- Email: [email protected]
Normal form and synchronization of strict-feedback chaotic systems
Chaos, Solitons and Fractals 22 (2004) 927–933 www.elsevier.com/locate/chaos
Normal form and synchronization of strict-feedback chaotic systems Feng Wang a, Shihua Chen b
a,*
, Minghai Yu b, Changping Wang
c
a College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, PR China c Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada B3H 3J5
Accepted 9 March 2004
Abstract This study concerns the normal form and synchronization of strict-feedback chaotic systems. We prove that, any strict-feedback chaotic system can be rendered into a normal form with a invertible transform and then a design procedure to synchronize the normal form of a non-autonomous strict-feedback chaotic system is presented. This approach needs only a scalar driving signal to realize synchronization no matter how many dimensions the chaotic system contains. Furthermore, the R€ ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming a strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The idea of synchronizing two identical chaotic systems with different initial conditions was first introduced by Pecora and Carrol [1]. Since then, synchronization in coupled chaotic systems has become a rapidly developing field in light of its great potential applications in secure communications, chemical reaction, modelling brain activity and so on [2,3]. A large varieties of approaches have been proposed for chaos synchronization such as the adaptive synchronization method [4], the sampled-data feedback synchronization method [5], the impulsive synchronization method [6], and many others [7–9]. Moreover, in the study of synchronizing nonlinear dynamics, several low-dimensional systems are frequently used as benchmark examples for verification and validation of a proposed theory, method and algorithm. These examples include Duffing oscillator, Van der Pol oscillator, R€ ossler system, Chua’s circuit, Lorenz system and Chen system [10–14]. It is very interesting to note that, many existing synchronization methods need several controllers to realize synchronization and all those chaotic systems mentioned above can be rewritten into a class of nonlinear systems in a so-called general strict-feedback form. This motivates the present work, which is to address two important issues concerning the strict-feedback chaotic system. One issue is to find the normal form of strict-feedback chaotic systems, the other is to find a scalar driving signal to realize synchronization. We prove that, any strict-feedback chaotic system can be rendered into a normal form with a invertible transform. Furthermore, a design procedure to synchronize the normal form of a non-autonomous strict-feedback chaotic system is presented. This approach needs only a scalar driving signal to realize synchronization no matter how many dimensions the chaotic system contains. In addition, the R€ ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming a
*
Corresponding author. E-mail address: [email protected] (S. Chen).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.03.010
928
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods.
2. The normal form of strict-feedback chaotic systems Many chaotic systems used in the study of nonlinear dynamics can be rewritten into a class of nonlinear systems in the form x_ 1 ¼ f1 ðx1 ; tÞx2 þ g1 ðx1 ; tÞ; x_ 2 ¼ f2 ðx1 ; x2 ; tÞx3 þ g2 ðx1 ; x2 ; tÞ; ......... x_ n1 ¼ fn1 ðx1 ; x2 ; . . . ; xn1 ; tÞxn þ gn1 ðx1 ; x2 ; . . . ; xn1 ; tÞ; x_ n ¼ fn ðx1 ; x2 ; . . . ; xn1 ; xn ; tÞ;
ð1Þ
where fi , gi ði ¼ 1; 2; . . . ; n 1Þ are sufficiently smooth nonlinear functions and fn is a continuous nonlinear function. It is often referred as a general strict-feedback system. If fi 6¼ 0 ði ¼ 1; 2; . . . ; n 1Þ, it is called a strict-feedback system. General strict-feedback systems and strict-feedback systems have been thoroughly studied during the past decades. The main works concentrated on controlling these systems as well as tracking any one-dimensional smooth trajectory [15,16]. Our aim in this section, however, is to find the normal form of strict-feedback chaotic systems. To this end, we first define two new variables u1 ¼ x1 ; u2 ¼ f1 ðx1 ; tÞx2 þ g1 ðx1 ; tÞ , F1 ðx1 ; x2 ; tÞ:
ð2Þ
Simple calculation can yields 1 X du2 o dxi ¼ f1 ðx1 ; tÞf2 ðx1 ; x2 ; tÞx3 þ þ f1 ðx1 ; tÞg2 ðx1 ; x2 ; tÞ , F2 ðx1 ; x2 ; x3 ; tÞ; F1 ðx1 ; x2 ; tÞ dt dt ox i i¼0
where x0 stands for t and
dx0 dt
ð3Þ
¼ 1. With the above notation, the first two equations of (1) can be rewritten as
u_ 1 ¼ u2 ; u_ 2 ¼ F2 ðx1 ; x2 ; x3 ; tÞ:
ð4Þ
We define the third variable as u3 ¼ F2 ðx1 ; x2 ; x3 ; tÞ:
ð5Þ
One can get 3 2 2 X du3 Y o dxi Y ¼ þ fi ðx1 ; . . . ; xi ; tÞg3 ðx1 ; x2 ; x3 ; tÞ , F3 ðx1 ; x2 ; x3 ; x4 ; tÞ: fi ðx1 ; . . . ; xi ; tÞx4 þ F2 ðx1 ; x2 ; x3 ; tÞ dt dt oxi i¼1 i¼0 i¼1
ð6Þ
With this notation, the first three equations of (1) can be rewritten as u_ 1 ¼ u2 ; u_ 2 ¼ u3 ; u_ 3 ¼ F3 ðx1 ; x2 ; x3 ; x4 ; tÞ:
ð7Þ
Generally, the k-th (3 6 i 6 n 1) variable is defined as uk ¼ Fk1 ðx1 ; x2 ; . . . ; xk ; tÞ;
ð8Þ
its time derivative is k k1 X duk Y o dxi ¼ fi ðx1 ; . . . ; xi ; tÞxkþ1 þ Fk1 ðx1 ; . . . ; xk ; tÞ ox dt dt i i¼1 i¼0
þ
k 1 Y i¼1
fi ðx1 ; . . . ; xi ; tÞgk ðx1 ; . . . ; xk ; tÞ , Fk ðx1 ; . . . ; xkþ1 ; tÞ:
ð9Þ
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
929
Then the first k equations of (1) can be transformed into u_ 1 ¼ u2 ; u_ 2 ¼ u3 ; ......
ð10Þ
u_ k1 ¼ uk ; u_ k ¼ Fk ðx1 ; . . . ; xkþ1 ; tÞ: The last step is to define the n-th variable as un ¼ Fn1 ðx1 ; . . . ; xn ; tÞ:
ð11Þ
Its time derivative is n n1 X dun Y o dxi fi ðx1 ; . . . ; xi ; tÞxn þ Fn1 ðx1 ; . . . ; xn ; tÞ ¼ : ox dt dt i i¼1 i¼0
ð12Þ
On the other hand, from the definitions of ui ði ¼ 1; 2; . . . ; nÞ one can find that xi ði ¼ 1; 2; . . . ; nÞ could be expressed as functions of t and ui ði ¼ 1; 2; . . . ; nÞ. Thus, the equation (12) could be written in the form dun ¼ F ðu1 ; . . . ; un ; tÞ; dt
ð13Þ
from which we have the following theorem: Theorem 2.1. Any strict-feedback chaotic system can be transformed into the normal form as follows: u_ 1 ¼ u2 ; u_ 2 ¼ u3 ; ...... u_ n1 ¼ un ; u_ n ¼ F ðu1 ; . . . ; un ; tÞ:
ð14Þ
3. Synchronizing strict-feedback chaotic systems via a scalar driving signal In this section, we will propose a design procedure to synchronize the normal form of the non-autonomous strictfeedback chaotic system. We design such a scalar controller U that the controlled strict-feedback chaotic system in normal form y_ 1 ¼ y2 ; y_ 2 ¼ y3 ; ......
ð15Þ
y_ n1 ¼ yn ; y_ n ¼ F ðy1 ; . . . ; yn ; tÞ þ U is synchronous with (14), the normal form of system (1). To this end, let U ¼ F ðu1 ; . . . ; un ; tÞ F ðy1 ; . . . ; yn ; tÞ þ U1 ;
ð16Þ
where U1 is waiting for determination. With this selection, the error system of (15) and (14) can be expressed as e_ ¼ Ae þ BU1 ;
ð17Þ
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where e ¼ ðe1 ; e2 ; . . . ; en Þ ¼ ðy1 u1 ; y2 u2 ; . . . ; yn un Þ> , B ¼ ð0; 0; . . . ; 1Þ> and 0 1 0 1 0 0 ... 0 B 0 0 1 0 ... 0 C B C A¼B . . . . . . . . . . .. ... ...C B C: @ 0 0 0 0 ... 1 A 0 0 0 0 ... 0 On the other hand, the matrix 0
½B
AB A2 B . . .
0 B 0 B An1 B ¼ B B... @ 0 1
0 0 ... 1 0
0 0 ... 0 0
... ... ... ... ...
0 1 ... 0 0
1 1 0 C C ...C C 0 A 0
is full rank, so the linear control theory [17] confirms that the single-input dynamic system (17) is controllable, i.e., all the eigenvalues are controllable. Thus, by the pole assigning method [17] we can select an appropriate feedback gain vector k ¼ ðk1 ; k2 ; . . . ; kn Þ> such that system (17) is globally asymptotically stable at zero with the state feedback control U1 ¼ ke, which implies that dynamical system (15) is synchronous with (14), the normal form of system (1).
4. Numerical simulation In order to demonstrate and verify the performance of the proposed method, numerical simulation with the R€ ossler chaotic system is presented below. In what follows, the fourth-order Runge–Kutta method is applied to solve the system of different equations with time step size equal to 0.001 in the numerical simulation. Suppose the master system is the R€ ossler chaotic system which is described by the following dynamical system [10]: x_ ¼ y z; y_ ¼ x þ ay;
ð18Þ
z_ ¼ b þ zðx cÞ; where a, b and c are system parameters. Obviously, it is not in the strict-feedback form. However, after a simple state transformation, i.e. letting u1 , u2 , u3 replace y, x, z respectively, system (18) can be rendered into the following desired strict-feedback form: u_ 1 ¼ u2 þ au1 ; u_ 2 ¼ u3 u1 ; u_ 3 ¼ b þ u3 ðu2 cÞ:
ð19Þ
The objective is to design such a scalar driving signal U that the controlled R€ ossler chaotic system y_ 1 ¼ y2 þ ay1 ; y_ 2 ¼ y3 y1 ; y_ 3 ¼ b þ y3 ðy2 cÞ þ U
ð20Þ
is synchronous with R€ ossler chaotic system (19). We can transform the system into the normal form following the procedure as above and then design the controller U by the pole assigning method. However, we can design a controller directly without transforming R€ ossler chaotic system into its normal form. First, let the error variables be e1 ¼ y1 u1 , e2 ¼ y2 u2 , e3 ¼ y3 u3 . Define the first variable z1 ¼ y1 u1 ¼ e1 : Its derivative along the solutions of system (19) and (20) is z_ 1 ¼ ae1 þ e2 :
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
931
We select the first partial Lyapunov function as 1 V1 ¼ z21 : 2 By simple calculation one can yield its time derivative along the solutions of system (19) and (20) as follows: V_1 ¼ z21 þ z1 ðða þ 1Þe1 þ e2 Þ:
ð21Þ
Now, we define the second variable z2 ¼ ða þ 1Þe1 þ e2 : With this notation, Eq. (21) can be rewritten as V_1 ¼ z21 þ z1 z2 : The second partial Lyapunov function is selected as 1 1 1 V2 ¼ V1 þ z22 ¼ z21 þ z22 : 2 2 2 Its time derivative along the solutions of system (19) and (20) is V_2 ¼ z21 z22 þ z2 ðða þ 1Þ2 e1 þ ða þ 2Þe2 e3 Þ:
ð22Þ
We define the third variable z3 ¼ ða þ 1Þ2 e1 þ ða þ 2Þe2 e3 : Thus, Eq. (22) can be transformed into V_2 ¼ z21 z22 þ z2 z3 :
ð23Þ
We form the Lyapunov function as 1 1 V ðtÞ ¼ V2 þ z23 ¼ ðz21 þ z22 þ z23 Þ: 2 2
ð24Þ
Its time derivative along the solutions of systems (19) and (20) is dV ðtÞ ¼ z21 z22 z23 þ z3 ðz2 þ z3 þ ða þ 1Þ2 e_ 1 þ ða þ 2Þ_e2 e_ 3 Þ dt ¼ z21 z22 z23 þ z3 ðða3 þ 3a2 þ 3aÞe1 þ ða2 þ 3a þ 4Þe2 ða þ 3Þe3 y3 ðy2 cÞ þ u3 ðu2 cÞ UÞ:
ð25Þ
Therefore, we have the following theorem: Theorem 4.1. If we design the scalar driving signal U as U ¼ ða3 þ 3a2 þ 3aÞe1 þ ða2 þ 3a þ 4Þe2 ða þ 3Þe3 y3 ðy2 cÞ þ u3 ðu2 cÞ;
ð26Þ
then the controlled R€ossler chaotic system (20) is globally asymptotically synchronous with R€ ossler chaotic system (19). Proof. Substituting (26) into (25), we have dV ðtÞ ¼ z21 z22 z23 ¼ 2V ðtÞ; dt from which we can yield V ðtÞ ¼ V ð0Þ expð2tÞ. This implies limt!þ1 zi ¼ 0 ði ¼ 1; 2; 3Þ, i.e. limt!þ1 ei ¼ 0 ði ¼ 1; 2; 3Þ. Thus, the controlled R€ ossler chaotic system (20) is globally asymptotically synchronous with R€ ossler chaotic system (19). h In the numerical simulation, the system parameters a ¼ b ¼ 15, c ¼ 5:7, with which the R€ ossler system behaves chaotically. The initial conditions are set to be u1 ð0Þ ¼ 3:0, u2 ð0Þ ¼ 4:0, u3 ð0Þ ¼ 5:0 and y1 ð0Þ ¼ 6:0, y2 ð0Þ ¼ 8:0, y3 ð0Þ ¼ 9:0. Fig. 1 presents the results of the numerical simulation. One can see that the synchronization of all variables is achieved successfully. Fig. 2 illustrates the control signal during the synchronization process.
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Fig. 1. Graph of synchronization errors: e1 ðtÞ ¼ y1 u1 ; e2 ðtÞ ¼ y2 u2 ; e3 ðtÞ ¼ y3 u3 :
Fig. 2. The control signal U during the chaos synchronization process.
5. Conclusion The strict-feedback chaotic system is studied. We prove that, any strict-feedback chaotic system can be rendered into a normal form with a invertible transform. A design procedure to synchronize the normal form of the strict-feedback chaotic system is presented. Furthermore, the R€ ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming the strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods.
Acknowledgements The work is supported by 973 Program of PR China (No. 2003CB415205) and the Opening Research Foundation of the Key Laboratory of Water and Sediment Science (Wuhan University), Ministry of Education, P.R China (No. 2003A001). The authors would like to thank the referees for their valuable comments.
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Pecora LM, Carroll TL. Synchronization in chaotic system. Phys Rev Lett 1990;64(8):821–4. Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; 1998. Vaneecek A, Celikovsk y S. Control systems: From linear analysis to synthesis of chaos. London: Prentice-Hall; 1996. Chen S, L€ u J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. Chen S, Liu J, Xie J, Lu J. Tracking control and synchronization of chaotic systems based upon sampled-data feedback. Chinese Phys 2002;11(3):233–7. Chen S, Yang Q, Wang CP. Impulsive control and synchronization of unified chaotic system. Chaos, Solitons & Fractals 2004;20:751–8. Hendrik R. Controlling chaotic systems with multiple strange attractors. Phys Lett A 2002;300:182–8. Sun JT. Some global synchronization criteria for coupled delay-systems via unidirectional linear error feedback approach. Chaos, Solitons & Fractals 2004;19:789–94. Kapitaniak T, Sekieta M, Ogorzalek M. Monotone synchronization of chaos. Int J Bifurcat Chaos 1996;6(1):211–7. R€ ossler OE. An equation for continous chaos. Phys Lett A 1976;7:397–8. Lorenz EN. Deterministic non-periodic flows. J Atmos Sci 1963;20:130–41. Ueta T, Chen G. Bifurcation analysis of Chen’s attractor. Int J Bifurcation and Chaos 2000;10:1917–31. Yassen MT. Adaptive control and synchronization of a modified Chua’s circuit system. Applied Mathematics and Computation 2003;135:113–28. Miguel S, Carlos A, Ricardo B. On recovering the parameters and velocity state of the Duffing’s oscillator. Phys Lett A 2003;308:47–53. Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12:1199–206. Wang C, Ge SS. Adaptive backstepping control of a class of chaotic systems. Int J Bifurcation Chaos 2001;11:1115–8. Decarlo RA. Linear systems. Englewood Cliffs, NJ: Prentice Hall; 1989.
Normal form and synchronization of strict-feedback chaotic systems Feng Wang a, Shihua Chen b
a,*
, Minghai Yu b, Changping Wang
c
a College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, PR China c Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada B3H 3J5
Accepted 9 March 2004
Abstract This study concerns the normal form and synchronization of strict-feedback chaotic systems. We prove that, any strict-feedback chaotic system can be rendered into a normal form with a invertible transform and then a design procedure to synchronize the normal form of a non-autonomous strict-feedback chaotic system is presented. This approach needs only a scalar driving signal to realize synchronization no matter how many dimensions the chaotic system contains. Furthermore, the R€ ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming a strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The idea of synchronizing two identical chaotic systems with different initial conditions was first introduced by Pecora and Carrol [1]. Since then, synchronization in coupled chaotic systems has become a rapidly developing field in light of its great potential applications in secure communications, chemical reaction, modelling brain activity and so on [2,3]. A large varieties of approaches have been proposed for chaos synchronization such as the adaptive synchronization method [4], the sampled-data feedback synchronization method [5], the impulsive synchronization method [6], and many others [7–9]. Moreover, in the study of synchronizing nonlinear dynamics, several low-dimensional systems are frequently used as benchmark examples for verification and validation of a proposed theory, method and algorithm. These examples include Duffing oscillator, Van der Pol oscillator, R€ ossler system, Chua’s circuit, Lorenz system and Chen system [10–14]. It is very interesting to note that, many existing synchronization methods need several controllers to realize synchronization and all those chaotic systems mentioned above can be rewritten into a class of nonlinear systems in a so-called general strict-feedback form. This motivates the present work, which is to address two important issues concerning the strict-feedback chaotic system. One issue is to find the normal form of strict-feedback chaotic systems, the other is to find a scalar driving signal to realize synchronization. We prove that, any strict-feedback chaotic system can be rendered into a normal form with a invertible transform. Furthermore, a design procedure to synchronize the normal form of a non-autonomous strict-feedback chaotic system is presented. This approach needs only a scalar driving signal to realize synchronization no matter how many dimensions the chaotic system contains. In addition, the R€ ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming a
*
Corresponding author. E-mail address: [email protected] (S. Chen).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.03.010
928
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods.
2. The normal form of strict-feedback chaotic systems Many chaotic systems used in the study of nonlinear dynamics can be rewritten into a class of nonlinear systems in the form x_ 1 ¼ f1 ðx1 ; tÞx2 þ g1 ðx1 ; tÞ; x_ 2 ¼ f2 ðx1 ; x2 ; tÞx3 þ g2 ðx1 ; x2 ; tÞ; ......... x_ n1 ¼ fn1 ðx1 ; x2 ; . . . ; xn1 ; tÞxn þ gn1 ðx1 ; x2 ; . . . ; xn1 ; tÞ; x_ n ¼ fn ðx1 ; x2 ; . . . ; xn1 ; xn ; tÞ;
ð1Þ
where fi , gi ði ¼ 1; 2; . . . ; n 1Þ are sufficiently smooth nonlinear functions and fn is a continuous nonlinear function. It is often referred as a general strict-feedback system. If fi 6¼ 0 ði ¼ 1; 2; . . . ; n 1Þ, it is called a strict-feedback system. General strict-feedback systems and strict-feedback systems have been thoroughly studied during the past decades. The main works concentrated on controlling these systems as well as tracking any one-dimensional smooth trajectory [15,16]. Our aim in this section, however, is to find the normal form of strict-feedback chaotic systems. To this end, we first define two new variables u1 ¼ x1 ; u2 ¼ f1 ðx1 ; tÞx2 þ g1 ðx1 ; tÞ , F1 ðx1 ; x2 ; tÞ:
ð2Þ
Simple calculation can yields 1 X du2 o dxi ¼ f1 ðx1 ; tÞf2 ðx1 ; x2 ; tÞx3 þ þ f1 ðx1 ; tÞg2 ðx1 ; x2 ; tÞ , F2 ðx1 ; x2 ; x3 ; tÞ; F1 ðx1 ; x2 ; tÞ dt dt ox i i¼0
where x0 stands for t and
dx0 dt
ð3Þ
¼ 1. With the above notation, the first two equations of (1) can be rewritten as
u_ 1 ¼ u2 ; u_ 2 ¼ F2 ðx1 ; x2 ; x3 ; tÞ:
ð4Þ
We define the third variable as u3 ¼ F2 ðx1 ; x2 ; x3 ; tÞ:
ð5Þ
One can get 3 2 2 X du3 Y o dxi Y ¼ þ fi ðx1 ; . . . ; xi ; tÞg3 ðx1 ; x2 ; x3 ; tÞ , F3 ðx1 ; x2 ; x3 ; x4 ; tÞ: fi ðx1 ; . . . ; xi ; tÞx4 þ F2 ðx1 ; x2 ; x3 ; tÞ dt dt oxi i¼1 i¼0 i¼1
ð6Þ
With this notation, the first three equations of (1) can be rewritten as u_ 1 ¼ u2 ; u_ 2 ¼ u3 ; u_ 3 ¼ F3 ðx1 ; x2 ; x3 ; x4 ; tÞ:
ð7Þ
Generally, the k-th (3 6 i 6 n 1) variable is defined as uk ¼ Fk1 ðx1 ; x2 ; . . . ; xk ; tÞ;
ð8Þ
its time derivative is k k1 X duk Y o dxi ¼ fi ðx1 ; . . . ; xi ; tÞxkþ1 þ Fk1 ðx1 ; . . . ; xk ; tÞ ox dt dt i i¼1 i¼0
þ
k 1 Y i¼1
fi ðx1 ; . . . ; xi ; tÞgk ðx1 ; . . . ; xk ; tÞ , Fk ðx1 ; . . . ; xkþ1 ; tÞ:
ð9Þ
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
929
Then the first k equations of (1) can be transformed into u_ 1 ¼ u2 ; u_ 2 ¼ u3 ; ......
ð10Þ
u_ k1 ¼ uk ; u_ k ¼ Fk ðx1 ; . . . ; xkþ1 ; tÞ: The last step is to define the n-th variable as un ¼ Fn1 ðx1 ; . . . ; xn ; tÞ:
ð11Þ
Its time derivative is n n1 X dun Y o dxi fi ðx1 ; . . . ; xi ; tÞxn þ Fn1 ðx1 ; . . . ; xn ; tÞ ¼ : ox dt dt i i¼1 i¼0
ð12Þ
On the other hand, from the definitions of ui ði ¼ 1; 2; . . . ; nÞ one can find that xi ði ¼ 1; 2; . . . ; nÞ could be expressed as functions of t and ui ði ¼ 1; 2; . . . ; nÞ. Thus, the equation (12) could be written in the form dun ¼ F ðu1 ; . . . ; un ; tÞ; dt
ð13Þ
from which we have the following theorem: Theorem 2.1. Any strict-feedback chaotic system can be transformed into the normal form as follows: u_ 1 ¼ u2 ; u_ 2 ¼ u3 ; ...... u_ n1 ¼ un ; u_ n ¼ F ðu1 ; . . . ; un ; tÞ:
ð14Þ
3. Synchronizing strict-feedback chaotic systems via a scalar driving signal In this section, we will propose a design procedure to synchronize the normal form of the non-autonomous strictfeedback chaotic system. We design such a scalar controller U that the controlled strict-feedback chaotic system in normal form y_ 1 ¼ y2 ; y_ 2 ¼ y3 ; ......
ð15Þ
y_ n1 ¼ yn ; y_ n ¼ F ðy1 ; . . . ; yn ; tÞ þ U is synchronous with (14), the normal form of system (1). To this end, let U ¼ F ðu1 ; . . . ; un ; tÞ F ðy1 ; . . . ; yn ; tÞ þ U1 ;
ð16Þ
where U1 is waiting for determination. With this selection, the error system of (15) and (14) can be expressed as e_ ¼ Ae þ BU1 ;
ð17Þ
930
F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933
where e ¼ ðe1 ; e2 ; . . . ; en Þ ¼ ðy1 u1 ; y2 u2 ; . . . ; yn un Þ> , B ¼ ð0; 0; . . . ; 1Þ> and 0 1 0 1 0 0 ... 0 B 0 0 1 0 ... 0 C B C A¼B . . . . . . . . . . .. ... ...C B C: @ 0 0 0 0 ... 1 A 0 0 0 0 ... 0 On the other hand, the matrix 0
½B
AB A2 B . . .
0 B 0 B An1 B ¼ B B... @ 0 1
0 0 ... 1 0
0 0 ... 0 0
... ... ... ... ...
0 1 ... 0 0
1 1 0 C C ...C C 0 A 0
is full rank, so the linear control theory [17] confirms that the single-input dynamic system (17) is controllable, i.e., all the eigenvalues are controllable. Thus, by the pole assigning method [17] we can select an appropriate feedback gain vector k ¼ ðk1 ; k2 ; . . . ; kn Þ> such that system (17) is globally asymptotically stable at zero with the state feedback control U1 ¼ ke, which implies that dynamical system (15) is synchronous with (14), the normal form of system (1).
4. Numerical simulation In order to demonstrate and verify the performance of the proposed method, numerical simulation with the R€ ossler chaotic system is presented below. In what follows, the fourth-order Runge–Kutta method is applied to solve the system of different equations with time step size equal to 0.001 in the numerical simulation. Suppose the master system is the R€ ossler chaotic system which is described by the following dynamical system [10]: x_ ¼ y z; y_ ¼ x þ ay;
ð18Þ
z_ ¼ b þ zðx cÞ; where a, b and c are system parameters. Obviously, it is not in the strict-feedback form. However, after a simple state transformation, i.e. letting u1 , u2 , u3 replace y, x, z respectively, system (18) can be rendered into the following desired strict-feedback form: u_ 1 ¼ u2 þ au1 ; u_ 2 ¼ u3 u1 ; u_ 3 ¼ b þ u3 ðu2 cÞ:
ð19Þ
The objective is to design such a scalar driving signal U that the controlled R€ ossler chaotic system y_ 1 ¼ y2 þ ay1 ; y_ 2 ¼ y3 y1 ; y_ 3 ¼ b þ y3 ðy2 cÞ þ U
ð20Þ
is synchronous with R€ ossler chaotic system (19). We can transform the system into the normal form following the procedure as above and then design the controller U by the pole assigning method. However, we can design a controller directly without transforming R€ ossler chaotic system into its normal form. First, let the error variables be e1 ¼ y1 u1 , e2 ¼ y2 u2 , e3 ¼ y3 u3 . Define the first variable z1 ¼ y1 u1 ¼ e1 : Its derivative along the solutions of system (19) and (20) is z_ 1 ¼ ae1 þ e2 :
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We select the first partial Lyapunov function as 1 V1 ¼ z21 : 2 By simple calculation one can yield its time derivative along the solutions of system (19) and (20) as follows: V_1 ¼ z21 þ z1 ðða þ 1Þe1 þ e2 Þ:
ð21Þ
Now, we define the second variable z2 ¼ ða þ 1Þe1 þ e2 : With this notation, Eq. (21) can be rewritten as V_1 ¼ z21 þ z1 z2 : The second partial Lyapunov function is selected as 1 1 1 V2 ¼ V1 þ z22 ¼ z21 þ z22 : 2 2 2 Its time derivative along the solutions of system (19) and (20) is V_2 ¼ z21 z22 þ z2 ðða þ 1Þ2 e1 þ ða þ 2Þe2 e3 Þ:
ð22Þ
We define the third variable z3 ¼ ða þ 1Þ2 e1 þ ða þ 2Þe2 e3 : Thus, Eq. (22) can be transformed into V_2 ¼ z21 z22 þ z2 z3 :
ð23Þ
We form the Lyapunov function as 1 1 V ðtÞ ¼ V2 þ z23 ¼ ðz21 þ z22 þ z23 Þ: 2 2
ð24Þ
Its time derivative along the solutions of systems (19) and (20) is dV ðtÞ ¼ z21 z22 z23 þ z3 ðz2 þ z3 þ ða þ 1Þ2 e_ 1 þ ða þ 2Þ_e2 e_ 3 Þ dt ¼ z21 z22 z23 þ z3 ðða3 þ 3a2 þ 3aÞe1 þ ða2 þ 3a þ 4Þe2 ða þ 3Þe3 y3 ðy2 cÞ þ u3 ðu2 cÞ UÞ:
ð25Þ
Therefore, we have the following theorem: Theorem 4.1. If we design the scalar driving signal U as U ¼ ða3 þ 3a2 þ 3aÞe1 þ ða2 þ 3a þ 4Þe2 ða þ 3Þe3 y3 ðy2 cÞ þ u3 ðu2 cÞ;
ð26Þ
then the controlled R€ossler chaotic system (20) is globally asymptotically synchronous with R€ ossler chaotic system (19). Proof. Substituting (26) into (25), we have dV ðtÞ ¼ z21 z22 z23 ¼ 2V ðtÞ; dt from which we can yield V ðtÞ ¼ V ð0Þ expð2tÞ. This implies limt!þ1 zi ¼ 0 ði ¼ 1; 2; 3Þ, i.e. limt!þ1 ei ¼ 0 ði ¼ 1; 2; 3Þ. Thus, the controlled R€ ossler chaotic system (20) is globally asymptotically synchronous with R€ ossler chaotic system (19). h In the numerical simulation, the system parameters a ¼ b ¼ 15, c ¼ 5:7, with which the R€ ossler system behaves chaotically. The initial conditions are set to be u1 ð0Þ ¼ 3:0, u2 ð0Þ ¼ 4:0, u3 ð0Þ ¼ 5:0 and y1 ð0Þ ¼ 6:0, y2 ð0Þ ¼ 8:0, y3 ð0Þ ¼ 9:0. Fig. 1 presents the results of the numerical simulation. One can see that the synchronization of all variables is achieved successfully. Fig. 2 illustrates the control signal during the synchronization process.
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Fig. 1. Graph of synchronization errors: e1 ðtÞ ¼ y1 u1 ; e2 ðtÞ ¼ y2 u2 ; e3 ðtÞ ¼ y3 u3 :
Fig. 2. The control signal U during the chaos synchronization process.
5. Conclusion The strict-feedback chaotic system is studied. We prove that, any strict-feedback chaotic system can be rendered into a normal form with a invertible transform. A design procedure to synchronize the normal form of the strict-feedback chaotic system is presented. Furthermore, the R€ ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming the strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods.
Acknowledgements The work is supported by 973 Program of PR China (No. 2003CB415205) and the Opening Research Foundation of the Key Laboratory of Water and Sediment Science (Wuhan University), Ministry of Education, P.R China (No. 2003A001). The authors would like to thank the referees for their valuable comments.
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