Impulsive synchronization of chaotic systems

Transcription

1 CHAOS 15, Impulsive synchronization of chaotic systems Chuandong Li a and Xiaofeng Liao College of Computer Science and Engineering, Chongqing University, China Xingyou Zhang College of Mathematics and Physics, Chongqing University, China and Institute of Fundamental Science, Massey University, Private Bag , Palmerston North, New Zealand Received 20 January 2004; accepted 9 March 2005; published online 27 April 2005 The issue of impulsive synchronization of a class of chaotic systems is investigated. Based on the impulsive theory and linear matrix inequality technique, some less conservative and easily verified criteria for impulsive synchronization of chaotic systems are derived. The proposed method is applied to the original Chua oscillators, and the corresponding synchronization conditions are obtained. Moreover, the boundary of the stable region is also estimated in terms of the equidistant impulse interval. The effectiveness of our method is shown by computer simulation American Institute of Physics. DOI: / Many synchronization schemes for chaotic systems have been reported in the literature (see Refs. 1 4 and 7 9, and the references therein). In these schemes (for drivingresponse systems), the continuous signals derived from the driving systems are almost always used to drive the response systems. However, the continuous signals transmitted along the channel may lead to several disadvantages such as, e.g., multichannel transmission and control complexity. Based on the impulsive control theory, 10 many researchers (e.g., Yang and Sun 15,16 ) have investigated the impulsive synchronization schemes and achieved the synchronization of several types of chaotic systems using only small control impulses. The strength of this scheme exists in the practicality in transmission and control because the synchronization signal is actually discrete. I. INTRODUCTION a Author to whom correspondence should be addressed. Electronic mail: licd@cqu.edu.cn, cd_licqu@163.com Over the last decade, synchronization of coupled chaotic system has attracted a great deal of attention due to its potential applications for secure communications 1 6 since the pioneering work of Pecora and Carroll. 7,8 Different regimes, namely, complete synchronization, phase synchronization, generalized synchronization, lag synchronization, and anticipative synchronization, etc., have been investigated for various chaotic systems such as, e.g., chaotic circuits, chaotic laser systems, pairs of neurons, chemical oscillators. A complete synchronization implies coincidence of states of interacting systems; a generalized synchronization is defined as the presence of some functional relation between the states of response and driver for drive response systems; phase synchronization means entrainment of phases of chaotic oscillators; lag synchronization appears as a coincidence of shifted-in-time states of two systems; and anticipative synchronization is that a dissipative chaotic system with a timedelayed feedback could drive an identical system in such a way that the driven system anticipates the driver by synchronizing with their future states. For more information about these chaotic systems and synchronization regimes, we refer the readers to the review monograph, 9 where the authors presented the main ideas involved in the field of synchronization of chaotic systems and relevant experimental applications in detail. Because impulsive control allows the stabilization and synchronization of chaotic systems using only small control impulses, it has been widely used to stabilize and synchronize chaotic systems, even though the chaotic behavior may follow unpredictable patterns. The main idea of impulsive control approach is to change the states of a system wherever some conditions are satisfied. These impulsively controlled systems are described by impulsive differential equations. Recently, several impulsive synchronization schemes have been reported in the literature. 10,15 Yang et al. 11 achieved the synchronization of two identical chaotic systems, i.e., Chua circuit, using the state variable at the fixed instant time as the impulsive signal. The authors investigated the impulsive synchronization criteria for coupled chaotic systems via unidirectional linear error feedback approach in Ref. 15. The stabilization and synchronization of Lorenz systems via impulsive signal are studied in Ref. 16. However, for many-dimension chaotic systems described by the ODE or DDE ordinary or delayed differential equations, the less conservative synchronization criteria such as, e.g., Theorem 2 in Ref. 14, are generally difficult to verify. Motivated by the aforementioned comments, we further investigate the impulsive synchronization of chaotic systems in this paper. Based on impulsive control theory and linear matrix inequality technique, some less conservative and easily verified criteria for impulsive synchronization of chaotic systems are derived. With the help of the LMI toolbox in MATLAB, we obtain the synchronization conditions conveniently, and consequently estimate the stable region of synchronized systems. The proposed method is also applied to /2005/152/023104/5/$ , American Institute of Physics

2 Li, Liao, and Zhang Chaos 15, the original Chua oscillators to illustrate the effectiveness. The rest of the paper is organized as follows. In the next section, the preliminaries relevant to the theory of impulsive control are presented. Some new synchronization criteria are obtained in Sec. III. The proposed method is applied to the original Chua oscillators in Sec. IV. Finally, conclusions are given in Sec. V. II. PRELIMINARIES Consider the general nonlinear system described by ẋt = ft,x, where f :R + R n R n is continuous, xr n is the state variable, and ẋ=dx/dt. Suppose that a discrete set i of time instants satisfies Let 0 t i i+1, i as i. Ui,x = x t=i x i + x i be the jump in the state variable at the time instant i, where x + i = lim xt and x i = lim xt. In general, for simplicity, it is assumed that x i =x i. Then, an impulsively + t i t i controlled system with the initial condition in time t 0 can be described by ẋ = ft,x, t i, x = Uix, t = i, xt 0 = x 0, t 0 =0,i = 1,2,.... This is also called an impulsive differential equation. 10 Several approaches, namely, the comparison-system approach, Lyapunov-type approach, and two-measure approach, etc., have been established to study the stability of Eq. 3. Furthermore, we assume that Eq. 3 satisfies ft,0=0 and Ui,0=0 for all i. To make the present paper self-contained, we restate some definitions and results involved in the comparison-system approach. Definition 1. Ref. 10 Let V:R + R n R + ; then, V is said to belong to class if 1 V is continuous in i 1, i R n and for each xr n,i =1,2,..., lim Vt,y = V + i,x exists. + t,y i,x 2 V is locally Lipschitzian in x. From this definition, we can see that V associated with impulsive system 3 is the analog of Lyapunov function for stability analysis of ODE. Because these Lyapunov-type functions are generally discontinuous, a generalized derivative should be defined, which is known as the right and upper Dini s derivative. Definition 2. For t,x i 1, i R n, the right and upper Dini s derivative of V with respect to time variable is defined as D + Vt,x lim h 0 + sup 1 h Vt + h,x + hft,x Vt,x. 4 To derive the asymptotical stability condition of system 3, the comparison approach is used. The main idea of this approach consists of constructing a comparison system such that its stability implies the stability of the considered system, and then deriving stability conditions of the considered system through this comparison system. However, it is not easy generally to construct an appropriate comparison system of the given impulsive system. Here, we restate directly a comparison system proposed in Ref. 10 for impulsive system 3 and the relevant results. 10,13 Definition 3. Ref. 10 Let V and assume that D + Vt,x gt,vt,x, t i, Vt,x + Ui,x i Vt,x, t = i, where g:r + R + R is continuous and gt,0=0, i :R + R + is nondecreasing. Then, the system = gt,, t i, i + = i i, t 0 = 0, is called the comparison system of system 3. Definition 4. S = x R n x, where denotes the Euclidean norm on R n. Definition 5. A function is said to belong to class if CR +,R +, 0=0, and x is strictly increasing in x. Theorem 1. Ref. 10. Assume that the following three conditions are satisfied: 1 V:R + S R +, 0, V, D + Vt,xgt,Vt,x, t i. 2 There exists a 0 0 such that xs 0 implies that x +Ui,xS for all i and Vt,x+Ui,x i Vt,x, t= i, xs 0. 3 xvt,xx on R + S, where.,.. Then, the stability properties of the trivial solution of the comparison system 6 imply the corresponding stability properties of the trivial solution of 3. Remark 1. This theorem plays an important role for investigating the convergence dynamics of system (3). Specifically, we can obtain some sufficient conditions for the stability of complex impulsive system (3) in virtue of the simple comparison system. The following asymptotical stability result, which was based on the above theorem, was reported in Ref. 13. Theorem 2. Ref. 13. Let gt,= t, C 1 R +,R +, i =d i, d i 0, i=1,2,....then, the origin 5 6 7

3 Impulsive Synchronization Chaos 15, of system 3 is asymptotically stable if the following conditions hold: 1 sup i d i exp i+1 i = 0 ; 8 2 there exists a 1 such that 2i+3 +lnd 2i+2 d 2i+1 2i+1 holds for all d 2i+3 d 2i+1 0, i=1,2,...; 3 t0; 4 there exists. and. in class such that x Vt,xx. III. CHAOS SYNCHRONIZATION VIA THE IMPULSES Consider a class of chaotic systems, which are described by the following ODE: ẋ = Ax + fx + u, 10 y = Cx, where xr n is the state variable, yr denotes the output of the system. AR nn, CR mn,ur n, and f :R n R n is a nonlinear function satisfying the following condition: There exists a positive scalar L such that for any x,yr n fx fy Lx y In order to synchronize system 10 called the driving system via the impulses, i.e., y=cx at instant time i, another system called the driven or response system is designed as x = Ax + fx + u, t i, x Bx x = PCx x, t = i,, 12 ỹ = Cx. where PR nm denotes the control gain. Let the error between the states of system 10 and 12 be e=x x. Then, we can easily obtain the following impulsive error system: ė = Ae + x,x, t i, 13 e = Be, t = i, where x,x = fx fx and B= PCb ij nn. Observation 1. From the analysis above, it follows that it is sufficient for synchronizing chaos that the origin of 13 is asymptotically stable. It is worth noting that the origin is one of the equilibria of system 13. Also, the origin is the unique equilibrium of system 13 because e=be implies e + i e i unless e + i =e i =0. Remark 2. Regarding the synchronizing signal, it is worth noting that y=cx is an artificial output of system (10), and it can be properly designed to feed the driven system (12). In the example illustrated in the next section, we let the matrix C be an nn diagonal matrix with the positive entries. Given C, our goal is to seek the control gain P such that the origin of (13) is asymptotically stable. For the stability of the origin of error system 13, we have the following results. Theorem 3. Let the impulses be equidistant and separated by interval. Suppose that there exists a symmetric and positive definite matrix QR nn, and constant scalars 0, k0, d0,1 such that 1 1 =A T Q+QA+L 2 I+1 Q 2 kq0, 2 2 =I+ PC T QI+ PC dq0, 3 k+lnd0, where I denotes the identity matrix. Then, the origin of the error system 13 is asymptotically stable. Proof. For t i, we construct a Lyapunov functional as follows: Vt,e = e T Qe, t i, 14 where Q is a symmetric and positive definite matrix, denoted as Q0. Let e= m Q and e= M Q, where m and M denote the smallest and largest eigenvalues of a square matrix, respectively. Then,.,.. Moreover 14 implies that evt,ex. The derivative of Vt,e along the solution of 13 is D + Vt,e = e T A T Q + QAe + T Qe + e T Q e T A T Q + QAe + T + 1 et Q T Qe e T A T Q + QA + L 2 I + 1 Q2 kqe + kvt,ee T 1 e + kvt,e kvt,e, where 1 =A T Q+QA+L 2 I+1/Q 2 kq0, and 0. For t= i, we have V i,e + PCe = e T I + PC T QI + PCe = e T I + PC T QI + PC dqe + dv i,e e T 2 e + dv i,e dv i,e where 2 =I+ PC T QI+ PC dq0. Let t=k, d i =d, i=1,2,.... Then, it follows from Theorem 2 that the origin of the error system 13 is asymptotically stable. Remark 3. We do not require that the matrix PC is symmetric. Moreover, we do not require that I+ PC1. Thus, our result can be used for a wider class of nonlinear systems as compared to Refs. 11 and 12. Remark 4. Let =1/L m Q M Q and A T Q+QA 1 Qor 1 = M Q 1 A T Q+QA. Then, we have A T Q + QA + L 2 I + 1 Q2 kq = QQ 1 A T Q + QA ki + L 2 Q Q 1 +2L MQ m Q kq. This implies that 1 0 holds if k = 1 +2L M Q/ m Q. Therefore, there exists k 1 +2L M Q/ m Q

4 Li, Liao, and Zhang Chaos 15, FIG. 3. Color online. Time response of the synchronization error system with K= 1.5, =0.01. FIG. 1. Color online. The double-scroll attractor of the origin Chua s oscillator. such that 1 0. Notice that from condition (3) in this theorem, the smaller k will result in a larger stable region. Hence, as compared to Refs , the stable region calculated using our criterion is larger, at least not smaller, than those using the criteria in Refs Remark 5. Notice that condition (1) can be rewritten as ATQ + QA + L2I ki 1 Q Q I This is a linear matrix inequality (LMI) with respect to Q, and can be solved quickly by interior-point algorithms. Moreover, to seek for the approximate largest stable region, we can calculate the following optimizing problem: max = lnd k s.t. 16 LMI 15 holds. condition 3) holds. FIG. 2. Color online. Estimate of the boundaries of stable region with different s used in 22. Let Q=I and =1 in Theorem 3; then, the following corollary holds. Corollary 1. Assume that there exist constant scalars K, k0, d0, 0, 1 such that 1 1 =A T +A+L 2 +1 ki0; 2 2 =I+ PC T I+ PC di0; 3 k+lnd0. Then, the origin of the error system 13 is asymptotically stable. IV. EXAMPLE: IMPULSIVE SYNCHRONIZATION OF CHUA CIRCUITS In this section, we study the impulsive synchronization of the original Chua s oscillators 17 by applying the theory presented in the previous section. The original and dimensionless form of a Chua s oscillator is given by 17 ẋ 1 = x 2 x 1 gx 1, ẋ 2 = x 1 x 2 + x 3, ẋ 3 = x 2, 17 where, are parameters and gx is the piecewise-linear characteristics of the Chua s diode, which is given by gx 1 = bx a bx 1 +1 x 1 1, 18 where ab0 are two constants. Let x T =x 1,x 2,x 3. Then, we can rewrite the Chua s oscillator equation 17 in the form where ẋ = Ax + fx, = b 0 0 A = 0.5a bx 1 +1 x 1 1 and fx In the following simulation, we choose the parameters of system 17 as =9.2156, = , a= , b = A fourth-order Runge Kutta method with step size 10 5 is used. The initial conditions for driving and driven systems are given by x0=0.15,0.1,0.2 T and x 0

5 Impulsive Synchronization Chaos 15, K= 1.5, =0.01 and for K= 1.8, =0.001, respectively. Figure 5 shows the unstable results outside the stable region for K= 1.5 and =0.1. FIG. 4. Color online. Time response of the synchronization error system with K= 1.8, = =0.5,0.3, 0.5 T, respectively. The phase diagram of the driving system is shown in Fig. 1, which is the Chua s double-scroll attractor. We choose that C = I, and P = KI, where K is the control gain. It follows from Theorem 3 that 2K0. In this case, it is easy to see that d = K From condition 1 in Theorem 3, i.e., LMI 15, we find the approximate smallest value of control gain, k= In fact, if we choose that =1, k= , then the LMI (15) has the feasible solution = Q with eigenvalues eigenq= T. Then, an estimate of the boundaries of the stable region is given by ln +lnk +12 0, 2 K Figure 2 shows the stable region for different s. The entire region below the curve corresponding to =1 is the predicted stable region. When, the stable region shrinks to a line K= 1. Figures 3 5 show the time response curves of synchronization error systems. Figure 3 and Figure 4 show the stable results within the stable region for FIG. 5. Color online. Time response of the synchronization error system with K= 1.5,=0.1. V. CONCLUSIONS In this paper, some impulsive synchronization criteria for a class of chaotic systems have been presented and a proposed impulsive scheme has been applied to the original Chua s oscillators. The main advantage of our results exists in the convenience of determining the control gain from the synchronization criterion. The effectiveness of the suggested method has been shown by the computer simulations. ACKNOWLEDGMENTS The authors are grateful to the editors and anonymous reviewers for their valuable suggestions. The work described in this paper was partially supported by the National Natural Science Foundation of China Grant No , the Doctorate Foundation of the Ministry of Education of China Grant No , and the Natural Science Foundation of Chongqing, China. 1 S. Hayes, C. Grebogi, and E. Ott, Communicating with chaos, Phys. Rev. Lett. 70, L. Kocarev and U. Parlitz, General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett. 74, N. J. Corron and D. W. Hahs, A new approach to communications using chaotic signals, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 40, T. Yang and L. O. Chua, Secure communication via chaotic parameter modulation, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 43, T. Yang, C. Wu, and L. Chua, Cryptography based on chaotic systems, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64, L. M. Pecora and T. L. Carroll, Driving systems with chaotic signals, Phys. Rev. A 44, S. Boccaletti, J. Kurths, G. Osipov, D. L. Vallares, and C. S. Zhou, The synchronization of chaotic systems, Phys. Rep. 366, T. Yang, Impulsive Control Theory Springer, Berlin, T. Yang and L. O.Chua, Impulsive stability for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, T. Yang, Impulsive control, IEEE Trans. Autom. Control 44, Z. G. Li, C. Y. Wen, and Y. C. Soh, Analysis and design of impulsive control systems, IEEE Trans. Autom. Control 46, J. T. Sun, Y. P.Zhang, and Q. D. Wu, Less conservative conditions for asymptotic stability of impulsive control systems, IEEE Trans. Autom. Control 48, J. T. Sun and Y. P. Zhang et al., Some simple impulsive synchronization criterion for coupled chaotic systems via unidirectional linear error feedback approach, Chaos, Solitons Fractals 19, J. T. Sun, Impulsive control for the stabilization and synchronization of Lorenz systems, Phys. Lett. A 298, L. P. Shil nikov, Chua s circuit: Rigorous results and future problems, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4,