A General Control Method for Inverse Hybrid Function Projective Synchronization of a Class of Chaotic Systems

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1 International Journal of Mathematical Analysis Vol. 9, 2015, no. 9, HIKARI Ltd, A General Control Method for Inverse Hybrid Function Projective Synchronization of a Class of Chaotic Systems Gasri Ahlem 1,2 and Adel Ouannas 2 1 Department of Mathematics and Computer Science Constantine University, Algeria 2 Department of Mathematics and Computer Science Tebessa University, Algeria Copyright c 2014 Gasri Ahlem and Adel Ouannas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, a new type of chaos synchronization called inverse hybrid function projective synchronization (IHFPS) is investigated for a class of continuous chaotic systems. Based on Lyapunov stability theory and nonlinear control method, a new controller can be designed to achieve inverse hybrid function projective synchronizationa for n D chaotic systems in continuous-time. Simulation example validate the derived synchronization result. Keywords: Hybrid function projective synchronization, inverse hybrid function projective synchronization, Lyapunov stability, nonlinear controllers, continuous-time chaotic systems 1 Introduction In recent years, chaos synchronization has become an active research area, due to its potential applications in secure communication [1]. Various pow-

2 430 Gasri Ahlem and Adel Ouannas erful methods for chaos synchronization have been investigated such as active control method [2], backstepping design approach [3] and sliding mode technique [4], etc. and so on have been successfully applied to chaotic and hyperchaotic systems. Many type of synchronization for chaotic systems have been presented such as anti-synchronization [5], generalized synchronization [6] and hybrid function projective synchronization [7], etc. However, hybrid function projective synchronization (HFPS) scheme for chaotic systems is extensively considered [8]. In HFPS, the master system and the slave system could be synchronized with a scaling function matrix and the unpredictability of the scaling function matrix in HFPS scheme can enhance the security in tele-communication. On the other hand, studying the inverse problem of HFPS which produce a new synchronization type called inverse hybrid function projective synchronization (IHFPS) is an attractive and important idea. In this paper, we introduce the inverse hybrid function projective synchronization type, (IHFPS), for n-dimensional general class of chaotic systems in continuous-time. To achieve IHFPS, we propose a general method based on new nonlinear controlllers. The theoritical result derived in this paper, is proved using Lyapunov stability theory. In order to verify the effectiveness of the proposed method, we apply it to 5-D continuous-time chaotic systems. This paper is organized as follows: in section 2, a description of the chaotic systems addressed in this paper is provided. In section 3, the definition of inverse hybrid function projective synchronzation is introduced. In section 4, our synchronization method is described. In section 5, numerical example is used to show the effectiveness of the proposed method. Finally, we make conclusion in section 6. 2 Master-slave systems description We consider the following master chaotic system described by ẋ i = a ij x j +... p n=1 p 1 =1 α (i) p 1...p n x p x pn n + γ i, i = 1, 2,..., n, (1) where (a ij ) R n n, ( ) α i p 1...p n R d d, (i = 1, 2,..., n), such that: α (i) = 0, α (i) = 0,..., α (i) = 0, (i = 1, 2,..., n), (γ i ) 1 i n are real numbers. As the slave system, we consider the following chaotic system described by ẏ i = b ij y j +... p n=1 p 1 =1 β (i) p 1...p n y p y pn n + δ i + u i, i = 1, 2,..., n, (2)

3 General control method 431 where (d ij ) R n n, ( ) β i p 1...p n R d d, (i = 1, 2,..., n), such that: β (i) = 0, β (i) = 0,..., β (i) = 0, i = 1, 2,..., n, (δ i ) 1 i n are real numbers and (u i ) 1 i n are controllers to be determined. 3 Definition of IHFPS First of all, we call the definition of hybrid function projective synchronous (HFPS) then we present the definition of inverse hybrid function projective synchronous (IHFPS) for master-slave chaotic systems given in Eqs. (1) and (2). Definition 1 The master system (1) and the slave system (2) are said to be hybrid function projective synchronized (HFPS), if there exists controllers (u i ) 1 i n and continuously differentiable bounded functions h i (t) 0, for all t, 1 i n, such that the synchronization errors e i = y i h i x i, 1 i n, satisfies that lim t + e i = 0, i = 1, 2,..., n. Remark 2 HFPS of chaotic dynamical systems, based on Lyapunov stability theory, have been studied and carried out, for example, in Ref. [7]. Definition 3 The master system (1) and the slave system (2) are said to be inverse hybrid function projective synchroniwed (IHFPS), if there exists controllers (u i ) 1 i n and continuously differentiable bounded functions h i (t) 0, for all t, 1 i n, such that the synchronization errors e i (t) = x i h i (t) y i, 1 i n, satisfies that lim t + e i = 0, i = 1, 2,..., n. 4 Synchronization Method The error system, according to definition 3, between the master system (1) and the slave system (2), can be derived as where ė i (t) = a ij e j (t) + φ i + ϕ i + ψ i h i (t) u i, i = 1, 2,..., n, (3) φ i =... p n=1 p 1 =1 α (i) p 1...p n x p x pn n, i = 1, 2,..., n, (4)

4 432 Gasri Ahlem and Adel Ouannas and ϕ i = h i (t)... p n=1 p 1 =1 β (i) p 1...p n y p y pn n, i = 1, 2,..., n, (5) ψ i = h i (t) (a ij b ij ) y j ḣi (t) y i + γ i h i (t) δ i, i = 1, 2,..., n. (6) To achieve inverse HFPS between the master system (1) and the slave system (2), we choose the control law as follow ( u i = 1 i 1 (2a ij + a ji ) e j (t) + k i e i (t) h i (t) j=i+1 a ji e j (t) + φ i + ϕ i + ψ i ) (7) where (k i ) 1 i n are control constants to be designed. By substituting Eq. (7) into Eq. (3), the error system can be written as, i = 1, 2,..., n, ė 1 (t) ė 2 (t). ė n (t) = a 11 k 1 a 21 + a 12 a n1 + a 1n (a 21 + a 12 ) a 22 k 2 a n2 + a 2n (a n1 + a 1n ) (a n2 + a 2n ) a nn k n e 1 (t) e 2 (t). e n (t) (8) Now, rewriting the error system described in Eq. (8) in the compact form ė (t) = ( L + K L T ) e (t), (9) where K = diag {(a 11 k 1 ),..., (a nn k n )} and L = (l ij ) n n such that l ij = { (aij + a ji ) if j < i 0 if j i (10) Theorem 4 If the control constants (k i ) 1 i n are chosen such that: k i > a ii, i = 1, 2,..., n. Then, the two systems (1) and (2) are globally inverse hybrid function projective synchronized under.the controllers (7).

5 General control method 433 Proof. Consider the Lyapunov function in the form: V (e (t)) = 1 2 et (t) e (t), then we get V (e (t)) = 1 (ėt (t) e (t) + e T (t) ė (t) ) 2 ( e T (t) ( L T + K T L ) e (t) + e T (t) ( L + K L ) T e (t) ) = 1 2 = e T (t) Ke (t) = n i=1 (a ii k i ) e 2 i (t) < 0. Thus, by Lyapunov stability theory it is immediate that lim t e i (t) = 0, i = 1, 2,..., n. We conclude that the systems (1) and (2) are globally inverse hybrid function projective synchronized. 5 Simulation Example In this section, an example of chaotic system is considered to validate the proposed chaos synchronization approach. The master system and the slave system are described as, respectively, ẋ 1 = a 1 (x 2 x 1 ) + x 2 x 3 x 4 x 5 ẋ 2 = a 2 (x 1 + x 2 ) x 1 x 3 x 4 x 5 ẋ 3 = x x 2 1, (11) ẋ 4 = a 3 x 4 + x 1 x 2 x 3 x 5 ẋ 5 = a 4 (x 5 x 4 ) a 5 x 1 + x 1 x 2 x 3 x 4 and ẏ 1 = a 1 (y 2 y 1 ) + y 2 y 3 y 4 y 5 + u 1 ẏ 2 = a 2 (y 1 + y 2 ) y 1 y 3 y 4 y 5 + u 2 ẏ 3 = y y u 3 ẏ 4 = a 3 y 4 + y 1 y 2 y 3 y 5 + u 4 ẏ 5 = a 4 (y 5 y 4 ) a 5 y 1 + y 1 y 2 y 3 y 4 + u 5, (12) where a 1, a 2, a 3, a 4, a 5 are bifuraction prameters and (u i ) 1 i 5 are the synchronization controllers. The system (11), has a chaotic attractor, when (a 1, a 2, a 3, a 4, a 5 ) = (37, 14.5, 10.5, 15, 9.5), [8]. According to our control method presented in section 2, the synchronization errors between the master system (11) and the slave system (12) can be derived as ė 1 (t) = a 1 (e 2 (t) e 1 (t)) + R 1 h 1 (t) u 1 ė 2 (t) = a 2 (e 1 (t) + e 2 (t)) + R i h 2 (t) u 2 ė 3 (t) = e 3 (t) + R 3 h 3 (t) u 3, (13) ė 4 (t) = a 3 e 4 (t) + R 4 h 4 (t) u 4 ė 5 (t) = a 4 (e 5 (t) e 4 (t)) a 5 e 1 (t) + R 5 h 5 (t) u 5 where

6 434 Gasri Ahlem and Adel Ouannas R 1 = x 2 x 3 x 4 x 5 h 1 (t) y 2 y 3 y 4 y 5 ḣ1 (t) y 1 R 2 = x 1 x 3 x 4 x 5 + h 2 (t) y 1 y 3 y 4 y 5 ḣ2 (t) y 2 R 3 = 0.1x 2 1 h 3 (t) (0.1y 2 1) ḣ3 (t) y 3 R 4 = x 1 x 2 x 3 x 5 h 4 (t) y 1 y 2 y 3 y 5 ḣ4 (t) y 4 R 5 = x 1 x 2 x 3 x 4 h 5 (t) y 1 y 2 y 3 y 4 ḣ5 (t) y 5, (14) and (h i (t)) 1 i 5 are the scaling functions. To achieve IHFPS between systems (11) and (12), the synchronization controllers are proposed as follow u 1 = 1 (a h 1 (t) 2e 2 (t) + a 5 e 5 (t) + R 1 ) u 2 = 1 (a h 2 (t) 1e 1 (t) + ke 2 (t) + R 2 ) u 3 = 1 R h 3 (t) 3, (15) u 4 = 1 (a h 4 (t) 4e 5 (t) R 4 ) u 5 = 1 R h 5 (t) 5 where k is a constant control to be designed. Then, the synchronization errors (13) can be described as ė 1 (t) ė 2 (t) ė 3 (t) ė 4 (t) ė 5 (t) = a 1 a 1 a a 5 a 2 a 1 a 2 k a 3 a 4 a a 4 a 4 e 1 (t) e 2 (t) e 3 (t) e 4 (t) e 5 (t), (16) and by using the same procedure of proof of Theorem 4, we can get the following result. Corollary 5 If the control constant k is chosen such that: k > a 2, then the two systems (11) and (12) are globally inverse hybrid function projective synchronized under the control law (15). Finally, we get the numeric result that is showen in Fig Conclusion In this paper, a new synchronization method was presented for some class of chaotic dynamical systems in continuous-time to investigate a new type of synchronization called inverse hybrid function projective synchronization (IHFPS). Numerical example was utilized to illustrate the effectiveness of the proposed method.

7 General control method 435 Figure 1: Time evolution of synchronization errors between systems (11) and (12). References [1] Mengue, A.D., Essimbi, B.Z.: Secure communication using chaotic synchronization in mutually coupled semiconductor lasers. Nonlinear Dyn. 70 (2012), [2] M.C. Ho and Y.C. Hung, Synchronization of two different chaotic systems using generalized active control. Physical Letters A, 301, 2002, [3] J. Zhao and J.Lu, Parameter identification and backstepping control of uncertain Lu system, Chaos, Solition and Fractals, 17, 2003; [4] H.T. Yau, Design of adaptive sliding mode controller for chaos synchronization with uncertainties, Chaos, Solition and Fractals, 22, 2004, [5] Y.W. Wang and Z.H. Guan, Generalized synchronization of continuous chaotic systems, Chaos, Solition and Fractals, 27, 2006,

8 436 Gasri Ahlem and Adel Ouannas [6] X.Zhang and H. Zhu, Anti-synchronization of two different hyperchaotic systems via active and adaptive control, International Journal Of Nonlinear Sciense, 6,, 2008, [7] X. Li, Generalized projective synchronization using nonlinear control method, International Journal Of Nonlinear Sience, 8, 2009, [8] Chun-Li Zhang and Jun-Min Li. Hybrid Function Projective Synchronization of Chaotic Systems with Uncertain Time-varying Parameters via Fourier Series Expansion. International Journal of Automation and Computing, 9(4), 2012, [9] K. Kemih, H. Bouraoui, M. Ghanes, R. Remmouche, A. Senouci, Passivitybased control of the new hyperchaotic system, Int. J. Modelling, identification Control, 17, 2012, Received: July 15, 2014; Publsihed: February 9, 2015