Commun Nonlinear Sci Numer Simulat

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1 Commun Nonlinear Sci Numer Simulat 14 29) Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: The function cascade synchronization scheme for discrete-time hyperchaotic systems Hong-Li An a,c, Yong Chen a,b,c, * a Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo , China b Institute of Theoretical Computing, East China Normal University, Shanghai 262, China c Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 1, China article info abstract Article history: Received 29 October 27 Received in revised form 3 March 2 Accepted 2 April 2 Available online 29 April 2 PACS: 5.45 In this paper, the function cascade synchronization scheme is proposed to investigate the discrete-time hyperchaotic systems. By choosing some different error functions and with the aid of symbolic numeric computation, the proposed scheme is applied to achieve the function cascade synchronization for two discrete-time hyperchaotic systems: the generalized Hénon map and the discrete-time Rössler system, respectively. Numerical simulations are used to verify the effectiveness and feasibility of the proposed technique. Ó 2 Elsevier B.V. All rights reserved. Keywords: Chaos synchronization Discrete-time hyperchaotic systems Function cascade synchronization 1. Introduction Since the pioneering wors of Pecora and Carroll [1], Ott et al. [2], chaos control and synchronization have attracted extensive attention from various areas [1 6]. Recently, great efforts have been devoted to realize chaos synchronization for two identical and different chaotic systems because of their great potential applications in secure communication, chemical and biological systems, telecommunications, system identification, etc. At the same time, many powerful methods have been proposed, such as OGY method [2], bacstepping design technique [7 9], linear and nonlinear feedbac approaches [1 13], adaptive control approach [4,14 16], active control approach [17,1], cascade synchronization method [19,2] as well as the function cascade synchronization method [21,22], etc., which are continuous-time synchronization schemes. In fact, many mathematical models of neural networs, biological process, physical process and chemical process, etc., were defined using discrete-time dynamical systems [23 25].Thus more and more attention has been paid to the chaos hyperchaos) synchronization in discrete-time dynamical systems [23 32]. In recent time, hyperchaotic systems have become the hot topic and many wors have been done for hyperchaotic synchronization [21,22,29 34]. A hyperchaotic system is defined as a chaotic system with at least two positive exponents, implying that its dynamics are expanded in several different directions simultaneously. It means that the hyperchaotic system has more complex dynamical behaviors, which can be used to improve the security of chaotic communication systems. Therefore, they are more suitable for some special engineering applications such as chaos-based encryption and secure communication. * Corresponding author. Address: Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo , China. address: ychen@sei.ecnu.edu.cn Y. Chen) /$ - see front matter Ó 2 Elsevier B.V. All rights reserved. doi:1.116/j.cnsns

2 H.-L. An, Y. Chen / Commun Nonlinear Sci Numer Simulat 14 29) In this paper, combined of function cascade synchronization scheme [21,22] and Q S synchronization method [3,31], the function cascade synchronization scheme for the continuous-time chaotic systems is successfully extended to the discretetime systems. By constructing function scalar controllers and choosing some different error functions, we can easily realize the function cascade synchronization for the discrete-time hyperchaotic systems with the aid of symbolic numeric computation. In order to verify its effectiveness of the proposed method, we apply it to two discrete-time hyperchaotic systems: the generalized Hénon map [23] and the discrete-time Rössler system [25], respectively. It is organized as follows: In Section 2, the function cascade synchronization method for the discrete-time hyperchaotic systems is introduced. In Section 3, the proposed scheme is applied to achieve the function cascade synchronization for two discrete-time hyperchaotic systems and numerical simulations are used to verify its effectiveness. In Section 4, conclusions are followed. 2. The function cascade synchronization scheme for discrete-time hyperchaotic systems The function cascade synchronization for the continuous-time chaotic system can be seen in Refs. [21,22]. Combined of the Q S synchronization method [3,31], we extend the function cascade synchronization method to the 3D discrete-time dynamical systems. It is defined as the following. Consider the hyperchaotic system: >< x 1 ð þ 1Þ ¼f ðx 1 ðþ; x 2 ðþ; x 3 ðþþ; x 2 ð þ 1Þ ¼gðx 1 ðþ; x 2 ðþ; x 3 ðþþ; ð1þ x 3 ð þ 1Þ ¼hðx 1 ðþ; x 2 ðþ; x 3 ðþþ: Firstly, copy the first two equations of 1) and we get a sub-response system: X 1 ð þ 1Þ ¼f ðx 1 ðþ; X 2 ðþ; x 3ðÞÞ þ u 1 ðþ; X 2 ð þ 1Þ ¼gðX 1ðÞ; X 2 ðþ; x 3ðÞÞ þ u 2 ðþ; ð2þ where variable x 3 is a corresponding signal afforded by the original system, u 1 ðþ and u 2 ðþ are the desired scalar controllers that mae the drive system 1) synchronize with the sub-response system 2) by choosing suitable scaling function factors. Now, we define the error functions e 1 ðþ ¼X 1 ðþ Q 1 ðx 1 ðþþx 1 ðþ, e 2 ðþ ¼X 2 ðþ Q 2ðx 2 ðþþx 2 ðþ: The error dynamical system between the drive system 1) and response system 2) is e 1 ð þ 1Þ ¼X 1 ð þ 1Þ Q 1 ðx 1 ð þ 1ÞÞx 1 ð þ 1Þ >< ¼ f ðx 1 ðþ; X 2 ðþ; x 3ðÞÞ Q 1 ðf ðx 1 ðþ; x 2 ðþ; x 3 ðþþþf ðx 1 ðþ; x 2 ðþ; x 3 ðþþ þ u 1 ðþ; e 2 ð þ 1Þ ¼X 2 ð þ 1Þ Q ð3þ 2ðx 1 ð þ 1ÞÞx 2 ð þ 1Þ ¼ gðx 1 ðþ; X 2 ðþ; x 3ðÞÞ Q 2 ðgðx 1 ðþ; x 2 ðþ; x 3 ðþþþgðx 1 ðþ; x 2 ðþ; x 3 ðþþ þ u 2 ðþ: We choose the scalar controllers u 1 ðþ and u 2 ðþ as u 1 ðþ ¼c 11 e 1 ðþþc 12 e 2 ðþ fðx 1 ðþ; X 2 ðþ; x 3ðÞÞ þ Q 1 ðf ðx 1 ðþ; x 2 ðþ; x 3 ðþþþf ðx 1 ðþ; x 2 ðþ; x 3 ðþþ; u 2 ðþ ¼c 21 e 1 ðþþc 22 e 2 ðþ gðx 1 ðþ; X 2 ðþ; x 3ðÞÞ þ Q 2 ðgðx 1 ðþ; x 2 ðþ; x 3 ðþþþgðx 1 ðþ; x 2 ðþ; x 3 ðþþ: ð4þ Substituting 4) into 3), we get the error system e 1 ð þ 1Þ ¼ C e 1ðÞ ; C ¼ c 11 c 12 : ð5þ e 2 ð þ 1Þ e 2 ðþ c 21 c 22 In fact, it is easy to prove that for any given parameters c ij ði; j ¼ 1; 2Þ in 5), there always exists at least one family of parameters c ij such that all eigenvalues of matrix C are in the unit disc for the origin, which indicates that the error system 5) is globally asymptotically stable. For example, the simplest case is < c ii < 1; c ij ¼ ði j and i; j ¼ 1; 2Þ: ð6þ Tae the Lyapunov function as L 1 ðþ ¼je 1 ðþj þ c 1 je 2 ðþj; c 1 > : ð7þ So we have the derivative of L 1 ðþ DL 1 ðþ ¼L 1 ð þ 1Þ L 1 ðþ ¼je 1 ð þ 1Þj þ c 1 je 2 ð þ 1Þj je 1 ðþj c 1 je 2 ðþj 6 ½jc 11 jþc 1 jc 21 j 1Šje 1 ðþj þ ½jc 12 jþc 1 jc 22 j c 1 Šje 2 ðþj: ðþ From ), we now if the parameters c 1 ; c ij ði; j ¼ 1; 2Þ satisfy jc 11 jþc 1 jc 21 j < 1; jc 12 jþc 1 jc 22 j < c 1 ; ð9þ then DL 1 ðþ is negative definite, denoting that the system is globally asymptotically stable and

3 1496 H.-L. An, Y. Chen / Commun Nonlinear Sci Numer Simulat 14 29) < : e 1 ðþ ¼ X 1 ðþ Q 1 ðx 1 ðþþx 1 ðþ ¼ ; e 2 ðþ ¼ X 2 ðþ Q 2ðx 2 ðþþx 2 ðþ ¼ : That is to say the synchronization is achieved between the discrete-time hyperchaotic systems 1) and 2) with the scalar controller ðu 1 ðþ; u 2 ðþþ given in 4). Secondly, copy another response system: X 2 ð þ 1Þ ¼gðX 1 ðþ; X 2 ðþ; X 3 ðþþ þ u 3 ðþ; ð11þ X 3 ð þ 1Þ ¼hðX 1 ðþ; X 2 ðþ; X 3 ðþþ þ u 4 ðþ; where X 1 ðþ is the drive variable corresponding to system 2) and u 3 ðþ; u 4 ðþ are the scalar controllers determined later. In order to achieve the discrete-time synchronization between systems 1) and 11), we mae the same analysis above. Defining the necessary error functions e 3 ðþ ¼X 2 ðþ Q 3 ðx 2 ðþþx 2 ðþ; e 4 ðþ ¼X 3 ðþ Q 4 ðþx 3 ðþ. The corresponding Lyapunov function is chosen as L 2 ðþ ¼je 3 ðþj þ c 2 je 4 ðþj; c 2 > : ð12þ ð1þ Here we set u 3 ðþ ¼c 31 e 3 ðþþc 32 e 4 ðþ gðx 1 ðþ; X 2 ðþ; X 3 ðþþ þ Q 3 ðgðx 1 ðþ; x 2 ðþ; x 3 ðþþþgðx 1 ðþ; x 2 ðþ; x 3 ðþþ; u 4 ðþ ¼c 41 e 3 ðþþc 42 e 4 ðþ hðx 1 ðþ; X 2 ðþ; X 3 ðþþ þ Q 4 ðhðx 1 ðþ; x 2 ðþ; x 3 ðþþþhðx 1 ðþ; x 2 ðþ; x 3 ðþþ; ð13þ then the derivative of Lyapunov function is DL 2 ðþ ¼L 2 ð þ 1Þ L 2 ðþ ¼je 3 ð þ 1Þj þ c 2 je 4 ð þ 1Þj je 3 ðþj c 2 je 4 ðþj 6 ½jc 31 jþc 2 jc 42 j 1Šje 3 ðþj þ ½jc 32 jþc 2 jc 42 j c 2 Šje 4 ðþj: ð14þ When the parameters c 2 ; c 31 ; c 32 ; c 41 ; c 42 satisfy jc 31 jþc 2 jc 42 j < 1; jc 32 jþc 2 jc 42 j < c 2 ; ð15þ DL 2 ðþ is negative and the error functions e 3 ðþ; e 4 ðþ approach to zero. That is to say the synchronization is achieved between systems 1) and 11). From the above, we now that the discrete-time hyperchaotic system 1) synchronize with the systems 2) and 11) with the controllers chosen in 4) and 13), i.e. X 1 ðþ Q 1 ðx 1 ðþþx 1 ðþ ¼ ; X 2 ðþ Q 3 ðx 2 ðþþx 2 ðþ ¼ ; So we can get X 1 ðþ Q 1 ðx 1 ðþþx 1 ðþ ¼ ; >< X 2 ðþ Q 3 ðx 2 ðþþx 2 ðþ ¼ ; X 3 ðþ Q 4 ðx 3 ðþþx 3 ðþ ¼ : X 2 ðþ Q 2ðx 2 ðþþx 2 ðþ ¼ ; X 3 ðþ Q 4 ðx 3 ðþþx 3 ðþ ¼ : ð16þ Therefore, we can predict that the function cascade synchronization is achieved for the discrete-time hyperchaotic system. Remar 1. In our scheme, we choose the error function e i ðþ ¼X i ðþ Q i ðx i ðþþx i ðþ as a special function form, so we name it the function cascade synchronization in discrete-time systems. When we set Q i ðx i ðþþ ¼ 1orQ i ðx i ðþþ ¼ a, the cascade synchronization methods nown [19,2] will appear, respectively. In addition, more parameters fc ij j1 6 i; j 6 4g are needed in our scheme, which enlarge the scope of the scalar controllers u i ðþ in the sense of synchronization. Therefore, our scheme is different from the nown techniques [7 9,23 34]. Remar 2. Although the ansätz of the error function here is general, how to choose an error function is the ey problem naturally. Even if the availability of computer symbolic systems lie Maple or Mathematica allows us to perform the complicated and tedious algebraic calculation and differential calculation on a computer, in general, it is very difficult, sometime impossible, to find all class of specified error function Q i ðx i ðþþ. As the calculation goes on, in order to drastically simplify the wor or mae the wor feasible to obtain controllers u i ðþ. Experience tells us that the error function Q i ðx i ðþþ is chosen as some special function forms, such as in the forms of tanh x; sechx, x n and P a n x n, e x, a trial-and-error basis. Remar 3. As is nown, a continuous system is a chaotic system that must be a three-dimensional dynamical system at least. However, a discrete-time system is a chaotic system that must be a two-dimensional dynamical system at least. Generally speaing, for the chaotic systems owning the same dimension and nonlinear property, the discrete-time system is always more complex than the continuous one, therefore, the method proposed in this paper is not a trivial extension to

4 H.-L. An, Y. Chen / Commun Nonlinear Sci Numer Simulat 14 29) discrete systems. In fact, the scheme can be applied to investigate the synchronization more than 3D dynamical systems and the tracing problems in the discrete-time systems. 3. Two examples of the function cascade synchronization for the discrete-time hyperchaotic systems In the following, we will apply the function cascade synchronization method to the discrete-time hyperchaotic systems: the generalized Hénon map [23] and the Rössler system [25], respectively. We choose the error functions in the forms of tanh x; sechx; x n and P a n x n. Numerical simulations are followed to illustrate the effectiveness of the proposed method The 3D generalized Hénon map The 3D generalized Hénon map [23] discovered by Hitzl and Zele is described as >< x 1 ð þ 1Þ ¼ bx 2 ðþ; x 2 ð þ 1Þ ¼1 þ x 3 ðþ ax 2 ðþ 2 ; x 3 ð þ 1Þ ¼x 1 ðþþbx 2 ðþ: ð17þ Here a; b are the control parameters of the discrete-time system. This system has a hyperchaotic attractor when a ¼ 1:7 and b ¼ :3. According to the above main idea, we copy 17) and obtain the first response system: X 1 ð þ 1Þ ¼ bx 2 ðþþu 1ðÞ; X 2 ð þ 1Þ ¼1 þ x 3ðÞ ax 2 ðþ2 þ u 2 ðþ; where x 3 ðþ is the state variable of the original system 17) and u 1 ðþ; u 2 ðþ are the control functions determined later. Let us define the Lyapunov function as L 1 ðþ ¼je 1 ðþj þ c 1 je 2 ðþj; c 1 > ; ð19þ where e 1 ðþ ¼X 1 ðþ x 1 ðþsech 2 x 1 ðþ; e 2 ðþ ¼X 2 ðþþð1 þ x 2ðÞÞx 2 ðþ.our aim is to find the controllers u 1 ðþ; u 2 ðþ that mae the drive system 17) globally asymptotically synchronize with the response system 1). The derivative of the corresponding Lyapunov function L 1 ðþ is DL 1 ðþ ¼L 1 ð þ 1Þ L 1 ðþ ¼je 1 ð þ 1Þj þ c 1 je 2 ð þ 1Þj je 1 ðþj c 1 je 2 ðþj ¼jX 1 ð þ 1Þ x 1 ð þ 1Þsech 2 x 1 ð þ 1Þj þ c 1 jx 2 ð þ 1Þþð1 þ x 2ð þ 1ÞÞx 2 ð þ 1Þj jx 1 ðþ x 1 ðþsech 2 x 1 ðþj c 1 jx 2 ðþþð1 þ x 2ðÞÞx 2 ðþj: We choose the controllers u 1 ðþ; u 2 ðþ as ð1þ ð2þ u 1 ðþ ¼bX 2 ðþ bx 2ðÞsechð bx 2 ðþþ þ c 11 ½X 1 ðþ x 1 ðþsech 2 x 1 ðþš þ c 12 ½X 2 ðþþx 2ðÞþx 2 ðþ 2 Š; u 2 ðþ ¼aX 2 ðþ 3 4x 3ðÞ x 3 ðþ 2 þ ax 2 ðþ 2 ½3 þ 2x 3 ðþ ax 2 ðþ 2 Š þ c 21 ½X 1 ðþ x 1 ðþsech 2 x 1 ðþš þ c 22 ½X 2 ðþþx 2ðÞþx 2 ðþ 2 Š: Substituting 17), 1) and 21) as well as the error functions e 1 ðþ and e 2 ðþ into 2), we can get DL 1 ðþ ¼L 1 ð þ 1Þ L 1 ðþ 6 ½jc 11 jþc 1 jc 21 j 1Šje 1 ðþj þ ½jc 12 jþc 1 jc 22 j c 1 Šje 2 ðþj: When setting the parameters c ij ði; j ¼ 1; 2Þ satisfy jc 11 jþc 1 jc 21 j < 1; jc 12 jþc 1 jc 22 j < c 1 ; ð21þ ð22þ ð23þ DL 1 ðþ is negative definite and the error functions e 1 ðþ; e 2 ðþ asymptotically tend to zero, which says that the synchronization is achieved between the system 17) and 1). Next we tae 17) as the drive system, another copied response system is X 2 ð þ 1Þ ¼1 þ X 3 ðþ ax 2 ðþ 2 þ u 3 ðþ; ð24þ X 3 ð þ 1Þ ¼X 1 ðþþbx 2 ðþþu 4 ðþ; where u 3 ðþ; u 4 ðþ are the controller functions to mae the system 17) synchronize with the system 24). We define the error functions in this form: e 3 ðþ ¼X 2 ðþ ax 2 ðþ; e 4 ðþ ¼X 3 ðþþx 3 ðþ 3 ð25þ

5 149 H.-L. An, Y. Chen / Commun Nonlinear Sci Numer Simulat 14 29) and the Lyapunov function is L 2 ðþ ¼je 3 ðþj þ c 2 je 4 ðþj: ð26þ For simplicity, we omit the calculation procedure and only give the final results of the controllers: u 3 ðþ ¼a½1 þ x 3 ðþš 1 X 3 ðþþa½x 2 ðþ 2 ax 2 ðþ 2 Šþc 31 ½X 2 ðþ ax 2 ðþš þ c 32 ½X 3 ðþþx 3 ðþ 3 Š; u 4 ðþ ¼ X 1 ðþ x 3 ðþ 3 3bx 1 ðþ 2 x 2 ðþ b½x 2 ðþþb 2 x 2 ðþ 3 Š 3b 2 x 1 ðþx 2 ðþ 2 þ c 41 ½X 2 ðþ ax 2 ðþš þ c 42 ½X 3 ðþþx 3 ðþ 3 Š: ð27þ With this choice, we can get the derivative of the Lyapunov function DL 2 ðþ ¼L 2 ð þ 1Þ L 2 ðþ ¼je 3 ð þ 1Þj þ c 2 je 4 ð þ 1Þj je 3 ðþj c 2 je 4 ðþj 6 ½jc 31 jþc 2 jc 42 j 1Šje 3 ðþj þ ½jc 32 jþc 2 jc 42 j c 2 Šje 4 ðþj ð2þ with the parameters satisfying jc 31 jþc 2 jc 42 j < 1; jc 32 jþc 2 jc 42 j < c 2 : ð29þ So we can get the error functions e 3 ðþ ¼ ; e 4 ðþ ¼, which denotes the synchronization is achieved between the systems 17) and 24). In this way, we get >< e 1 ðþ ¼ jx 1 ðþ x 1 ðþsech 2 x 1 ðþj ¼ ; e 3 ðþ ¼ jx 2 ðþ ax 2 ðþj ¼ ; e 4 ðþ ¼ jx 3 ðþþx 3 ðþ 3 j¼: So we predict that the function cascade synchronization is achieved for the discrete-time Hénon map. In the following, in order to verify the effectiveness of the above controllers, we draw the numerical simulation figures. The parameters are taen as ða; bþ ¼ð1:7; 3Þ; c 1 ¼ :5; c 2 ¼ :5; c 11 ¼ 1 3 ; c 12 ¼ :2; c 21 ¼ 1; c 22 ¼ :5; c 31 ¼ :5; c 32 ¼ :1; c 41 ¼ :1; c 42 ¼ :1, which satisfy 23) and 29), and the initial values are chosen as ðx 1 ðþ; x 2 ðþ; x 3 ðþþ ¼ ð:2; :7; :6Þ and ðx 1 ðþ; X 2 ðþ; X 3 ðþþ ¼ ð:5; :1; :Þ, respectively. Fig. 1 displays the discrete-time hyperchaotic attractors of the drive system and response system. The states of the error functions are displayed in Fig. 2a c D discrete-time Rössler system ð3þ Tae the 3D discrete-time Rössler system [25] as >< x 1 ð þ 1Þ ¼3:x 1 ðþ½1 x 1 ðþš :5½x 3 ðþþ:35š½1 2x 2 ðþš; x 2 ð þ 1Þ ¼3:7x 2 ðþ½1 x 2 ðþš þ :2x 3 ðþ; x 3 ð þ 1Þ ¼:1½1 1:9x 1 ðþš½ðx 3 ðþþ:35þð1 2x 2 ðþþ 1Š: The steps are similar to the generalized Hénon map, we consider the first copied response system as X 1 ð þ 1Þ ¼3:X 1 ðþ½1 X 1 ðþš :5½x 3 ðþþ:35š½1 2X 2 ðþš þ u 1ðÞ; X 2 ð þ 1Þ ¼3:7X 2 ðþ½1 X 2 ðþš þ :2x 3ðÞþu 2 ðþ: ð31þ ð32þ Here X 1 ðþ; X 2 ðþ; x 3ðÞ are the state variables and u 1 ðþ; u 2 ðþ are the external controllers designed later. x3x3).4.2 x1x1) x2x2) Fig. 1. Hyperchaotic attractors for the discrete-time generalized Hénon map. The blac and small figure is for the drive system and the big one is the attractor for the response system. For the special error functions, the attractor for the response system is extended outside a little compared with the drive system. However, it is obvious that the function cascade synchronization is achieved from this figure.

6 H.-L. An, Y. Chen / Commun Nonlinear Sci Numer Simulat 14 29) a b c error1.2.1 error error Fig. 2. The states of the hyperchaotic synchronization error functions: a) e 1ðÞ ¼X 1ðÞ x 1ðÞsech 2 x 1ðÞ, b) e 3ðÞ ¼X 2ðÞ ax 2ðÞ, c) e 4ðÞ ¼X 3ðÞþx 3ðÞ 3. We define the Lyapunov function L 1 ðþ is the same as 19) L 1 ðþ ¼je 1 ðþj þ c 1 je 2 ðþj; c 1 > with e 1 ðþ ¼X 1 ðþþx 1 ðþ 2 ; e 2 ðþ ¼X 2 ðþþx 2ðÞ tanh x 2 ðþ: When we choose the controllers as u 1 ðþ ¼: þ :133x 1 ðþþ:1225x 2 ðþþ:425x 3 ðþ 14:573x 1 ðþ 2 :1225x 2 ðþ 2 :25x 3 ðþ 2 :35X 2 ðþþ3:x 1ðÞ½X 1 ðþ 1Š þ :76x 1 ðþx 2 ðþx 3 ðþ½x 1 ðþ 1Šþ:1x 2 ðþx 3 ðþ½1 x 2 ðþš½x 3 ðþþ:7š; u 2 ðþ ¼3:7X 2 ðþ½x 2 ðþ 1Šþc 21½X 1 ðþþx 1 ðþ 2 Š þ c 22 ½X 2 ðþ x 2 ðþ tanh x 2 ðþš :2x 3 ðþþtanh½3:7x 2 ðþ 2 3:7x 2 ðþ :2x 3 ðþš½3:7x 2 ðþ 2 3:7x 2 ðþ :2x 3 ðþš: ð33þ Substituting the drive system 31), the response system 32) and the controllers 33) into the derivative of the Lyapunov function DL 1 ðþ, we can get the result DL 1 ðþ ¼L 1 ð þ 1Þ L 1 ðþ 6 ½jc 11 jþc 1 jc 21 j 1Šje 1 ðþj þ ½jc 12 jþc 1 jc 22 j c 1 Šje 2 ðþj: If we set the parameters satisfying jc 11 jþc 1 jc 21 j < 1; jc 12 jþc 1 jc 22 j < c 1 we can now that DL 1 ðþ 6. So we predict that the synchronization between system 31) and 32) is achieved and the error functions e 1 ðþ; e 2 ðþ asymptotically tend to zero. In the following we depict another system as the response system of 31) in this form: X 2 ð þ 1Þ ¼3:7X 2 ðþ½1 X 2 ðþš þ :2X 3 ðþþu 3 ðþ; ð34þ X 3 ð þ 1Þ ¼:1½1 1:9X 1 ðþš½ðx 3 ðþþ:35þð1 2X 2 ðþþ 1Šþu 4 ðþ: According to the Lyapunov stable theory, we choose the Lyapunov function L 2 ðþ as L 2 ðþ ¼je 3 ðþj þ c 2 je 4 ðþj; c 2 > ; where the error functions are in the style of e 3 ðþ ¼X 2 ðþ x 2 ðþtanh 2 x 2 ðþ; e 4 ðþ ¼X 3 ðþ ax 3 ðþ. With the aid of symbolic computation system Maple, we can easily wor out the controllers desired u 3 ðþ and u 4 ðþ that mae the system 31) globally asymptotically synchronize with the system 34). Here we omit the heavy the calculation and just give the final results: u 3 ðþ ¼3:7X 2 ðþ½x 2 ðþ 1Š :2X 3 ðþþc 31 ½X 2 ðþ x 2 ðþtanh 2 x 2 ðþš þ c 32 ½X 3 ðþ ax 3 ðþš þ tanh 2 ½3:7x 2 ðþ 2 3:7x 2 ðþ :2x 3 ðþš½3:7x 2 ðþ 3:7x 2 ðþ 2 :2x 3 ðþš; u 4 ðþ ¼:65ð1 aþ :1235½X 1 ðþ ax 1 ðþš þ :7½X 2 ðþ ax 2 ðþš :1½X 3 ðþ ax 3 ðþš :133½X 1 ðþx 2 ðþ ax 1 ðþx 2 ðþš þ :19½X 1 ðþx 3 ðþ ax 1 ðþx 3 ðþš þ :2½X 2 ðþx 3 ðþ ax 2 ðþx 3 ðþš þ c 41 ½X 2 ðþ x 2 ðþtanh 2 x 2 ðþš þ c 42 ½X 3 ðþ ax 3 ðþš :3½X 1 ðþx 2 ðþx 3 ðþ ax 1 ðþx 2 ðþx 3 ðþš: When the parameters are chosen as jc 31 jþc 2 jc 42 j < 1; jc 32 jþc 2 jc 42 j < c 2, the derivative of the Lyapunov function DL 2 ðþ satisfies DL 2 ðþ ¼L 2 ð þ 1Þ L 2 ðþ ¼je 3 ð þ 1Þj þ c 2 je 4 ð þ 1Þj je 3 ðþj c 2 je 4 ðþj 6 ½jc 31 jþc 2 jc 42 j 1Šje 3 ðþj þ ½jc 32 jþc 2 jc 42 j c 2 Šje 4 ðþj 6 : ð35þ So the system 31) asymptotically synchronizes with the system 34), that is to say e 3 ðþ ¼ and e 4 ðþ ¼. Therefore, we have

7 15 H.-L. An, Y. Chen / Commun Nonlinear Sci Numer Simulat 14 29) x3x3) x2x2) x1x1).6 Fig. 3. Hyperchaotic attractors for the discrete-time Rössler system. The more blac figure is the attractor for the drive system and the other figure is for the response system. a b c error error error Fig. 4. The states of the hyperchaotic synchronization error functions: a) e 1ðÞ ¼X 1ðÞþx 1ðÞ 2, b) e 3ðÞ ¼X 2ðÞ x 2ðÞtanh 2 x 2ðÞ, c) e 4ðÞ ¼X 3ðÞ ax 3ðÞ. >< e 1 ðþ ¼ jx 1 ðþþx 1 ðþ 2 j¼; e 3 ðþ ¼ jx 2 ðþ x 2 ðþtanh 2 x 2 ðþj ¼ ; e 4 ðþ ¼ jx 3 ðþ ax 3 ðþj ¼ ; which indicate the function cascade synchronization is achieved for the discrete-time Rössler system. Numerical simulation results are used in order to verify the effectiveness of the proposed scheme for the Rössler system. The initial values are chosen as ðx 1 ðþ; x 2 ðþ; x 3 ðþþ ¼ ð:1; :2; :5Þ and ðx 1 ðþ; X 2 ðþ; X 3 ðþþ ¼ ð:3; :5; :2Þ and the parameters as c 1 ¼ 1:2; c 2 ¼ :3; c 11 ¼ :5; c 12 ¼ :2; c 21 ¼ 1=3; c 22 ¼ :5; c 31 ¼ :5; c 32 ¼ :1; c 41 ¼ :1; c 42 ¼ :1. Fig. 3 displays the discrete-time hyperchaotic attractors for the Rössler system. Fig. 4a c is the error function figures. 4. Conclusions In summary, this paper has successfully extended the cascade synchronization from the continuous-time systems to the discrete-time dynamical systems. The proposed method are used to realize the discrete-time hyperchaotic synchronization of the 3D generalized Hénon map and the 3D Rössler system, respectively. It is necessary to point out that here we use the scheme presented to study hyperchaotic system, generally speaing, the method by us is effective to synchronize the discrete-time chaotic system, whether it is hyperchaotic system or not. Numerical simulations are implemented to show the effectiveness of the method. Our scheme contains many parameters c ij which provide a larger choice scope in the controllers. In addition, as for the error functions, we can have many different choices, such as a, tanh x, x n and some simple polynomial P an x n, P a n tanh n x, etc., which allow us to adjust these desired scaling error functions to obtain the reliable results needed. So we predict that our scheme is also a powerful method and different form the former synchronization methods. Whether this scheme can be applied to complex networs and fractional dimension will be further studied. Acnowledgements The author would lie to than the referees very much for their careful reading of the manuscript and many valuable suggestions. The wor is supported by the National Natural Science Foundation of China Grant No ), Shanghai Leading Academic Discipline Project No. B412), Zhejiang Provincial Natural Science Foundations of China Grant No. Y6456), ð36þ

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