368 (2007) 379–382.
2446 [9] [10] [11] [12]
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446 C. Tao, C. Yang, et al., Speed feedback control of chaotic system, Chaos, Solitons and Fractals 23 (2005) 259–263. S. Jiang, L. Tian, X. Wang, The control of an optical hyper-chaotic system, Chaos, Solitons and Fractals 34 (2007) 1560–1566. M.T. Yassen, Chaos control of chaotic dynamical systems using backstepping design, Chaos, Solitons and Fractals 27 (2006) 537–548. G.-L. Cai, Z.-M. Tan, W.-H. Zhou, et al., Dynamical analysis of a new chaotic system and its chaotic control, Acta Physica Sinica 56 (2007) 6230–6237.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system Congxu Zhu School of Information Science and Engineering, Central South University, Changsha, 410083, China
article
info
Article history: Received 17 January 2008 Accepted 15 January 2009 Keywords: Chaotic system Stabilization of unstable equilibrium points Dislocated feedback control Enhancing feedback control Speed feedback control
abstract This paper studies the stabilization of unstable equilibrium points in a new chaotic system. Three different methods, the dislocated feedback control method, the enhancing feedback control method and the speed feedback control method, are used to stabilize unstable equilibrium points in the new chaotic system. On the basis of the Routh–Hurwitz theorem and the linearization model of a system at the adjacent equilibrium point, the conditions of stabilization are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the three different methods. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Dynamic chaos is a very interesting non-linear effect, which has been intensively studied since Lorenz [1] found the first canonical chaotic attractor in 1963. The effect is very common; it has been detected in a large number of dynamic systems of various physical natures. However, this effect is usually undesirable in practice, and it restricts the operating range of many electronic and mechanical devices. Recently, control of this kind of complex dynamical system has attracted a great deal of attention within the engineering community. Research efforts have investigated the chaos control in many physical chaotic systems [2–11]. Different control strategies for stabilizing chaos have been proposed, such as adaptive control, time delay control, and fuzzy control. Generally speaking, there are two main approaches for controlling chaos: feedback control and non-feedback control. The feedback control approach offers many advantages such as robustness and computational complexity over the non-feedback control method. Very recently, Cai et al. presented a new chaotic dynamical system [12]. It is a three-dimensional autonomous system described by
x˙ = a(y − x) y˙ = bx + cy − xz z˙ = x2 − hz
(1)
where a, b, c and h are system parameters. System (1) has a chaotic attractor as shown in Fig.√ 1 when a =√20, b = 14, c = 10.6, h = √ 2.8. It is obvious √ that system (1) has three unstable equilibrium points O1 (0, 0, 0), O2 ( h(b + c ), h(b + c ), b+c ) and O3 (− h(b + c ), − h(b + c ), b + c ). The aim of this article is to discuss three novel feedback control strategies for the new chaotic system (1). The rest of the paper is organized as follows. Section 2 presents the dislocated feedback control strategy; the enhancing feedback control strategy and the speed feedback control strategy are presented in Sections 3 and 4, respectively. Section 5 concludes the paper.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.127
2442
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
Fig. 1. The new chaotic attractor.
Fig. 2. The state of system (2) converging to O1 (0, 0, 0) when k = 25, and x(0) = 20, y(0) = 20, z (0) = 20.
2. Dislocated feedback control strategies For the ordinary feedback control, the system variable is often multiplied by a coefficient as the feedback gain is added to the right-hand side of the corresponding equation. If such a variable is added to the right-hand side of another equation, then this method is called dislocated feedback control. Theorem 1. Let the controlled new chaotic system be
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − kx z˙ = x2 − 2.8z
(2)
where k is the feedback coefficient; when k > 24.6, system (2) will gradually converge to the unstable equilibrium point O1 (0, 0, 0). Proof. The Jacobi matrix of system (2) as regards the unstable equilibrium point O1 (0, 0, 0) is
−20 J = 14 − k 0
20 10.6 0
0 0 −2.8
!
and then the characteristic equation is
λ3 + c1 λ2 + c2 λ + c3 = 0 where c1 = 12.2,
c2 = 20k − 465.68,
c3 = 56k − 1377.6.
Obviously, when k > 24.6, c1 > 0, c2 > 0, c3 > 0 and c1 c2 − c3 > 0. According to the Routh–Hurwitz theorem, system (2) will gradually converge to the unstable equilibrium point O1 (0, 0, 0). The proof is thus completed (see Fig. 2).
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
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Fig. 3. The state of system (3) converging to O2 (8.2994, 8.2994, 24.6) when k = 16, and x(0) = 20, y(0) = 20, z (0) = 20.
Furthermore, if the controlled new chaotic system is
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − k(x − α) z˙ = x2 − 2.8z
(3)
when α = 8.2994, k > 13.3729, system (3) will gradually converge to another unstable equilibrium point O2 (8.2994, 8.2994, 24.6) (see Fig. 3). For demonstrating this conclusion, we do the following transformations: x1 = x − α , y1 = y − α , z1 = z − β . If α = 8.2994, β = 24.6, then system (3) has the following form:
x˙ 1 = 20(y1 − x1 ) y˙ 1 = (−10.6 − k)x1 + 10.6y1 − x1 z1 − 8.2994z1 z˙ = x2 + 16.5988x − 2.8z 1 1 1 1
(30 )
when k > 13.3729; one can easily prove that system (30 ) will gradually converge to unstable equilibrium point O1 (0, 0, 0), that is system (3) will gradually converge to the unstable equilibrium point O2 (8.2994, 8.2994, 24.6). Similarly, if α = −8.2994, β = 24.6, system (3) will gradually converge to another unstable equilibrium point O3 (−8.2994, −8.2994, 24.6). In addition, if the controlled new chaotic system is
x˙ = 20(y − x) − ky y˙ = 14x + 10.6y − xz z˙ = x2 − 2.8z
(4)
when k > 4.8571, then system (4) will gradually converge to unstable equilibrium point O1 (0, 0, 0). 3. Enhancing feedback control strategies It is difficult to control a complex system by means of only one feedback variable, and in such cases the feedback gain is always very large. So we consider using multiple variables multiplied by a proper coefficient as the feedback gain. This method is called enhancing feedback control. Theorem 2. For the particular structure of the new chaotic system, the control method is considered as follows:
x˙ = 20(y − x) − kx y˙ = 14x + 10.6y − xz − ky z˙ = x2 − 2.8z
(5)
when k > 17.9736; system (5) will converge to the unstable equilibrium point O1 (0, 0, 0). Proof. The Jacobi matrix of system (5) as regards the unstable equilibrium point O1 (0, 0, 0) is
−20 − k J =
14 0
20 10.6 − k 0
0 0 −2.8
and then the characteristic equation is
λ3 + c1 λ2 + c2 λ + c3 = 0
!
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C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
Fig. 4. The state of system (5) converging to O1 (0, 0, 0) when k = 18, and x(0) = 20, y(0) = 10, z (0) = 25.
where c1 = 2k + 12.2,
c2 = k2 + 15k − 465.68,
c3 = k2 + 9.4k − 492.
It can easily be seen that when k > 17.9736, we can obtain c1 > 0, c2 > 0, c3 > 0 and c1 c2 − c3 > 0. According to the Routh–Hurwitz theorem, system (5) will gradually converge to the unstable equilibrium point O1 (0, 0, 0) (see Fig. 4). In addition, if the controlled new chaotic system is
x˙ = 20(y − x) − k(x − α) y˙ = 14x + 10.6y − xz − k(y − α) z˙ = x2 − 2.8z
(6)
when k > 6.1, and α = 8.2994 or α = −8.2994, it can thus be demonstrated that the system is controlled to the points O2 (8.2994, 8.2994, 24.6) and O3 (−8.2994, −8.2994, 24.6) respectively by a similar method. Considering the case of the ordinary feedback control
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − ky z˙ = x2 − 2.8z
(7)
when k > 24.6, it can be demonstrated that system (7) will gradually converge to the unstable equilibrium point O1 (0, 0, 0). Obviously, the infimum of the enhancing feedback coefficient for the new chaotic system is smaller than that of the ordinary feedback coefficient. 4. Speed feedback control strategies For the feedback control, the independent variable of a system function is often multiplied by a coefficient as the feedback gain, so the method is called displacement feedback control. Similarly, if the derivative of an independent variable is multiplied by a coefficient as the feedback gain, it is called speed feedback control. Theorem 3. Let the controlled new chaotic system be
x˙ = 20(y − x) y˙ = 14x + 10.6y − xz − kx˙ z˙ = x2 − 2.8z
(8)
where k is the feedback coefficient; when k > 1.03, system (8) will gradually converge to the unstable equilibrium point O2 (8.2994, 8.2994, 24.6). Proof. Let x1 = x − α , y1 = y − α , z1 = z − β ; if α = 8.2994 and β = 24.6, then one can transform system (8) into the following form:
x˙ 1 = 20(y1 − x1 ) y˙ 1 = (20k − 10.6)x1 + (10.6 − 20k)y1 − α z1 − x1 z1 z˙ = x2 + 2α x − 2.8z . 1 1 1
(9)
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446
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Fig. 5. The state of system (8) converging to O2 (8.2994, 8.2994, 24.6) when k = 1.5, and x(0) = 20, y(0) = 5, z (0) = 25.
The Jacobi matrix of (9) as regards the unstable equilibrium point O1 (0, 0, 0) is
−20 J = 20k − 10.6 2α
20 10.6 − 20k 0
0
!
−α −2.8
and then the characteristic equation is
λ3 + c1 λ2 + c2 λ + c3 = 0 where c1 = 20k + 12.2,
c2 = 56k + 26.32,
c3 = 40α 2 = 2755.2.
Obviously, when k > 1.03, then c1 > 0, c2 > 0, c3 > 0 and c1 c2 − c3 > 0. According to the Routh–Hurwitz theorem, system (9) will gradually converge to the point O1 (0, 0, 0), that is system (8) will gradually converge to the unstable equilibrium point O2 (8.2994, 8.2994, 24.6). The proof is thus completed (see Fig. 5). Similarly, if α = −8.2994 and β = 24.6, when k > 1.03, the system (8) can converge to the unstable equilibrium point O3 (−8.2994, −8.2994, 24.6). But the new chaotic system cannot converge to the point O1 (0, 0, 0) under this speed feedback control strategy. Obviously, the infimum of the speed feedback coefficient for the new chaotic system is also smaller than that of the ordinary feedback coefficient. 5. Conclusions This article adopts the dislocated feedback control method, the enhancing feedback control method and the speed feedback control method to research stabilization of unstable equilibrium points in a new chaotic system. On the basis of the Routh–Hurwitz theorem and the linearization model of a system at the adjacent equilibrium point, the author gets the sufficient conditions for achieving stabilization of unstable equilibrium points in the new chaotic system theoretically. It should be noted that, comparing enhancing feedback control and speed feedback control with ordinary feedback control, the coefficients of the former are smaller than the latter, so the complexity and cost are reduced. Numerical simulations show the effectiveness of the three different methods. Acknowledgement This work was supported by the Chinese Provincial Natural Science Foundation of Hunan Province (No: 06JJ5098). References [1] [2] [3] [4] [5]
E.N. Lorenz, Deterministic non-periodic flows, Journal of the Atmospheric Sciences 20 (1963) 130–141. E. Ott, C. Grebogi, Controlling chaos, Physical Review Letters 64 (1990) 1196–1199. A. Alasty, H. Salarieh, Controlling the chaos using fuzzy estimation of OGY and Pyragas controllers, Chaos, Solitons and Fractals 26 (2005) 379–392. X. Liao, P. Yu, Chaos control for the family of Rössler systems using feedback controllers, Chaos, Solitons and Fractals 29 (2006) 91–107. H.-T. Yau, C.-L. Chenc, Chaos control of Lorenz systems using adaptive controller with input saturation, Chaos, Solitons and Fractals 34 (2007) 1567–1574. [6] H.N. Agiza, Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons and Fractals 13 (2002) 341–352. [7] H.N. Agiza, Chaos synchronization of Lü dynamical system, Nonlinear Analysis 58 (2004) 11–20. [8] L. Shen, M. Wang, Adaptive control of chaotic systems based on a single layer neural network, Physics Letters A 368 (2007) 379–382.
2446 [9] [10] [11] [12]
C. Zhu / Nonlinear Analysis 71 (2009) 2441–2446 C. Tao, C. Yang, et al., Speed feedback control of chaotic system, Chaos, Solitons and Fractals 23 (2005) 259–263. S. Jiang, L. Tian, X. Wang, The control of an optical hyper-chaotic system, Chaos, Solitons and Fractals 34 (2007) 1560–1566. M.T. Yassen, Chaos control of chaotic dynamical systems using backstepping design, Chaos, Solitons and Fractals 27 (2006) 537–548. G.-L. Cai, Z.-M. Tan, W.-H. Zhou, et al., Dynamical analysis of a new chaotic system and its chaotic control, Acta Physica Sinica 56 (2007) 6230–6237.