- Email: [email protected]
Global synchronization of two parametrically excited systems using active control
Chaos, Solitons and Fractals 28 (2006) 428–436 www.elsevier.com/locate/chaos
Global synchronization of two parametrically excited systems using active control Youming Lei
a,*
, Wei Xu a, Jianwei Shen a, Tong Fang
b
a b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China Accepted 31 May 2005
Abstract In this paper, we apply an active control technique to synchronize a kind of two parametrically excited chaotic systems. Based on Lyapunov stability theory and Routh–Hurwitz criteria, some generic sufficient conditions for global asymptotic synchronization are obtained. Illustrative examples on synchronization of two Duffing systems subject to a harmonic parametric excitation and that of two parametrically excited chaotic pendulums are considered here. Numerical simulations show the validity and feasibility of the proposed method. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the idea of synchronizing two identical autonomous chaotic systems under different initial conditions was first introduced by Pecora and Carroll in 1990 [1], chaos synchronization, as an important topic in nonlinear science, has been widely investigated in many fields, such as physical [2], chemical and ecological science [3,4], secure communications [5], etc. Hence, a lot of approaches have been proposed for the synchronization of autonomous chaotic systems with linear or nonlinear feedback control [6–10], adaptive control [11–16], back-stepping design [17,18], or active control [19–26]. In fact, the parametrically excited systems for modeling the behavior of many engineering systems, such as offshore platforms, buildings under earthquakes and so on have been widely explored [27–30]. Many complex phenomena of this kind of nonlinear dynamic systems have been demonstrated. The different chaotic motions and their attractive zones in parameter space have been described by numerical calculations and experiments, forming a great number of references on parametrically excited systems. However, to our knowledge, comparing with autonomous chaotic systems, synchronization of two identical chaotic systems with parametric excitations is seldom considered. In this paper, based on Lyapunov stability theory and Routh–Hurwitz criteria, we apply active control to synchronize two parametrically excited chaotic systems globally. The paper is organized as follows: In Section 2, the chaotic system considered in this work is presented and some criteria for global chaos synchronization are established. In Section 3, global synchronization of two identical chaotic Duffing systems subject to a harmonic parametric excitation is *
Corresponding author. Tel./fax: +86 29 8849 4404. E-mail addresses: [email protected] (Y. Lei), [email protected] (W. Xu).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.05.043
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
429
derived and numerical simulations are done to validate the proposed synchronization approach. In Section 4, it is shown that the proposed method can be extended to synchronize two parametrically excited chaotic pendulums. Finally, some conclusions are drawn in Section 5. 2. Synchronization principle Consider a parametrically excited chaotic system described by x_ ¼ AðtÞx þ f ðxÞ;
ð1Þ
n
n
where x(t) 2 R is an n-dimensional state vector of the system, A(t) 2 R is a time-periodic matrix for the system parameter, and f : Rn ! Rn is a nonlinear part of the system (1), which is considered as a driving system. A responding system with the same form of Eq. (1) with a controller u(t) 2 Rn added to its right side is introduced as follows: y_ ¼ AðtÞy þ f ðyÞ þ uðtÞ;
ð2Þ
n
where y(t) 2 R denotes the responding state vector. The synchronization problem is how to design the controller u(t), which would synchronize the states of both the driving and the responding systems. If we define the error vector as e = y x, the dynamic equation of synchronization error can be expressed as e_ ¼ AðtÞe þ f ðyÞ f ðxÞ þ uðtÞ.
ð3Þ
Hence, the objective of synchronization is to make limt!+1ke(t)k = 0. The problem of synchronization between the driving and the responding systems can be transformed into a problem of how to realize the asymptotical stabilization of the error system (3). So our aim is to design a controller u(t) to make the dynamical system (3) asymptotically stable at the origin. We define the active control function u(t) as uðtÞ ¼ Be f ðyÞ þ f ðxÞ;
ð4Þ
n
where B 2 R is a constant feedback gain matrix. Then the error dynamical system (3) can be rewritten as e_ ¼ Me;
ð5Þ
n
where M = A(t) B and M 2 R . The following theorem gives the sufficient condition for the system (5) to be globally asymptotically stable. Theorem 1. If there exists the feedback gain matrix B such that the eigenvalues of the matrix M are negative real or complex with negative real parts, then the error dynamical system (5) is globally asymptotically stable at the origin, thus implying that the two systems (1) and (2) are globally asymptotically synchronized. Remark 1. The proof of the theorem is obvious. Assuming that the parameters of the driving and the response systems are known and the states of both systems are measurable. We can achieve the synchronization by selecting an active control function u(t) to make the eigenvalues of the matrix M be negative real or complex with negative real parts. Then the states of the responding system and driving systems are globally asymptotically synchronized. 3. Synchronization of two Duffing systems subject to a harmonic parametric excitation In this section, the proposed method is applied to synchronize two identical chaotic Duffing systems subject to a harmonic parametric excitation and some global asymptotic synchronization conditions are first obtained and then further verified by numerical simulations. 3.1. Synchronization law Consider a Duffing system subject to a harmonic parametric excitation in the form [31] €x þ a_x x þ x3 ¼ lx sin Xt;
ð6Þ
where a and l denote the damping coefficient and the amplitude of the parametric excitation, respectively, and sin Xt represents a harmonic parametric excitation with period T = 2p/X.
430
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
The driving Duffing system and the responding Duffing system can be described, respectively, as x_ 1 ¼ x2 ; x_ 2 ¼ ax2 þ x1 x31 þ lx1 sin Xt; y_ 1 ¼ y 2 þ u1 ðtÞ; y_ 2 ¼ ay 2 þ y 1 y 31 þ ly 1 sin Xt þ u2 ;
ð7Þ ð8Þ
where xi, yi (i = 1, 2) are state variables of the driving and the responding system, respectively, and u1(t), u2(t) are two control functions to be determined. In order to ascertain the control functions, subtracting Eq. (7) from Eq. (8), we obtain y_ 1 x_ 1 ¼ ðy 2 x2 Þ þ u1 ; ð9Þ y_ 2 x_ 2 ¼ ð1 þ l sin XtÞðy 1 x1 Þ aðy 2 x2 Þ y 31 þ x31 þ u2 . Let ei = xi yi (i = 1, 2), then the error system can be rewritten as e_ 1 ¼ e2 þ u1 ðtÞ; e_ 2 ¼ ke1 ae2 y 31 þ x31 þ u2 ðtÞ; where k = 1 + l sin Xt. We define the active control functions u1(t) and u2(t) as follows: u1 ðtÞ ¼ V 1 ðtÞ; u2 ðtÞ ¼ y 31 x31 þ V 2 ðtÞ; Hence the error system (10) becomes e_ 1 ¼ e2 þ V 1 ðtÞ; e_ 2 ¼ ke1 ae2 þ V 2 ðtÞ.
ð10Þ
ð11Þ
ð12Þ
The system (12) describes the error dynamics and can be interpreted as a control problem where the system to be controlled is a linear system with a control input V1(t) and V2(t) as functions of e1 and e2. As long as these feedbacks stabilize the system, e1 and e2 converge to zero as time t goes to infinity. This implies that two Duffing systems are synchronized with active control. There are many possible choices for the control V1(t) and V2(t). We choose e1 V 1 ðtÞ ; ð13Þ ¼B V 2 ðtÞ e2 a b where B ¼ is a 2 · 2 constant matrix. Hence the error system (12) can be rewritten as c d e_ 1 e1 ¼ AðtÞ ; ð14Þ e_ 2 e2 a 1þb where AðtÞ ¼ is the coefficient matrix. cþk d a According to Lyapunov stability theory and Routh–Hurwitz criteria, if ( a þ d a < 0; ð15Þ ð1 þ bÞðc þ kÞ < 0; then the eigenvalues of the error system (14) must be negative real or complex with negative real parts. From Theorem 1, the error system will be stable and the two Duffing systems subject to a harmonic parametric excitation are globally asymptotic synchronized. 3.2. Numerical simulation In this subsection, numerical simulations are given to verify the proposed method. In these numerical simulations, the sixth-order Runge–Kutta method is used to solve the Duffing system with the adaptive step-size algorithm. The phase portrait of the chaotic attractor associated with Duffing system (6) for a = 0.2, l = 0.5 and X = 1.0 is given in Fig. 1, for which the corresponding Poincare map is given in Fig. 2, respectively.
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
431
Fig. 1. The phase portrait of the chaotic Duffing system, x/p versus x_ .
Fig. 2. The Poincare map of the chaotic attractor.
Let a = 0.2, l = 0.5 and X = 1.0, and suppose that a particular form of the matrix B is given by 0 2 B¼ . 2 0 For this particular choice, the conditions (15) are satisfied and the eigenvalues of the error system must be negative real or complex with negative real parts, thus leading to the synchronization of two chaotic Duffing systems. Without lost of generality, the initial values of the driving system (7) and the responding system (8) are taken as x1 = 1.0, x2 = 2.1, y1 = 0.81, y2 = 1.4 for the given parameters, respectively. The simulation results are illustrated in Fig. 3 for e1 = y1 x1 and in Fig. 4 for e2 = y2 x2. In these figures, it can be seen that the synchronization error will converge to zero finally and two identical systems from different initial values indeed achieve chaos synchronization.
4. Synchronization of two parametrically excited chaotic pendulums In this section, the proposed method is extended to synchronize two parametrically excited chaotic pendulums, for which the error dynamical system can be transformed into the form of Eq. (3) and some global asymptotic synchronization conditions are obtained and further verified by numerical simulations.
432
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
Fig. 3. The graph of error e1 between two chaotic Duffing systems with active control.
Fig. 4. The graph of error e2 between two chaotic Duffing systems with active control.
4.1. Synchronization law The system under consideration can be described by [29] €x þ b_x þ ð1 þ p cos xtÞ sin x ¼ 0;
ð16Þ
where x is the measure of the angular displacement, and the forcing is represented by the term p cos xt with period T = 2p/x. Consider two identical pendulums as follows: x_ 1 ¼ x2 ; ð17Þ x_ 2 ¼ bx2 ð1 þ p cos xtÞ sin x1 ; y_ 1 ¼ y 2 þ u1 ðtÞ; ð18Þ y_ 2 ¼ by 2 ð1 þ p cos xtÞ sin y 1 þ u2 ðtÞ; where xi, yi (i = 1, 2) are state variables of the driving and the responding system, respectively, and u1(t), u2(t) are two control functions to be determined. In order to ascertain the control functions, subtracting Eq. (17) from Eq. (18), we obtain y_ 1 x_ 1 ¼ ðy 2 x2 Þ þ u1 ; ð19Þ y_ 2 x_ 2 ¼ bðy 2 x2 Þ ð1 þ p cos xtÞðsin y 1 sin x1 Þ þ u2 .
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
Let ei = xi yi (i = 1, 2), then the error system (19) can be rewritten as e_ 1 ¼ e2 þ u1 ðtÞ; e_ 2 ¼ be2 þ kðsin y 1 sin x1 Þ þ u2 ðtÞ;
433
ð20Þ
where k = (1 + p cos xt). The system (20) does not agree with the form of Eq. (3), so the above stated active control technique cannot be used directly. It follows from the differential mean-value theorem that sin y 1 sin x1 ¼ cos n ðy 1 x1 Þ;
n 2 ðx1 ; y 1 Þ; suppose x1 < y 1 .
Substituting Eq. (21) into Eq. (20), we have e_ 1 ¼ e2 þ u1 ðtÞ; e_ 2 ¼ be2 þ ke1 cos n þ u2 ðtÞ.
ð21Þ
ð22Þ
Hence the system (22) now has the same form of Eq. (3) describing the error dynamics and can be interpreted as a control problem for a linear system with control inputs u1(t) and u2(t) as functions of e1 and e2. As long as these feedbacks stabilize the system (22), e1 and e2 converge to zero as time t goes to infinity. This implies that two parametrically excited pendulums (17) and (18) are synchronized with the active control. Among various possible choices for the control u1(t) and u2(t), we choose u1 ðtÞ e1 ; ð23Þ ¼B e2 u2 ðtÞ a b where B ¼ is a 2 · 2 constant matrix. Hence the error system (22) can be rewritten as c d e_ 1 e1 ¼ AðtÞ ; ð24Þ e_ 2 e2 a 1þb is the coefficient matrix. where AðtÞ ¼ c þ k cos n d b According to Lyapunov stability theory and Routh–Hurwitz criteria, if a þ d b < 0; ð1 þ bÞðc þ k cos nÞ < 0;
ð25Þ
then the eigenvalues of the error system (24) must be negative real or complex with negative real parts. In this case, the error system will be asymptotically stable and the two parametrically excited chaotic pendulums are globally asymptotically synchronized. 4.2. Numerical simulation In this subsection, numerical simulations are given to verify the proposed method. In these numerical simulations, the sixth-order Runge–Kutta method is used to solve the pendulums with adaptive step-size algorithm. The phase portrait of chaotic attractor associated with the pendulum (16) for b = 0.1, p = 2.0 and x = 2.0 is given in Fig. 5, while the corresponding Poincare map is given in Fig. 6, respectively. Let parameters b = 0.1, p = 2.0 and x = 2.0, and suppose that a particular form of the matrix B is given by 1.0 1.2 B¼ . 4.0 0 For this particular choice, the conditions (25) are satisfied and the eigenvalues of the error system must be negative real or complex with negative real parts, thus leading to the synchronization of two chaotic pendulums. Without losing any generality, the initial conditions of the driving system (17) and the responding system (18) are taken as x1 = 0.3, x2 = 0.8, y1 = 2.6, y2 = 1.0 for the given parameters, respectively. The simulation results are illustrated in Fig. 7 for e1 = y1 x1 and Fig. 8 for e2 = y2 x2. In these figures, it can be seen that the synchronization error will converge to zero finally and two identical pendulums under different initial conditions indeed achieve chaos synchronization.
434
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
Fig. 5. The phase portrait of the chaotic pendulum, x/p versus x_ .
Fig. 6. The Poincare map of the chaotic attractor.
Fig. 7. The graph of error e1 between two chaotic pendulums with active control.
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
435
Fig. 8. The graph of error e2 between two chaotic pendulums with active control.
5. Conclusions Based on Lyapunov stability theory and Routh–Hurwitz criteria, this paper offers some sufficient conditions for global asymptotic synchronization between two parametrically excited chaotic systems by the active control method. The controller can be easily designed on the basis of these conditions to ensure the global chaos synchronization. How to apply these sufficient conditions to systems governed by Eq. (1) are demonstrated with two illustrative examples. Numerical results show that the proposed active control method is very effective. Acknowledgments The authors are grateful for the support of the National Natural Science Foundation of China (Grant Nos. 10472091 and 10332030) and the support of the center for high performance computing of NPU. The first author would also like to express his gratitude to the support of Youth for NPU Teachers Scientific and Technological Innovation Foundation and the Doctorate Creation Foundation of Northwestern Polytechnical University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64(8):821–4. Lakshmanan M, Murali K. Chaos in nonlinear oscillators: controlling and synchronization. Singapore: World Scientific; 1996. Han SK, Kerrer C, Kuramoto Y. Dephasing and bursting in coupled neural oscillators. Phys Rev Lett 1995;75:3190–3. Blasius B, Huppert A, Stone L. Complex dynamics and phase synchronization in spatially extended ecological system. Nature 1999;399:354–9. Feki M. An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons & Fractals 2003;18:141–8. Zhou T, Lu¨ J, Chen G, Tang Y. Synchronization stability of three chaotic systems with linear coupling. Phys Lett A 2002;301:231–40. Sun J, Zhang Y, Qiao F, Wu Q. Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach. Chaos, Solitons & Fractals 2004;19:1049–55. Yu Y, Zhang S. The synchronization of linearly bidirectional coupled chaotic systems. Chaos, Solitons & Fractals 2004;22:189–97. Chen H-K. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons & Fractals 2005;23:1245–51. Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. Chen S, Lu¨ J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. Wang Y, Guan Z-H, Wen X. Adaptive synchronization for Chen chaotic system with fully unknown parameters. Chaos, Solitons & Fractals 2004;19:899–903. Elabbasy EM, Agiza HN, El-Dessoky MM. Adaptive synchronization of Lu¨ system with uncertain parameters. Chaos, Solitons & Fractals 2004;21:657–67.
436
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
[15] Park JH. Adaptive synchronization of Rossler system with uncertain parameters. Chaos, Solitons & Fractals 2005;25:333–8. [16] Yassen MT. Adaptive control and synchronization of a modified ChuaÕs circuit system. Appl Math Comput 2001;135(1):113–28. [17] Bowong S, Moukam Kakmeni FM. Synchronization of uncertain chaotic systems via backstepping approach. Chaos, Solitons & Fractals 2004;21:999–1011. [18] Zhang J, Li C, Zhang H, Yu J. Chaos synchronization using single variable feedback based on backstepping method. Chaos, Solitons & Fractals 2004;21:1183–93. [19] Bai EW, Lonngren KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. [20] Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4. [21] Agiza HN, Yassen MT. Synchronization of Rossler and chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [22] Ho MC, Hung YC. Synchronization of two different systems by using generalized active control. Phys Lett A 2002;301:424–8. [23] Codreanu S. Synchronization of spatiotemporal nonlinear dynamical systems by an active control. Chaos, Solitons & Fractals 2003;15:507–10. [24] Zhang H, Ma XK. Synchronization of uncertain chaotic systems with parameters perturbation via active control. Chaos, Solitons & Fractals 2004;21:39–47. [25] Uc¸ar A, Lonngren KE, Bai EW. Synchronization of the coupled FitzHugh–Nagumo systems. Chaos, Solitons & Fractals 2004;20:1085–90. [26] Yassen MT. Chaos synchronization between two different chaotic systems using active control. Chaos, Solitons & Fractals 2005;23:131–40. [27] Leven RW, Pompe B, Wilke C, Koch BP. Experiments on periodic and chaotic motions of a parametrically forced pendulum. Physica D 1985;16:371–84. [28] van de Water W, Hoppenbrouwers M. Unstable periodic orbits in the parametrically excited pendulum. Phys Rev A 1991;44:6388–98. [29] Bishop SR, Clifford MJ. Zones of chaotic behavior in the parametrically excited pendulum. J Sound Vib 1996;189:142–7. [30] Zhang Y, Hu SQ, Du GH. Chaos synchronization of two parametrically excited pendulums. J Sound Vib 1999;223:247–54. [31] Lei Y, Xu W, Xu Y, Fang T. Chaos control by harmonic excitation with proper random phase. Chaos, Solitons & Fractals 2004;21:1175–81.
Global synchronization of two parametrically excited systems using active control Youming Lei
a,*
, Wei Xu a, Jianwei Shen a, Tong Fang
b
a b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China Accepted 31 May 2005
Abstract In this paper, we apply an active control technique to synchronize a kind of two parametrically excited chaotic systems. Based on Lyapunov stability theory and Routh–Hurwitz criteria, some generic sufficient conditions for global asymptotic synchronization are obtained. Illustrative examples on synchronization of two Duffing systems subject to a harmonic parametric excitation and that of two parametrically excited chaotic pendulums are considered here. Numerical simulations show the validity and feasibility of the proposed method. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the idea of synchronizing two identical autonomous chaotic systems under different initial conditions was first introduced by Pecora and Carroll in 1990 [1], chaos synchronization, as an important topic in nonlinear science, has been widely investigated in many fields, such as physical [2], chemical and ecological science [3,4], secure communications [5], etc. Hence, a lot of approaches have been proposed for the synchronization of autonomous chaotic systems with linear or nonlinear feedback control [6–10], adaptive control [11–16], back-stepping design [17,18], or active control [19–26]. In fact, the parametrically excited systems for modeling the behavior of many engineering systems, such as offshore platforms, buildings under earthquakes and so on have been widely explored [27–30]. Many complex phenomena of this kind of nonlinear dynamic systems have been demonstrated. The different chaotic motions and their attractive zones in parameter space have been described by numerical calculations and experiments, forming a great number of references on parametrically excited systems. However, to our knowledge, comparing with autonomous chaotic systems, synchronization of two identical chaotic systems with parametric excitations is seldom considered. In this paper, based on Lyapunov stability theory and Routh–Hurwitz criteria, we apply active control to synchronize two parametrically excited chaotic systems globally. The paper is organized as follows: In Section 2, the chaotic system considered in this work is presented and some criteria for global chaos synchronization are established. In Section 3, global synchronization of two identical chaotic Duffing systems subject to a harmonic parametric excitation is *
Corresponding author. Tel./fax: +86 29 8849 4404. E-mail addresses: [email protected] (Y. Lei), [email protected] (W. Xu).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.05.043
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
429
derived and numerical simulations are done to validate the proposed synchronization approach. In Section 4, it is shown that the proposed method can be extended to synchronize two parametrically excited chaotic pendulums. Finally, some conclusions are drawn in Section 5. 2. Synchronization principle Consider a parametrically excited chaotic system described by x_ ¼ AðtÞx þ f ðxÞ;
ð1Þ
n
n
where x(t) 2 R is an n-dimensional state vector of the system, A(t) 2 R is a time-periodic matrix for the system parameter, and f : Rn ! Rn is a nonlinear part of the system (1), which is considered as a driving system. A responding system with the same form of Eq. (1) with a controller u(t) 2 Rn added to its right side is introduced as follows: y_ ¼ AðtÞy þ f ðyÞ þ uðtÞ;
ð2Þ
n
where y(t) 2 R denotes the responding state vector. The synchronization problem is how to design the controller u(t), which would synchronize the states of both the driving and the responding systems. If we define the error vector as e = y x, the dynamic equation of synchronization error can be expressed as e_ ¼ AðtÞe þ f ðyÞ f ðxÞ þ uðtÞ.
ð3Þ
Hence, the objective of synchronization is to make limt!+1ke(t)k = 0. The problem of synchronization between the driving and the responding systems can be transformed into a problem of how to realize the asymptotical stabilization of the error system (3). So our aim is to design a controller u(t) to make the dynamical system (3) asymptotically stable at the origin. We define the active control function u(t) as uðtÞ ¼ Be f ðyÞ þ f ðxÞ;
ð4Þ
n
where B 2 R is a constant feedback gain matrix. Then the error dynamical system (3) can be rewritten as e_ ¼ Me;
ð5Þ
n
where M = A(t) B and M 2 R . The following theorem gives the sufficient condition for the system (5) to be globally asymptotically stable. Theorem 1. If there exists the feedback gain matrix B such that the eigenvalues of the matrix M are negative real or complex with negative real parts, then the error dynamical system (5) is globally asymptotically stable at the origin, thus implying that the two systems (1) and (2) are globally asymptotically synchronized. Remark 1. The proof of the theorem is obvious. Assuming that the parameters of the driving and the response systems are known and the states of both systems are measurable. We can achieve the synchronization by selecting an active control function u(t) to make the eigenvalues of the matrix M be negative real or complex with negative real parts. Then the states of the responding system and driving systems are globally asymptotically synchronized. 3. Synchronization of two Duffing systems subject to a harmonic parametric excitation In this section, the proposed method is applied to synchronize two identical chaotic Duffing systems subject to a harmonic parametric excitation and some global asymptotic synchronization conditions are first obtained and then further verified by numerical simulations. 3.1. Synchronization law Consider a Duffing system subject to a harmonic parametric excitation in the form [31] €x þ a_x x þ x3 ¼ lx sin Xt;
ð6Þ
where a and l denote the damping coefficient and the amplitude of the parametric excitation, respectively, and sin Xt represents a harmonic parametric excitation with period T = 2p/X.
430
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
The driving Duffing system and the responding Duffing system can be described, respectively, as x_ 1 ¼ x2 ; x_ 2 ¼ ax2 þ x1 x31 þ lx1 sin Xt; y_ 1 ¼ y 2 þ u1 ðtÞ; y_ 2 ¼ ay 2 þ y 1 y 31 þ ly 1 sin Xt þ u2 ;
ð7Þ ð8Þ
where xi, yi (i = 1, 2) are state variables of the driving and the responding system, respectively, and u1(t), u2(t) are two control functions to be determined. In order to ascertain the control functions, subtracting Eq. (7) from Eq. (8), we obtain y_ 1 x_ 1 ¼ ðy 2 x2 Þ þ u1 ; ð9Þ y_ 2 x_ 2 ¼ ð1 þ l sin XtÞðy 1 x1 Þ aðy 2 x2 Þ y 31 þ x31 þ u2 . Let ei = xi yi (i = 1, 2), then the error system can be rewritten as e_ 1 ¼ e2 þ u1 ðtÞ; e_ 2 ¼ ke1 ae2 y 31 þ x31 þ u2 ðtÞ; where k = 1 + l sin Xt. We define the active control functions u1(t) and u2(t) as follows: u1 ðtÞ ¼ V 1 ðtÞ; u2 ðtÞ ¼ y 31 x31 þ V 2 ðtÞ; Hence the error system (10) becomes e_ 1 ¼ e2 þ V 1 ðtÞ; e_ 2 ¼ ke1 ae2 þ V 2 ðtÞ.
ð10Þ
ð11Þ
ð12Þ
The system (12) describes the error dynamics and can be interpreted as a control problem where the system to be controlled is a linear system with a control input V1(t) and V2(t) as functions of e1 and e2. As long as these feedbacks stabilize the system, e1 and e2 converge to zero as time t goes to infinity. This implies that two Duffing systems are synchronized with active control. There are many possible choices for the control V1(t) and V2(t). We choose e1 V 1 ðtÞ ; ð13Þ ¼B V 2 ðtÞ e2 a b where B ¼ is a 2 · 2 constant matrix. Hence the error system (12) can be rewritten as c d e_ 1 e1 ¼ AðtÞ ; ð14Þ e_ 2 e2 a 1þb where AðtÞ ¼ is the coefficient matrix. cþk d a According to Lyapunov stability theory and Routh–Hurwitz criteria, if ( a þ d a < 0; ð15Þ ð1 þ bÞðc þ kÞ < 0; then the eigenvalues of the error system (14) must be negative real or complex with negative real parts. From Theorem 1, the error system will be stable and the two Duffing systems subject to a harmonic parametric excitation are globally asymptotic synchronized. 3.2. Numerical simulation In this subsection, numerical simulations are given to verify the proposed method. In these numerical simulations, the sixth-order Runge–Kutta method is used to solve the Duffing system with the adaptive step-size algorithm. The phase portrait of the chaotic attractor associated with Duffing system (6) for a = 0.2, l = 0.5 and X = 1.0 is given in Fig. 1, for which the corresponding Poincare map is given in Fig. 2, respectively.
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
431
Fig. 1. The phase portrait of the chaotic Duffing system, x/p versus x_ .
Fig. 2. The Poincare map of the chaotic attractor.
Let a = 0.2, l = 0.5 and X = 1.0, and suppose that a particular form of the matrix B is given by 0 2 B¼ . 2 0 For this particular choice, the conditions (15) are satisfied and the eigenvalues of the error system must be negative real or complex with negative real parts, thus leading to the synchronization of two chaotic Duffing systems. Without lost of generality, the initial values of the driving system (7) and the responding system (8) are taken as x1 = 1.0, x2 = 2.1, y1 = 0.81, y2 = 1.4 for the given parameters, respectively. The simulation results are illustrated in Fig. 3 for e1 = y1 x1 and in Fig. 4 for e2 = y2 x2. In these figures, it can be seen that the synchronization error will converge to zero finally and two identical systems from different initial values indeed achieve chaos synchronization.
4. Synchronization of two parametrically excited chaotic pendulums In this section, the proposed method is extended to synchronize two parametrically excited chaotic pendulums, for which the error dynamical system can be transformed into the form of Eq. (3) and some global asymptotic synchronization conditions are obtained and further verified by numerical simulations.
432
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
Fig. 3. The graph of error e1 between two chaotic Duffing systems with active control.
Fig. 4. The graph of error e2 between two chaotic Duffing systems with active control.
4.1. Synchronization law The system under consideration can be described by [29] €x þ b_x þ ð1 þ p cos xtÞ sin x ¼ 0;
ð16Þ
where x is the measure of the angular displacement, and the forcing is represented by the term p cos xt with period T = 2p/x. Consider two identical pendulums as follows: x_ 1 ¼ x2 ; ð17Þ x_ 2 ¼ bx2 ð1 þ p cos xtÞ sin x1 ; y_ 1 ¼ y 2 þ u1 ðtÞ; ð18Þ y_ 2 ¼ by 2 ð1 þ p cos xtÞ sin y 1 þ u2 ðtÞ; where xi, yi (i = 1, 2) are state variables of the driving and the responding system, respectively, and u1(t), u2(t) are two control functions to be determined. In order to ascertain the control functions, subtracting Eq. (17) from Eq. (18), we obtain y_ 1 x_ 1 ¼ ðy 2 x2 Þ þ u1 ; ð19Þ y_ 2 x_ 2 ¼ bðy 2 x2 Þ ð1 þ p cos xtÞðsin y 1 sin x1 Þ þ u2 .
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
Let ei = xi yi (i = 1, 2), then the error system (19) can be rewritten as e_ 1 ¼ e2 þ u1 ðtÞ; e_ 2 ¼ be2 þ kðsin y 1 sin x1 Þ þ u2 ðtÞ;
433
ð20Þ
where k = (1 + p cos xt). The system (20) does not agree with the form of Eq. (3), so the above stated active control technique cannot be used directly. It follows from the differential mean-value theorem that sin y 1 sin x1 ¼ cos n ðy 1 x1 Þ;
n 2 ðx1 ; y 1 Þ; suppose x1 < y 1 .
Substituting Eq. (21) into Eq. (20), we have e_ 1 ¼ e2 þ u1 ðtÞ; e_ 2 ¼ be2 þ ke1 cos n þ u2 ðtÞ.
ð21Þ
ð22Þ
Hence the system (22) now has the same form of Eq. (3) describing the error dynamics and can be interpreted as a control problem for a linear system with control inputs u1(t) and u2(t) as functions of e1 and e2. As long as these feedbacks stabilize the system (22), e1 and e2 converge to zero as time t goes to infinity. This implies that two parametrically excited pendulums (17) and (18) are synchronized with the active control. Among various possible choices for the control u1(t) and u2(t), we choose u1 ðtÞ e1 ; ð23Þ ¼B e2 u2 ðtÞ a b where B ¼ is a 2 · 2 constant matrix. Hence the error system (22) can be rewritten as c d e_ 1 e1 ¼ AðtÞ ; ð24Þ e_ 2 e2 a 1þb is the coefficient matrix. where AðtÞ ¼ c þ k cos n d b According to Lyapunov stability theory and Routh–Hurwitz criteria, if a þ d b < 0; ð1 þ bÞðc þ k cos nÞ < 0;
ð25Þ
then the eigenvalues of the error system (24) must be negative real or complex with negative real parts. In this case, the error system will be asymptotically stable and the two parametrically excited chaotic pendulums are globally asymptotically synchronized. 4.2. Numerical simulation In this subsection, numerical simulations are given to verify the proposed method. In these numerical simulations, the sixth-order Runge–Kutta method is used to solve the pendulums with adaptive step-size algorithm. The phase portrait of chaotic attractor associated with the pendulum (16) for b = 0.1, p = 2.0 and x = 2.0 is given in Fig. 5, while the corresponding Poincare map is given in Fig. 6, respectively. Let parameters b = 0.1, p = 2.0 and x = 2.0, and suppose that a particular form of the matrix B is given by 1.0 1.2 B¼ . 4.0 0 For this particular choice, the conditions (25) are satisfied and the eigenvalues of the error system must be negative real or complex with negative real parts, thus leading to the synchronization of two chaotic pendulums. Without losing any generality, the initial conditions of the driving system (17) and the responding system (18) are taken as x1 = 0.3, x2 = 0.8, y1 = 2.6, y2 = 1.0 for the given parameters, respectively. The simulation results are illustrated in Fig. 7 for e1 = y1 x1 and Fig. 8 for e2 = y2 x2. In these figures, it can be seen that the synchronization error will converge to zero finally and two identical pendulums under different initial conditions indeed achieve chaos synchronization.
434
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
Fig. 5. The phase portrait of the chaotic pendulum, x/p versus x_ .
Fig. 6. The Poincare map of the chaotic attractor.
Fig. 7. The graph of error e1 between two chaotic pendulums with active control.
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
435
Fig. 8. The graph of error e2 between two chaotic pendulums with active control.
5. Conclusions Based on Lyapunov stability theory and Routh–Hurwitz criteria, this paper offers some sufficient conditions for global asymptotic synchronization between two parametrically excited chaotic systems by the active control method. The controller can be easily designed on the basis of these conditions to ensure the global chaos synchronization. How to apply these sufficient conditions to systems governed by Eq. (1) are demonstrated with two illustrative examples. Numerical results show that the proposed active control method is very effective. Acknowledgments The authors are grateful for the support of the National Natural Science Foundation of China (Grant Nos. 10472091 and 10332030) and the support of the center for high performance computing of NPU. The first author would also like to express his gratitude to the support of Youth for NPU Teachers Scientific and Technological Innovation Foundation and the Doctorate Creation Foundation of Northwestern Polytechnical University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64(8):821–4. Lakshmanan M, Murali K. Chaos in nonlinear oscillators: controlling and synchronization. Singapore: World Scientific; 1996. Han SK, Kerrer C, Kuramoto Y. Dephasing and bursting in coupled neural oscillators. Phys Rev Lett 1995;75:3190–3. Blasius B, Huppert A, Stone L. Complex dynamics and phase synchronization in spatially extended ecological system. Nature 1999;399:354–9. Feki M. An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons & Fractals 2003;18:141–8. Zhou T, Lu¨ J, Chen G, Tang Y. Synchronization stability of three chaotic systems with linear coupling. Phys Lett A 2002;301:231–40. Sun J, Zhang Y, Qiao F, Wu Q. Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach. Chaos, Solitons & Fractals 2004;19:1049–55. Yu Y, Zhang S. The synchronization of linearly bidirectional coupled chaotic systems. Chaos, Solitons & Fractals 2004;22:189–97. Chen H-K. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons & Fractals 2005;23:1245–51. Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. Chen S, Lu¨ J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. Wang Y, Guan Z-H, Wen X. Adaptive synchronization for Chen chaotic system with fully unknown parameters. Chaos, Solitons & Fractals 2004;19:899–903. Elabbasy EM, Agiza HN, El-Dessoky MM. Adaptive synchronization of Lu¨ system with uncertain parameters. Chaos, Solitons & Fractals 2004;21:657–67.
436
Y. Lei et al. / Chaos, Solitons and Fractals 28 (2006) 428–436
[15] Park JH. Adaptive synchronization of Rossler system with uncertain parameters. Chaos, Solitons & Fractals 2005;25:333–8. [16] Yassen MT. Adaptive control and synchronization of a modified ChuaÕs circuit system. Appl Math Comput 2001;135(1):113–28. [17] Bowong S, Moukam Kakmeni FM. Synchronization of uncertain chaotic systems via backstepping approach. Chaos, Solitons & Fractals 2004;21:999–1011. [18] Zhang J, Li C, Zhang H, Yu J. Chaos synchronization using single variable feedback based on backstepping method. Chaos, Solitons & Fractals 2004;21:1183–93. [19] Bai EW, Lonngren KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. [20] Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4. [21] Agiza HN, Yassen MT. Synchronization of Rossler and chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [22] Ho MC, Hung YC. Synchronization of two different systems by using generalized active control. Phys Lett A 2002;301:424–8. [23] Codreanu S. Synchronization of spatiotemporal nonlinear dynamical systems by an active control. Chaos, Solitons & Fractals 2003;15:507–10. [24] Zhang H, Ma XK. Synchronization of uncertain chaotic systems with parameters perturbation via active control. Chaos, Solitons & Fractals 2004;21:39–47. [25] Uc¸ar A, Lonngren KE, Bai EW. Synchronization of the coupled FitzHugh–Nagumo systems. Chaos, Solitons & Fractals 2004;20:1085–90. [26] Yassen MT. Chaos synchronization between two different chaotic systems using active control. Chaos, Solitons & Fractals 2005;23:131–40. [27] Leven RW, Pompe B, Wilke C, Koch BP. Experiments on periodic and chaotic motions of a parametrically forced pendulum. Physica D 1985;16:371–84. [28] van de Water W, Hoppenbrouwers M. Unstable periodic orbits in the parametrically excited pendulum. Phys Rev A 1991;44:6388–98. [29] Bishop SR, Clifford MJ. Zones of chaotic behavior in the parametrically excited pendulum. J Sound Vib 1996;189:142–7. [30] Zhang Y, Hu SQ, Du GH. Chaos synchronization of two parametrically excited pendulums. J Sound Vib 1999;223:247–54. [31] Lei Y, Xu W, Xu Y, Fang T. Chaos control by harmonic excitation with proper random phase. Chaos, Solitons & Fractals 2004;21:1175–81.