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Impulsive control of projective synchronization in chaotic systems
Physics Letters A 372 (2008) 3228–3233 www.elsevier.com/locate/pla
Impulsive control of projective synchronization in chaotic systems Manfeng Hu a,∗ , Yongqing Yang a,b , Zhenyuan Xu a a School of Science, Jiangnan University, Wuxi 214122, China b School of Automation, Southeast University, Nanjing 210096, China
Received 10 December 2007; received in revised form 16 January 2008; accepted 28 January 2008 Available online 2 February 2008 Communicated by A.R. Bishop
Abstract Scaling factor of projective synchronization in coupled partially linear chaotic systems is hardly predictable. To control projective synchronization of chaotic systems in a preferred way, an impulsive control scheme is introduced to direct the scaling factor onto a desired value. The control approach is derived from the impulsive differential equation theory. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results. Furthermore, some interesting and surprising numerical results are discussed. © 2008 Published by Elsevier B.V. PACS: 05.45.Xt Keywords: Projective synchronization; Impulsive differential equation; Chaos; Partially linear system
1. Introduction Chaotic synchronization has been a subject of intense study and is considered to be a fundamental mechanism behind a variety of behaviors in nature during the last decade [1–4]. The original synchronization technique was developed by Pecora and Carroll [1]. In their seminal paper, they addressed the synchronization of chaotic systems in the drive-response coupling scheme. A chaotic system, called the driver (or master), generates a signal which is sent over a channel to a responser (or slaver), which uses this signal to synchronize itself with the master. The signal is usually one of the coordinates of the drive chaotic system. For example, consider the often studied example of synchronization, the Lorenz system x˙ = 10(y − x), y˙ = (28 − z)x − y, z˙ = xy − 8/3z. It is well known that the Lorenz system exhibits identical synchronization in the x coordinate, synchronization in the y coordinate, while z is not a synchronizing coordinate at all [5] (in fact, the authors of Ref. [6] showed that the Lorenz system can exhibit identical synchronization by * Corresponding author. Tel.: +86 510 85910837 (work); fax: +86 510 85913660. E-mail address: [email protected] (M. Hu).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.01.054
selectively using only parts of driving signals z). The authors of [7] have showed that the Lorenz system exhibits projective synchronization (PS), characterized by a scaling factor α, provided the whole signals z is used to drive the response system. In [8], the author has showed that it is hard to estimate the scaling factor because it is dependent on not only the initial condition but also the underlying chaotic dynamics. From the application point of view, it is necessary to control the system in order to offer the occasion to select and direct the solution of synchronization in a defined way. In [8–10], the authors introduced three types feedback controller to the drive system to conduct the scaling factor onto a desired value, respectively. In [11], the authors used a simple and physically available controller proposed firstly in [12] to direct the scaling factor onto a desired value. In [13], the authors further extend the projective synchronization feature to general nonlinear systems rather than partially linear systems by using a controller to the response system under the consideration for potential application of secure communications [14,15]. The authors [16–18] also extend the concept of PS to the one of full state hybrid projective synchronization (FSHPS) in two different chaotic systems (even in two different order chaotic systems). Some recent results regarding the issue of PS have reported in [19–24].
M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
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 [25–37]. The impulsive control technique is also suitable to deal with systems which cannot endure continuous disturbance. Using this method the response system receives the information from the drive system only in discrete times and the amount of conveyed information is, therefore, decreased. However, this is suitable in practice because of reduced control cost. Motivated by the aforementioned comments, the main aim of this Letter is to further study the PS in partially linear chaotic systems coupled through the z variable by adopting the impulsive control method to direct the scaling factor onto a desired value. The organization of the remaining part is as follows. In Section 2, the preliminaries relevant to the theory of impulsive control are presented. Some PS criteria are obtained in Section 3. Results of simulation on Lorenz system are given in Section 4. Finally, conclusions are drawn in Section 5. 2. Basic theory of impulsive differential equations To make the present Letter self-contained, we address some basic results in this section which is useful in this Letter. These results can be found in [26,29,31,32]. Consider the general nonlinear system described by x˙ = f (t, x),
(1)
n where x ∈ R n is the state variable, x˙ ≡ dx dt and f : R+ × R → n R is continuous. Suppose that a discrete set {tk } of time instants satisfies
0 t0 < t1 < · · · < tk < tk+1 < · · · ,
tk → ∞ as
k → ∞.
Ik (x) = δx|t=tk ≡ x tk+ − x tk−
(2)
be the “jump” in the state variable at time instant tk , where x(tk+ ) = limt→t + x(t), x(tk− ) = limt→t − x(t). In general, for k
simplicity, it is assumed that x(tk− ) = x(tk ). Then, the impulsive control system with the initial condition in time t0 can be described by x˙ = f (t, x),
t = tk ,
δx = Ik (x),
t = tk ,
Definition 2. For (t, x) ∈ (tk−1 , tk ] × R n , the right and upper Dini’s derivative of V ∈ V0 is defined as D + V (t, x) ≡ lim sup h→0+
1 V t + h, x + hf (t, x) − V (t, x) . h
(4)
Since the impulsive control system in Eq. (3) is an nthorder impulsive differential equation, instead of investigating the stability of Eq. (3), it is convenient to investigate that of a first-order impulsive differential equation which is given by the following comparison system. Definition 3. Let V ∈ V0 and assume that D + V (t, x) g t, V (t, x) , t = tk , V t, x + Ik (x) Ψk V (t, x) , t = tk ,
(5)
where g : R+ × R+ → R is continuous and g(t, 0) = 0, Ψk : R+ → R+ is nondecreasing. Then, the system w˙ = g(t, w), t = tk , w tk+ = Ψk w(tk ) , t = tk , w t0+ = w0 0
(6)
is the comparison system of (3). Definition 4. Sρ = x ∈ R n : x < ρ , where · denotes the Euclidean norm on
(7) Rn .
Definition 5. A function a is said to belong to class κ if a ∈ C[R+ , R+ ], a(0) = 0 and a(x) is strictly increasing in x. Theorem 1. Assume that the following three conditions are satisfied:
Let
k
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k = 1, 2, 3, . . . ,
x(t0 ) = x0 .
(3)
Assumptions. f (t, 0) = 0 and Ik (0) = 0 for all k. Definition 1. Let V : R+ × R n → R+ , then V is said to belong to class V0 if (1) V is continuous in (tk−1 , tk ] × R n , and for each x ∈ R n , k = 1, 2, 3, . . . , lim(t,y)→(t + ,x) V (t, y) = V (tk+ , x) exists; k (2) V is locally Lipschitzian in x.
(1) V : R+ × Sρ → R+ , ρ > 0, V ∈ V0 , D + V (t, x) g(t, V (t, x)), t = tk . (2) There exists a ρ0 > 0 such that x ∈ Sρ0 implies that x + Ik (x) ∈ Sρ for all k and V (t, x + Ik (x)) Ψk (V (t, x)), t = tk , x ∈ Sρ0 . (3) b(x) V (t, x) a(x) on R+ × Sρ , where a(·), b(·) ∈ κ. Then the stability properties of the trivial solution of the comparison system (6) imply the corresponding stability properties of the trivial solution of (3). ˙ Theorem 2. Let g(t, ω) = λ(t)ω, λ ∈ C 1 [R+ , R+ ], Ψk (ω) = dk ω, dk 0, k = 1, 2, . . . . Then the origin of system (3) is asymptotically stable if the following conditions hold: (1) supk {dk exp[λ(tk+1 ) − λ(tk )]} = ε0 < ∞; (2) there exists a γ > 1 such that λ(t2k+3 ) + ln(γ d2k+2 × d2k+1 ) λ(t2k+1 ) holds for all d2k+2 d2k+1 = 0, k = 1, 2, . . . ;
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M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
(3) λ˙ (t) 0; (4) there exist a(·) and b(·) in class κ such that b(x) V (t, x) a(x). 3. Projective synchronization via the impulsive control Partially linear system, such as Lorenz system, Lü system, hyper-chaotic Lü system, is defined by a set of ordinary differential equations in which the state vector can be broken into two parts (u, z), where the equation for z is nonlinearly related to the other variable, while the equation for the rate of charge of the vector u is linearly related to u through a matrix M(z) that can be depend on the variable z: ˙ = M(z) · u, u(t) z˙ = f (u, z).
(8)
δe = Be = Ik (e),
t = tk .
(12)
We have translated the problem of directing the scaling factor α onto the desired value of (11) into the stability problem of origin of the error system (12). Theorem 3. Let the impulses be equidistant and separated by interval . Q ∈ R n×n is a symmetric and positive definite matrix. Constant r > 0 is an upper bound of the maximum eigenvalue of MT (z)Q + QM(z). Suppose that there exist constant scalars d 0, γ > 1 such that (i) Ω1 = (I + B)T Q(I + B) − dQ 0, (ii) λmin2r(Q) + ln(γ d 2 ) 0, where I denotes the identity matrix. Then the origin of the error system (12) is asymptotically stable, i.e., system (11) realizes projective synchronization with desired scaling factor α.
A general condition for the occurrence of projective synchronization has been reported [38] in two arbitrary dimensional chaotic systems (8) coupled through the variable z:
Proof. Let us construct a Lyapunov function
u˙ d = M(z) · ud ,
V (t, e) = eT Qe,
z˙ = f (ud , z),
where Q is a symmetric and positive definite matrix. Let b(e) = λmin (Q)e2 and a(e) = λmax (Q)e2 , where λmin and λmax denote the smallest and largest eigenvalues of a square matrix, respectively. Then a(·) and b(·) belong to κ. Moreover (13) implies that b(e) V (t, e) a(e). For t = tk , the derivative of V (t, e) along the solution of (12) is D + V (t, e) = eT MT (z)Q + QM(z) e r r (14) eT Qe = V (t, e). λmin (Q) λmin (Q)
u˙ r = M(z) · ur
(9)
the subscripts d and r stand for the drive system and response system respectively, ud = (u1d , u2d , . . . , und )T , ur = (u1r , u2r , . . . , unr )T ∈ R n , z ∈ R 1 is a one-dimensional coupling variable, which is the same in both the drive and the response systems. The matrix M(z) is only dependent on the variable z that is nonlinearly related to the variables in ud : ⎤ ⎡ m11 (z) m12 (z) · · · m1n (z) ⎢ m21 (z) m22 (z) · · · m2n (z) ⎥ . M(z) = ⎢ (10) .. .. ⎥ .. ⎣ ... . . . ⎦ mn1 (z)
mn2 (z)
···
mnn (z)
We shall use the abbreviation M, mij to denote M(z) and mij (z) respectively, for simplicity in the later discussion. If there exists a constant α (α = 0) such that limt→∞ ur − αud = 0, then the projective synchronization between the drive system and response systems is achieved, and we call α as “scaling factor”. It is well known that the coupled systems (9) can achieve projective synchronization, but the scaling factor α is unpredictable. For directing the scaling factor onto the desired value α, we introduce the impulses at instant time tk in response system of (9): u˙ d = M(z) · ud , z˙ = f (ud , z), u˙ r = M(z) · ur ,
t = tk ,
δur = B(ur − αud ),
t = tk .
(11)
(13)
For t = tk , we have V (tk , e + Be) = eT (I + B)T Q(I + B)e = eT (I + B)T Q(I + B) − dQ e + dV (tk , e) = eT Ω1 e + dV (tk , e) dV (tk , e). ˙ = Let λ(t)
r λmin (Q) ,
(15)
dk = d, k = 1, 2, . . . . Then, it follows
from Theorem 2 that the origin of the error system (12) is asymptotically stable. This completes the proof. 2 Let Q = I in Theorem 3, the following corollary holds. Corollary 1. Let constant r > 0 be an upper bound of the maximum eigenvalue of MT (z) + M(z). Assume that there exist constants d 0, γ > 1 such that (i) Ω1 = (I + B)T (I + B) − dI 0, (ii) 2r + ln(γ d 2 ) 0,
Let e = (e1 , e2 , . . . , en )T , ei = uir − αuid , i = 1, 2, . . . , n denote the projective synchronization errors of (11), then we have
then the origin of the error system (12) is asymptotically stable.
e˙ = M(z)e,
Remark 1. In Refs. [8–10] and [11], the authors have studied the issue of controlling projective synchronization in coupled partially linear systems by using continuous synchronization
z˙ = f (ud , z),
t = tk ,
M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
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schemes, in which driving signals are transmitted continuously to the response systems such that the PS with desired scaling factors can be realized. Furthermore, in this Letter, we have considered the same issue by using the impulsive synchronization scheme, in which only samples of state variables are used to realize PS. The proposed impulsive projective synchronization scheme is more practicable than the existed ones [8–11] in application because of the reduced control cost. 4. Example and simulations In this section, one example is given for illustration the above impulsive projective synchronization criteria. Meanwhile, some interesting and surprising numerical results are also discussed. Example. Lorenz system is described by (a)
x˙ = σ (y − x), y˙ = (μ − z)x − y, z˙ = xy − ρz
(16)
with three real parameters σ = 10, μ = 28, ρ = 8/3. Taking −σ σ z as the coupling variable, we have M(z) = μ−z −1 in the form of (9) which satisfying −σ − 1 = −11 < 0. According to the criterion derived in [38], projective synchronization occurs in the coupled Lorenz systems but the scaling factor is hardly predictable. To manipulate the scaling factor onto the desired value, we use the impulsive control method in the form of (11), i.e., X˙ = M(z) · X, z˙ = xy − ρz, ˙˜ = M(z) · X, ˜ X
t = tk , ˜ δ X = Be = Ik (e), t = tk , (x, y)T , X˜
(17)
= e = X˜ − αX, α is a desired where X = value. −σ σ For M(z) = μ−z −1 , after a simple calculation, we have λ1,2 MT (z) + M(z) = λ1,2 (z) = −(1 + σ ) ±
(x, ˜ y) ˜ T,
(b) Fig. 1. (a) z vs t , (b) λ1,2 vs z.
boundaries of the stable region is given by ln γ + 2 ln(k + 1)2 , −2 k 0. (20) 2r Fig. 2 shows the stable region for PS with different γ . The whole region under the curve corresponding to γ = 1 is the predicted stable region. The stable region shrinks to a line k = −1 when γ → +∞. Fig. 3 shows the stable results within the stable region for k = −1.1 and = 0.15. The time response of desired scaling factors plotted in Fig. 4 shows that the aim of the control is realized. It is worthy noting that the condition of impulsive synchronization is only a conservative sufficient condition. In other words, the true stable region in actual application is larger than that predicted in Fig. 2. It can be found that the length of impulsive interval can be arbitrarily selected in many numerical 0 −
(1 − σ )2 + (σ + μ − z)2 .
(18)
In simulations, without of generalization, we pick (x(0), y(0), z(0), x(0), ˜ y(0)) ˜ = (1, 2, 3, 2, 1). From Fig. 1(a), we can see that the bounded variable z varies in (0, 50). Then we get the relation between λ1,2 (z) and z (see Fig. 1(b)). So we can select r = 23.2053 (λmax = 23.2053 as z = 5) such that the condition λ1,2 (z) r is satisfied. We choose B as k 0 B= . (19) 0 k It is easy to see that if d (k + 1)2 0, the condition (1) of Corollary 1 is satisfied. When d = (k + 1)2 , an estimate of the
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M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
Fig. 2. Estimate of the boundaries of stable regions for PS with different γ used in (20).
Fig. 3. Time response of the PS error, initial value x(0), y(0), z(0), x(0), ˜ y(0) ˜ = (1, 2, 3, 2, 1), integration step = 0.01, impulsive interval = 0.15.
Fig. 4. Time response of the scaling factor with impulsive interval = 0.15 and feedback gain k = −1.1.
Fig. 5. Time response of the scaling factor with impulsive interval = 1 and feedback gain k = −1.1.
Fig. 6. Time response of the scaling factor with impulsive interval = 1 and feedback gain k = −1.
experiments done on Lorenz system. For instance, when taking = 1, the other conditions are same as those in Figs. 3 and 4, the PS with desired scaling factors α = −3 can also be realized (see Fig. 5). An attempt has been made here to explore the reasons why so interesting things can happen. On the one hand, the coupled variable z is the main cause to realize PS in coupled partially linear systems, however, the ultimate state of the synchronization is hard to predict. On the other hand, impulsive signals, derived from samples of the state variables of the drive and response systems with desired constant α, play a key role to direct the synchronization behavior to the desired one. From Figs. 4 and 5, one can easily see that each impulses signals added to the response system direct the scaling factor to the desired one. These numerical simulations indicate that any width of impulsive interval can be used to implement PS with desired scaling factor in coupled partially linear systems. The freedom to choose the discrete time instants with impulsive signals therefore leads to a large flexibility in applications, especially in chaos communication.
M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
There exists, in addition, a surprising result. In numerical simulations, taking feedback gain k = −1, the other conditions are same as those in Figs. 3 and 4, the PS with desired scaling factors α = −3 can be realized after only one impulse effects (see Fig. 6). But, from the viewpoint of mathematics, the feedback gain k cannot take −1 as shown in Eq. (20) or Fig. 2. In summary, new theory of impulsive differential equation is needed for deeply understanding these two experiments results. 5. Conclusions In the present Letter, we have investigated the issue on directing the scaling factor of PS in coupled partially linear systems onto desired values via an impulsive control method. Some theoretic conditions are obtained to guarantee that the PS with desired value can be realized via impulses. The main advantage of our results is that the width of impulsive interval can be much more bigger, which will be easily applied to real situations. Acknowledgements The authors thank the anonymous reviewers for their helpful comments and suggestions. This work was jointly supported by Postdoctoral Science Foundation under Grant 20070420958, Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 0702001B and Pre-research Funds of Jiangnan University. References [1] L.M. Perora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [2] G. Chen, X. Dong, From Chaos to Order: Perspective, Methodologies and Applications, World Scientific, Singapore, 1998. [3] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, Phys. Rep. 366 (2002) 1.
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[4] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Science, Cambridge Univ. Press, Cambridge, 2001. [5] C. Tresser, P.A. Worfolk, H. Bass, Chaos 5 (1995) 693. [6] M. Itoh, T. Yang, L.O. Chua, Int. J. Bifur. Chaos 11 (2001) 551. [7] R. Mainieri, J. Rehacek, Phys. Rev. Lett. 82 (1999) 3042. [8] D. Xu, Phys. Rev. E 63 (2001) 27201. [9] D. Xu, Z. Li, R. Bishop, Chaos 11 (2001) 439. [10] D. Xu, C.Y. Chee, Phys. Rev. E 66 (2002) 046218. [11] M.F. Hu, Z.Y. Xu, Nonlinear Anal.: Real Word Appl. (2007), doi:10.1016/ j.nonrwa.2007.03.005. [12] D. Huang, Phys. Rev. Lett. 93 (2004) 214101. [13] D. Xu, Z. Li, Int. J. Bifur. Chaos 12 (2002) 1395. [14] J.A. Lu, X.Q. Wu, J.H. Lü, Phys. Lett. A 305 (2002) 365. [15] Y.W. Liu, G.M. Ge, H. Zhao, Y.H. Wang, Phys. Rev. E 62 (2000) 7898. [16] M.F. Hu, Z.Y. Xu, R. Zhang, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 456. [17] M.F. Hu, Z.Y. Xu, R. Zhang, A.H. Hu, Phys. Lett. A 361 (2007) 231. [18] M.F. Hu, Z.Y. Xu, R. Zhang, A.H. Hu, Phys. Lett. A 365 (2007) 315. [19] Q. Jia, Phys. Lett. A 370 (2007) 40. [20] Y. Tang, J.A. Fang, Phys. Lett. A 372 (2007) 1816. [21] Q.J. Zhang, J.A. Lu, Phys. Lett. A 372 (2007) 1416. [22] X.Y. Wang, Y.J. He, Phys. Lett. A 372 (2007) 435. [23] R.H. Li, W. Xu, S. Li, Phys. Lett. A 367 (2007) 199. [24] T.M. Hoang, M. Nakagawa, Phys. Lett. A 365 (2007) 407. [25] T. Yang, Impulsive Control Theory, Springer, Berlin, 2001. [26] T. Yang, Int. J. Comput. Cognition 2 (2004) 81, http://www.YangSky. com/yangijcc.htm. [27] T. Yang, C.M. Yang, L.B. Yang, Phys. Lett. A 232 (1997) 356. [28] T. Yang, L.B. Yang, C.M. Yang, Phys. Lett. A 226 (1997) 349. [29] Z.G. Li, C.Y. Wen, Y.C. Soh, IEEE Trans. Automat. Control 46 (2001) 894. [30] J.T. Sun, Y.P. Zhang, Q.D. Wu, IEEE Trans. Automat. Control 48 (2003) 829. [31] C.D. Li, X.F. Liao, X.Y. Zhang, Chaos 15 (2005) 023104. [32] C.D. Li, X.F. Liao, X.F. Yang, T.W. Huang, Chaos 15 (2005) 043103. [33] Z. Yang, D. Xu, IEEE Trans. Automat. Control 52 (2007) 1448. [34] C.Y. Wen, Y. Ji, Z.G. Li, Phys. Lett. A 361 (2007) 108. [35] R. Zhang, M.F. Hu, Z.Y. Xu, Phys. Lett. A 364 (2007) 239. [36] X.W. Liu, S.M. Zhong, Phys. Lett. A 370 (2007) 260. [37] Y. Li, X.F. Liao, C.D. Li, T.W. Huang, D.G. Yang, Phys. Lett. A 366 (2007) 52. [38] Z. Li, D. Xu, Phys. Lett. A 282 (2001) 175.
Impulsive control of projective synchronization in chaotic systems Manfeng Hu a,∗ , Yongqing Yang a,b , Zhenyuan Xu a a School of Science, Jiangnan University, Wuxi 214122, China b School of Automation, Southeast University, Nanjing 210096, China
Received 10 December 2007; received in revised form 16 January 2008; accepted 28 January 2008 Available online 2 February 2008 Communicated by A.R. Bishop
Abstract Scaling factor of projective synchronization in coupled partially linear chaotic systems is hardly predictable. To control projective synchronization of chaotic systems in a preferred way, an impulsive control scheme is introduced to direct the scaling factor onto a desired value. The control approach is derived from the impulsive differential equation theory. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results. Furthermore, some interesting and surprising numerical results are discussed. © 2008 Published by Elsevier B.V. PACS: 05.45.Xt Keywords: Projective synchronization; Impulsive differential equation; Chaos; Partially linear system
1. Introduction Chaotic synchronization has been a subject of intense study and is considered to be a fundamental mechanism behind a variety of behaviors in nature during the last decade [1–4]. The original synchronization technique was developed by Pecora and Carroll [1]. In their seminal paper, they addressed the synchronization of chaotic systems in the drive-response coupling scheme. A chaotic system, called the driver (or master), generates a signal which is sent over a channel to a responser (or slaver), which uses this signal to synchronize itself with the master. The signal is usually one of the coordinates of the drive chaotic system. For example, consider the often studied example of synchronization, the Lorenz system x˙ = 10(y − x), y˙ = (28 − z)x − y, z˙ = xy − 8/3z. It is well known that the Lorenz system exhibits identical synchronization in the x coordinate, synchronization in the y coordinate, while z is not a synchronizing coordinate at all [5] (in fact, the authors of Ref. [6] showed that the Lorenz system can exhibit identical synchronization by * Corresponding author. Tel.: +86 510 85910837 (work); fax: +86 510 85913660. E-mail address: [email protected] (M. Hu).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.01.054
selectively using only parts of driving signals z). The authors of [7] have showed that the Lorenz system exhibits projective synchronization (PS), characterized by a scaling factor α, provided the whole signals z is used to drive the response system. In [8], the author has showed that it is hard to estimate the scaling factor because it is dependent on not only the initial condition but also the underlying chaotic dynamics. From the application point of view, it is necessary to control the system in order to offer the occasion to select and direct the solution of synchronization in a defined way. In [8–10], the authors introduced three types feedback controller to the drive system to conduct the scaling factor onto a desired value, respectively. In [11], the authors used a simple and physically available controller proposed firstly in [12] to direct the scaling factor onto a desired value. In [13], the authors further extend the projective synchronization feature to general nonlinear systems rather than partially linear systems by using a controller to the response system under the consideration for potential application of secure communications [14,15]. The authors [16–18] also extend the concept of PS to the one of full state hybrid projective synchronization (FSHPS) in two different chaotic systems (even in two different order chaotic systems). Some recent results regarding the issue of PS have reported in [19–24].
M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
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 [25–37]. The impulsive control technique is also suitable to deal with systems which cannot endure continuous disturbance. Using this method the response system receives the information from the drive system only in discrete times and the amount of conveyed information is, therefore, decreased. However, this is suitable in practice because of reduced control cost. Motivated by the aforementioned comments, the main aim of this Letter is to further study the PS in partially linear chaotic systems coupled through the z variable by adopting the impulsive control method to direct the scaling factor onto a desired value. The organization of the remaining part is as follows. In Section 2, the preliminaries relevant to the theory of impulsive control are presented. Some PS criteria are obtained in Section 3. Results of simulation on Lorenz system are given in Section 4. Finally, conclusions are drawn in Section 5. 2. Basic theory of impulsive differential equations To make the present Letter self-contained, we address some basic results in this section which is useful in this Letter. These results can be found in [26,29,31,32]. Consider the general nonlinear system described by x˙ = f (t, x),
(1)
n where x ∈ R n is the state variable, x˙ ≡ dx dt and f : R+ × R → n R is continuous. Suppose that a discrete set {tk } of time instants satisfies
0 t0 < t1 < · · · < tk < tk+1 < · · · ,
tk → ∞ as
k → ∞.
Ik (x) = δx|t=tk ≡ x tk+ − x tk−
(2)
be the “jump” in the state variable at time instant tk , where x(tk+ ) = limt→t + x(t), x(tk− ) = limt→t − x(t). In general, for k
simplicity, it is assumed that x(tk− ) = x(tk ). Then, the impulsive control system with the initial condition in time t0 can be described by x˙ = f (t, x),
t = tk ,
δx = Ik (x),
t = tk ,
Definition 2. For (t, x) ∈ (tk−1 , tk ] × R n , the right and upper Dini’s derivative of V ∈ V0 is defined as D + V (t, x) ≡ lim sup h→0+
1 V t + h, x + hf (t, x) − V (t, x) . h
(4)
Since the impulsive control system in Eq. (3) is an nthorder impulsive differential equation, instead of investigating the stability of Eq. (3), it is convenient to investigate that of a first-order impulsive differential equation which is given by the following comparison system. Definition 3. Let V ∈ V0 and assume that D + V (t, x) g t, V (t, x) , t = tk , V t, x + Ik (x) Ψk V (t, x) , t = tk ,
(5)
where g : R+ × R+ → R is continuous and g(t, 0) = 0, Ψk : R+ → R+ is nondecreasing. Then, the system w˙ = g(t, w), t = tk , w tk+ = Ψk w(tk ) , t = tk , w t0+ = w0 0
(6)
is the comparison system of (3). Definition 4. Sρ = x ∈ R n : x < ρ , where · denotes the Euclidean norm on
(7) Rn .
Definition 5. A function a is said to belong to class κ if a ∈ C[R+ , R+ ], a(0) = 0 and a(x) is strictly increasing in x. Theorem 1. Assume that the following three conditions are satisfied:
Let
k
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k = 1, 2, 3, . . . ,
x(t0 ) = x0 .
(3)
Assumptions. f (t, 0) = 0 and Ik (0) = 0 for all k. Definition 1. Let V : R+ × R n → R+ , then V is said to belong to class V0 if (1) V is continuous in (tk−1 , tk ] × R n , and for each x ∈ R n , k = 1, 2, 3, . . . , lim(t,y)→(t + ,x) V (t, y) = V (tk+ , x) exists; k (2) V is locally Lipschitzian in x.
(1) V : R+ × Sρ → R+ , ρ > 0, V ∈ V0 , D + V (t, x) g(t, V (t, x)), t = tk . (2) There exists a ρ0 > 0 such that x ∈ Sρ0 implies that x + Ik (x) ∈ Sρ for all k and V (t, x + Ik (x)) Ψk (V (t, x)), t = tk , x ∈ Sρ0 . (3) b(x) V (t, x) a(x) on R+ × Sρ , where a(·), b(·) ∈ κ. Then the stability properties of the trivial solution of the comparison system (6) imply the corresponding stability properties of the trivial solution of (3). ˙ Theorem 2. Let g(t, ω) = λ(t)ω, λ ∈ C 1 [R+ , R+ ], Ψk (ω) = dk ω, dk 0, k = 1, 2, . . . . Then the origin of system (3) is asymptotically stable if the following conditions hold: (1) supk {dk exp[λ(tk+1 ) − λ(tk )]} = ε0 < ∞; (2) there exists a γ > 1 such that λ(t2k+3 ) + ln(γ d2k+2 × d2k+1 ) λ(t2k+1 ) holds for all d2k+2 d2k+1 = 0, k = 1, 2, . . . ;
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(3) λ˙ (t) 0; (4) there exist a(·) and b(·) in class κ such that b(x) V (t, x) a(x). 3. Projective synchronization via the impulsive control Partially linear system, such as Lorenz system, Lü system, hyper-chaotic Lü system, is defined by a set of ordinary differential equations in which the state vector can be broken into two parts (u, z), where the equation for z is nonlinearly related to the other variable, while the equation for the rate of charge of the vector u is linearly related to u through a matrix M(z) that can be depend on the variable z: ˙ = M(z) · u, u(t) z˙ = f (u, z).
(8)
δe = Be = Ik (e),
t = tk .
(12)
We have translated the problem of directing the scaling factor α onto the desired value of (11) into the stability problem of origin of the error system (12). Theorem 3. Let the impulses be equidistant and separated by interval . Q ∈ R n×n is a symmetric and positive definite matrix. Constant r > 0 is an upper bound of the maximum eigenvalue of MT (z)Q + QM(z). Suppose that there exist constant scalars d 0, γ > 1 such that (i) Ω1 = (I + B)T Q(I + B) − dQ 0, (ii) λmin2r(Q) + ln(γ d 2 ) 0, where I denotes the identity matrix. Then the origin of the error system (12) is asymptotically stable, i.e., system (11) realizes projective synchronization with desired scaling factor α.
A general condition for the occurrence of projective synchronization has been reported [38] in two arbitrary dimensional chaotic systems (8) coupled through the variable z:
Proof. Let us construct a Lyapunov function
u˙ d = M(z) · ud ,
V (t, e) = eT Qe,
z˙ = f (ud , z),
where Q is a symmetric and positive definite matrix. Let b(e) = λmin (Q)e2 and a(e) = λmax (Q)e2 , where λmin and λmax denote the smallest and largest eigenvalues of a square matrix, respectively. Then a(·) and b(·) belong to κ. Moreover (13) implies that b(e) V (t, e) a(e). For t = tk , the derivative of V (t, e) along the solution of (12) is D + V (t, e) = eT MT (z)Q + QM(z) e r r (14) eT Qe = V (t, e). λmin (Q) λmin (Q)
u˙ r = M(z) · ur
(9)
the subscripts d and r stand for the drive system and response system respectively, ud = (u1d , u2d , . . . , und )T , ur = (u1r , u2r , . . . , unr )T ∈ R n , z ∈ R 1 is a one-dimensional coupling variable, which is the same in both the drive and the response systems. The matrix M(z) is only dependent on the variable z that is nonlinearly related to the variables in ud : ⎤ ⎡ m11 (z) m12 (z) · · · m1n (z) ⎢ m21 (z) m22 (z) · · · m2n (z) ⎥ . M(z) = ⎢ (10) .. .. ⎥ .. ⎣ ... . . . ⎦ mn1 (z)
mn2 (z)
···
mnn (z)
We shall use the abbreviation M, mij to denote M(z) and mij (z) respectively, for simplicity in the later discussion. If there exists a constant α (α = 0) such that limt→∞ ur − αud = 0, then the projective synchronization between the drive system and response systems is achieved, and we call α as “scaling factor”. It is well known that the coupled systems (9) can achieve projective synchronization, but the scaling factor α is unpredictable. For directing the scaling factor onto the desired value α, we introduce the impulses at instant time tk in response system of (9): u˙ d = M(z) · ud , z˙ = f (ud , z), u˙ r = M(z) · ur ,
t = tk ,
δur = B(ur − αud ),
t = tk .
(11)
(13)
For t = tk , we have V (tk , e + Be) = eT (I + B)T Q(I + B)e = eT (I + B)T Q(I + B) − dQ e + dV (tk , e) = eT Ω1 e + dV (tk , e) dV (tk , e). ˙ = Let λ(t)
r λmin (Q) ,
(15)
dk = d, k = 1, 2, . . . . Then, it follows
from Theorem 2 that the origin of the error system (12) is asymptotically stable. This completes the proof. 2 Let Q = I in Theorem 3, the following corollary holds. Corollary 1. Let constant r > 0 be an upper bound of the maximum eigenvalue of MT (z) + M(z). Assume that there exist constants d 0, γ > 1 such that (i) Ω1 = (I + B)T (I + B) − dI 0, (ii) 2r + ln(γ d 2 ) 0,
Let e = (e1 , e2 , . . . , en )T , ei = uir − αuid , i = 1, 2, . . . , n denote the projective synchronization errors of (11), then we have
then the origin of the error system (12) is asymptotically stable.
e˙ = M(z)e,
Remark 1. In Refs. [8–10] and [11], the authors have studied the issue of controlling projective synchronization in coupled partially linear systems by using continuous synchronization
z˙ = f (ud , z),
t = tk ,
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schemes, in which driving signals are transmitted continuously to the response systems such that the PS with desired scaling factors can be realized. Furthermore, in this Letter, we have considered the same issue by using the impulsive synchronization scheme, in which only samples of state variables are used to realize PS. The proposed impulsive projective synchronization scheme is more practicable than the existed ones [8–11] in application because of the reduced control cost. 4. Example and simulations In this section, one example is given for illustration the above impulsive projective synchronization criteria. Meanwhile, some interesting and surprising numerical results are also discussed. Example. Lorenz system is described by (a)
x˙ = σ (y − x), y˙ = (μ − z)x − y, z˙ = xy − ρz
(16)
with three real parameters σ = 10, μ = 28, ρ = 8/3. Taking −σ σ z as the coupling variable, we have M(z) = μ−z −1 in the form of (9) which satisfying −σ − 1 = −11 < 0. According to the criterion derived in [38], projective synchronization occurs in the coupled Lorenz systems but the scaling factor is hardly predictable. To manipulate the scaling factor onto the desired value, we use the impulsive control method in the form of (11), i.e., X˙ = M(z) · X, z˙ = xy − ρz, ˙˜ = M(z) · X, ˜ X
t = tk , ˜ δ X = Be = Ik (e), t = tk , (x, y)T , X˜
(17)
= e = X˜ − αX, α is a desired where X = value. −σ σ For M(z) = μ−z −1 , after a simple calculation, we have λ1,2 MT (z) + M(z) = λ1,2 (z) = −(1 + σ ) ±
(x, ˜ y) ˜ T,
(b) Fig. 1. (a) z vs t , (b) λ1,2 vs z.
boundaries of the stable region is given by ln γ + 2 ln(k + 1)2 , −2 k 0. (20) 2r Fig. 2 shows the stable region for PS with different γ . The whole region under the curve corresponding to γ = 1 is the predicted stable region. The stable region shrinks to a line k = −1 when γ → +∞. Fig. 3 shows the stable results within the stable region for k = −1.1 and = 0.15. The time response of desired scaling factors plotted in Fig. 4 shows that the aim of the control is realized. It is worthy noting that the condition of impulsive synchronization is only a conservative sufficient condition. In other words, the true stable region in actual application is larger than that predicted in Fig. 2. It can be found that the length of impulsive interval can be arbitrarily selected in many numerical 0 −
(1 − σ )2 + (σ + μ − z)2 .
(18)
In simulations, without of generalization, we pick (x(0), y(0), z(0), x(0), ˜ y(0)) ˜ = (1, 2, 3, 2, 1). From Fig. 1(a), we can see that the bounded variable z varies in (0, 50). Then we get the relation between λ1,2 (z) and z (see Fig. 1(b)). So we can select r = 23.2053 (λmax = 23.2053 as z = 5) such that the condition λ1,2 (z) r is satisfied. We choose B as k 0 B= . (19) 0 k It is easy to see that if d (k + 1)2 0, the condition (1) of Corollary 1 is satisfied. When d = (k + 1)2 , an estimate of the
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Fig. 2. Estimate of the boundaries of stable regions for PS with different γ used in (20).
Fig. 3. Time response of the PS error, initial value x(0), y(0), z(0), x(0), ˜ y(0) ˜ = (1, 2, 3, 2, 1), integration step = 0.01, impulsive interval = 0.15.
Fig. 4. Time response of the scaling factor with impulsive interval = 0.15 and feedback gain k = −1.1.
Fig. 5. Time response of the scaling factor with impulsive interval = 1 and feedback gain k = −1.1.
Fig. 6. Time response of the scaling factor with impulsive interval = 1 and feedback gain k = −1.
experiments done on Lorenz system. For instance, when taking = 1, the other conditions are same as those in Figs. 3 and 4, the PS with desired scaling factors α = −3 can also be realized (see Fig. 5). An attempt has been made here to explore the reasons why so interesting things can happen. On the one hand, the coupled variable z is the main cause to realize PS in coupled partially linear systems, however, the ultimate state of the synchronization is hard to predict. On the other hand, impulsive signals, derived from samples of the state variables of the drive and response systems with desired constant α, play a key role to direct the synchronization behavior to the desired one. From Figs. 4 and 5, one can easily see that each impulses signals added to the response system direct the scaling factor to the desired one. These numerical simulations indicate that any width of impulsive interval can be used to implement PS with desired scaling factor in coupled partially linear systems. The freedom to choose the discrete time instants with impulsive signals therefore leads to a large flexibility in applications, especially in chaos communication.
M. Hu et al. / Physics Letters A 372 (2008) 3228–3233
There exists, in addition, a surprising result. In numerical simulations, taking feedback gain k = −1, the other conditions are same as those in Figs. 3 and 4, the PS with desired scaling factors α = −3 can be realized after only one impulse effects (see Fig. 6). But, from the viewpoint of mathematics, the feedback gain k cannot take −1 as shown in Eq. (20) or Fig. 2. In summary, new theory of impulsive differential equation is needed for deeply understanding these two experiments results. 5. Conclusions In the present Letter, we have investigated the issue on directing the scaling factor of PS in coupled partially linear systems onto desired values via an impulsive control method. Some theoretic conditions are obtained to guarantee that the PS with desired value can be realized via impulses. The main advantage of our results is that the width of impulsive interval can be much more bigger, which will be easily applied to real situations. Acknowledgements The authors thank the anonymous reviewers for their helpful comments and suggestions. This work was jointly supported by Postdoctoral Science Foundation under Grant 20070420958, Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 0702001B and Pre-research Funds of Jiangnan University. References [1] L.M. Perora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [2] G. Chen, X. Dong, From Chaos to Order: Perspective, Methodologies and Applications, World Scientific, Singapore, 1998. [3] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, Phys. Rep. 366 (2002) 1.
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