Chin. Phys. B Vol. 19, No. 2 (2010) 020511 The existence of generalized synchronisation of three bidirectionally coupled chaotic systems? Hu Ai-Hua(胡爱花)a)b)? , Xu Zhen-Yuan(徐振源)a) , and Guo Liu-Xiao(过榴晓)a) a)School of Science, Jiangnan University, Wuxi 214122, China b)School of Information Technology, Jiangnan University, Wuxi 214122, China (Received 18 February 2009; revised manuscript received 5 August 2009) The existence of two types of generalized synchronisation is studied. The model considered here includes three bidirectionally coupled chaotic systems, and two of them denote the driving systems, while the rest stands for the response system. Under certain conditions, the existence of generalised synchronisation can be turned to a problem of compression ?xed point in the family of Lipschitz functions. In addition, theoretical proofs are proposed to the exponential attractive property of generalised synchronisation manifold. Numerical simulations validate the theory. Keywords: generalised synchronisation manifold, compression ?xed point, exponential attractive property PACC: 0545 1. Introduction Since the pioneer work of Pecora and Carroll,[1] chaos synchronisation has attracted much attention due to its potential applications in various ?elds.[2?9] At present, several types of chaos synchronisation have been revealed, such as complete synchronisa- tion (CS), phase synchronisation (PS), lag synchro- nisation (LS), anticipated synchronisation (AS) and generalized synchronisation (GS). GS therein is an in- teresting and more important topic, which includes many synchronisation phenomena observed in labora- tory experiments.[10,11] In 1995, Rulkov et al.[12] ?rst described the GS phenomenon and presented the idea of mutual false nearest neighbours to detect the GS. In 1996, Abarbanel et al.[13] suggested the auxiliary system method to study GS in driving-response sys- tems, and the theory about this method was given in Ref. [14]. Recently, Hramov et al.[15] proposed a modi?ed system approach to study the GS. Based on the modi?ed system method, we have presented some theoretical results about GS in Refs. [16]–[20], however, the models in these references included only two chaotic systems. In this paper, we will further consider the existence of GS, and the model investigated here consists of three bidirection- ally coupled chaotic systems Y , Z, and X. Because this kind of model is close to the dynamical network, which exists everywhere in the real world, the research of the GS of three systems is more valuable. The pa- per proposes theoretical results about two types of GS, when the modi?ed systems Y and Z are chaotic, and system X collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbits. 2. The existence of the GS mani- fold Consider the following three bidirectionally cou- pled chaotic systems: dy dt = A1y + g(y) + K1(x ? y) + K′ 1(z ? y), (1) dz dt = A2z + h(z) + K2(x ? z) + K′ 2(y ? z), (2) dx dt = A3x + f(x) + K3(y ? x) + K′ 3(z ? x), (3) where y ∈ Rn y , z ∈ Rn z , x ∈ Rn x, g(y), h(z) and f(x) are smooth vector functions, Ai, Ki, K′ i, (i = 1, 2, 3) are n * n constant matrices. De?nition 1[13] Given two dynamical systems X and Y , if there exists a manifold S = {(X, Y ), X = Φ(Y )}, which includes at least one attractor, then X and Y carry out GS, and Φ is called the GS map. For three dynamical systems X, Y and Z, there are two expressions of GS manifold. One is X = ?Project supported by the National Natural Science Foundation of China (Grant No. 60575038), the Youth Foundation of Jiangnan University (Grant No. 314000-52210756) and the Program for Innovative Research Team of Jiangnan University. ?Corresponding author. E-mail: aihuahu@126.com c ? 2010 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/journals/cpb http://cpb.iphy.ac.cn 020511-1 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 Φ(Y, Z), in this case Y and Z are the driving systems, X is the response system; the other is X = Φ(Y ), and Z = Ψ(Y ), while Y is the driving system, and X and Z are the response systems. According to the dynamical property of the mod- i?ed Y and Z systems, one has dy dt = A1y + g(y) ? (K1 + K′ 1)y, (4) dz dt = A2z + h(z) ? (K2 + K′ 2)z, (5) and the modi?ed X system dx dt = A3x + f(x) ? (K3 + K′ 3)x. (6) The existence of two kinds of GS is theoretically proved, when the modi?ed Y and Z systems are chaotic, and the modi?ed X system collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbits. Based on the above, the GS manifold here is expressed as X = Φ(Y, Z). 2.1. The ?rst kind of GS Without loss of generality, suppose the system (6) has an equilibrium point x0 = 0 (If x0 ?= 0, we will use linear transformation to make 0 to be its equilibrium point), and the systems (4) and (5) are chaotic. Then the system (6) can be rewritten as dx dt = A3x ? (K3 + K′ 3)x + f′ (0)x + f(x) ? f′ (0)x = Bx + F(x), where B = A3 ? K3 ? K′ 3 + f′ (0), F(x) = f(x) ? f′ (0)x. In addition, all the eigenvalues of matrix B have the negative real part. Now the system (3) can be denoted as follows: dx dt = Bx + F(x) + K3y + K′ 3z. (7) Firstly, we introduce a useful lemma given in Ref. [21]. Lemma 1[21] Suppose ?(t), ψ(t), and ω(t) are continuous functions de?ned on [a, b], ω(t) > 0, while ψ(x) is a monotonous non-negative and non- decreasing function. If ?(x) ≤ ψ(x) + ∫ x a ω(s)?(s)ds, then ?(x) ≤ ψ(x) e ∫ x a ω(s)ds . Lemma 2 Suppose g(y) and h(z) in the systems (1) and (2) satisfy Lipschitz condition, i.e., |g(y1) ? g(y2)| ≤ L |y1 ? y2|, |h(z1) ? h(z2)| ≤ L |z1 ? z2|, where L is a positive constant, | ? | denotes vector norm; e(A1?K1?K′ 1)t ≤ M1 e?t , e(A2?K2?K′ 2)t ≤ M1 e?t , where ? > 0, t > 0, denotes matrix or operator norm. When t ≤ τ, y(τ) = η, z(τ) = χ, the solutions y(t) = y(t; τ, η, χ, x(.)), z(t) = z(t; τ, η, χ, x(.)) of the systems (1) and (2) exist for any continuous function x : (?∞, τ] → U ? Rn x on (?∞, τ], then for any η, η′ , χ, χ′ , x, x′ , the following formulae are satis?ed: |y(t; τ, η, χ, x(.)) ? y′ (t; τ, η′ , χ′ , x′ (.))| ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y ? y′ | + ∥K1∥ |x ? x′ | + ∥K′ 1∥ |z ? z′ |}ds, |z(t; τ, η, χ, x(.)) ? z′ (t; τ, η′ , χ′ , x′ (.))| ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |z ? z′ | + ∥K2∥ |x ? x′ | + ∥K′ 2∥ |y ? y′ |}ds. (8) Proof If dy dt = A1y + g(y) + K1(x ? y) + K′ 1(z ? y), y(τ) = η, dy′ dt = A1y′ + g(y′ ) + K1(x′ ? y′ ) + K′ 1(z′ ? y), y′ (τ) = η′ , then we have d(y ? y′ ) dt = (A1 ? K1 ? K′ 1)(y ? y′ ) + g(y) ? g(y′ ) + K1(x ? x′ ) + K′ 1(z ? z′ ), y(τ) ? y′ (τ) = η ? η′ . 020511-2 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 The solution of the equation can be described as follows: y ? y′ = e(A1?K1?K′ 1)(τ?t) (η ? η′ ) + ∫ τ t e(A?K1?K′ 1)(s?t) [(g(y) ? g(y′ )) + K1(x ? x′ ) + K′ 1(z ? z′ )]ds. Consequently, we have |y ? y′ | ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y ? y′ | + ∥K1∥ |x ? x′ | + ∥K′ 1∥ |z ? z′ |}ds. Another inequality in formulae (8) will be attained as the same. Based on the above, we ?rst present the theoretical result about the existence of the ?rst kind of GS manifold as Theorem 1. Theorem 1 Suppose the following conditions are satis?ed: in the systems (1), (2), and (7) y ∈ Rn y , z ∈ Rn z , and x ∈ Rn x, F : Rn x → Rn x, g : Rn y → Rn y , h : Rn z → Rn z are su?ciently smooth functions, and |F(x) + K3y + K′ 3z ? F(x′ ) ? K3y′ ? K′ 3z′ | ≤ λ[|x ? x′ | + |y ? y′ | + |z ? z′ |], (9) |F(x) + K3y + K′ 3z| ≤ N, (10) eBt ≤ M e?βt , (11) when t ≤ τ, y(τ) = η ∈ Rn y , we have t ≤ τ, z(τ) = χ ∈ Rn z . Based on Lemma 2, the solutions y(t) = y(t; τ, η, χ, x), z(t) = y(t; τ, η, χ, x) of the systems (1) and (2) exist on (?∞, τ). For any η, η′ , χ, χ′ , the inequality (8) holds. Additionally, |x| ≤ D, MN/β ≤ D, λM/β + λM(1 + ?)M1(∥K1∥ + ∥K2∥)/? β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K′ 12∥) < 1, λM(1 + ?)M1 β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K12∥) ≤ ?, (12) where λ, M, M1, N, β, ?, D, ? are all non-negative constants, ∥K′ 12∥ = max{∥K′ 1∥ , ∥K′ 2∥}. Then there exists a GS manifold among systems (1), (2), and (7): S = {(x, y, z) |x = σ(y, z), ?∞ < t < ∞, y ∈ Rn y , z ∈ Rn z }, (13) which satis?es |σ(y, z)| ≤ D, and |σ(y, z) ? σ(y′ , z′ )| ≤ ?(|y ? y′ | + |z ? z′ |). Proof Suppose FD,? is a family of Lipschitz function satisfying |σ(y, z)| ≤ D, |σ(y, z) ? σ(y′ , z′ )| ≤ ?(|y ? y′ | + |z ? z′ |). (14) De?ne the distance ∥σ1 ? σ2∥ = Supy∈Rn y ,z∈Rn z |σ1(y, z) ? σ2(y, z)|. It is easy to verify that FD,? is a complete metric space. Let t ≤ τ, and y(τ) = η ∈ Rn y , then the solutions y(t) = y(t; τ, η, χ) and z(t) = z(t; τ, η, χ) of the systems (1) and (2) exist on (?∞, τ). The operator G is de?ned as follows: G(σ(τ, η)) = ∫ τ ?∞ eB(τ?s) [F(σ(y(s), z(s))) + K3y(s) + K′ 3z(s)]ds, where y(s) = y(s; τ, η, χ) and z(s) = z(s; τ, η, χ). We will prove that the G maps FD,? on FD,?, and it is a contraction map: |G(σ(τ, η))| ≤ ∫ τ ?∞ e?β(τ?s) MN ds = MN β ≤ D. Suppose σ, σ′ are two maps satisfying formulae (14), η, η′ ∈ Rn y . Let y(t) = y(t; τ, η, χ), y′ (t) = y(t; τ, η′ , χ′ ); from expression (8), we have |y ? y′ | ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y ? y′ | + ∥K1∥ |x ? x′ | + ∥K′ 1∥ |z ? z′ |}ds 020511-3 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y1 ? y2| + ∥K1∥ |σ(y, z) ? σ′ (y′ , z′ )| + ∥K′ 1∥ |z ? z′ |}ds ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y1 ? y2| + ∥K1∥ [∥σ ? σ′ ∥ +?(|y ? y′ | + |z ? z′ |)] + ∥K′ 1∥ |z ? z′ |}ds ≤ M1 e?(τ?t) |η ? η′ | + M1 ∥K1∥ ∥σ ? σ′ ∥ e?(τ?t) /? + M1 ∫ τ t e?(s?t) {L |y1 ? y2| + ∥K1∥ ?(|y ? y′ | + |z ? z′ |) + ∥K′ 1∥ ∥ |z ? z′ |}ds. Then we obtain e?(t?τ) |y ? y′ | ≤ M1 |η ? η′ | + M1 ∥K1∥ ∥σ ? σ′ ∥ /? + M1 ∫ τ t e?(s?τ) {L |y ? y′ | + ∥K1∥ ?(|y ? y′ | + |z ? z′ |) + ∥K′ 1∥ |z ? z′ |}ds. (15) Similarly, we also have e?(t?τ) |z ? z′ | ≤ M1 |χ ? χ′ | + M1 ∥K2∥ ∥σ ? σ′ ∥ /? + M1 ∫ τ t e?(s?τ) {L |z ? z′ | + ∥K2∥ ?(|y ? y′ | + |z ? z′ |) + ∥K′ 2∥ |y ? y′ |}ds. (16) Adding formula (15) to formula (16) yields e?(t?τ) (|y ? y′ | + |z ? z′ |) ≤ M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /? + M1 ∫ τ t e?(s?τ) {L(|y ? y′ | + |z ? z′ |) + (∥K1∥ + ∥K2∥)?(|y ? y′ | + |z ? z′ |) + ∥K′ 12∥ |z ? z′ | + ∥K′ 12∥ |y ? y′ |}ds, where ∥K′ 12∥ = max{∥K′ 1∥ , ∥K′ 2∥}. Based on Lemma 1, the following formulae are satis?ed e?(t?τ) (|y ? y′ | + |z ? z′ |) ≤ {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?} e[M1(L+∥K1∥?+∥K2∥?+∥K′ 12∥](τ?t) , |y ? y′ | + |z ? z′ | ≤ {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?} e[M1(L+∥K1∥?+∥K2∥?+∥K′ 12∥)??](τ?t) . Consequently, the following expression can be obtained |G(σ(τ, η)) ? G(σ′ (τ, η′ ))| = ∫ τ ?∞ eB(τ?s) [F(σ(y(s), z(s))) + K3y(s) + K3z(s) ? F(σ′ (y′ (s), z′ (s))) ? K3y′ (s) ? K′ 3z(s)]ds ≤ ∫ τ ?∞ Mλ e?β(τ?s) [|σ(y(s), z(s)) ? σ′ (y′ (s), z′ (s))| + |y(s) ? y′ (s)| + |z(s) ? z′ (s)|]ds ≤ λM ∫ τ ?∞ e?β(τ?s) [∥σ ? σ′ ∥ + (1 + ?)(|y(s) ? y′ (s)| + |z(s) ? z′ (s)|]ds ≤ λM β ∥σ ? σ′ ∥ + λM(1 + ?) ∫ τ ?∞ e?β(τ?s) {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?} * e[M1(L+∥K1∥?+∥K2∥?+∥K′ 12∥)??](τ?s) ds ≤ λM β ∥σ ? σ′ ∥ + λM(1 + ?) β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K′ 12∥) {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?. Then the operator G is a Lipschitz contraction map on FD,?, and has a ?xed point, i.e., there exists a GS manifold S = {(σ(y, z), y, z)}. 020511-4 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 In addition, the manifold S is invariant. Suppose (x0, y0, z0) ∈ S, x0 = σ(y0, z0), for equations dy dt = A1y + g(y) + K1[σ(y, z) ? y] + K′ 1[z ? y], y(t0) = y0, dz dt = A2z + h(z) + K2[σ(y, z) ? z] + K′ 2[y ? z], z(t0) = z0. Let x(t) = σ(y(t), z(t)); there exists the unique solution of Eq. (7) xb(t) = ∫ t ?∞ eB(t?s) [F(σ(y(s), z(s))) + K2y(s) + K′ 2z(s)]ds. Moreover, xb(t) is bounded when t → ∞. It means that (x(t), y(t), z(t)) ∈ S, then S is invariant. Theorem 2 Under the conditions given in Theorem 1, GS manifold is exponentially attractive. Especially, when x(t), y(t), and z(t) are the solutions of the systems (1), (2) and (7) respectively, then |x(t) ? σ(y(t), z(t))| ≤ M |x(t0) ? σ(y(t0), z(t0))| e(?β+Mλ)(t?t0) , where Mλ < β. Proof The solution of Eq. (7) can be written as x(t) = eB(t?t0) x(t0) + ∫ t t0 eB(t?s) [F(x) + K3y + K′ 3z]ds, and on S, σ(y, z) = ∫ t ?∞ eB(t?s) [F(σ(y(s), z(s))) + K3y(s) + K′ 3z(s)]ds, then ξ(t) = x(t) ? σ(y, z) = eB(t?t0) x(t0) + ∫ t t0 eB(t?s) [F(x) + K3y + K′ 3z]ds ? ∫ t ?∞ eB(t?s) [F(σ(y(s), z(s))) + K3y(s) + K′ 3z(s)]ds, ξ(t) ? eB(t?t0) ξ(t0) = ∫ t t0 eB(t?s) [F(x(s)) ? F(σ(y(s), z(s)))]ds. Consequently, ξ(t) = x(t) ? σ(y(t), z(t)) = eB(t?t0) |ξ(t0)| + ∫ t t0 eB(t?s) [F(x(s)) ? F(σ(y(s), z(s)))]ds, |ξ(t)| ≤ e?β(t?t0) M |ξ(t0)| + λM ∫ t t0 e?β(t?s) |x(s) ? σ(y(s), z(s))| ds, eβ(t?t0) |ξ(t)| ≤ M |ξ(t0)| + λM ∫ t t0 e?β(t0?s) |ξ(s)| ds. Based on Lemma 1, we have eβ(t?t0) |ξ(t)| ≤ M |ξ(t0)| eMλ(t?t0) , then |ξ(t)| ≤ M |ξ(t0)| e(?β+Mλ)(t?t0) . 2.2. The second kind of GS In this subsection, consider that the modi?ed response system (6) has an asymptotically stable periodic solution ? x(t) = ? x(t + T), T > 0; and the systems (4) and (5) are chaotic. We introduce another two lemmas in Ref. [22] to prove the existence of this kind of GS manifold. Lemma 3[22] For a linear periodic di?erential equation described as dx dt = A(t)x, (17) 020511-5 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 where A(t) is a continuous function, A(t + T) = A(t), T > 0. Then there exists a continuous function Z(t) with period T, and equation (17) can be transformed into Eq. (18) using w(t) = Z(t)x, dw dt = Bw, (18) where B is a constant matrix. Lemma 4[22] Suppose the real parts of the eigenvalues of the matrix B in the equality (18) are β1, β2, ..., βn; the eigenvalues of (A(t)+A? (t))/2 are α1(t), α2(t), ..., αn(t); then there exists a unitary matrix S(t) = (Si,j(t)) such that βi = 1 T ∫ T 0 n ∑ j=1 |Si,j(t)| 2 αi,j(t)dt, i = 1, 2,n. Letting e(t) = x(t) ? ? x(t), we have de dt = A3x + f(x) ? K3x ? K′ 3x ? [A3 ? x + f(? x) ? K3 ? x ? K′ ? x3] + K3y + K′ 3z = A3e ? K3e ? K′ 3e + Df(? x)e + f(x) ? f(? x) ? Df(? x)e + K3y + K′ 3z = A(t)e + F(x, ? x, e) + K3y + K′ 3z, (19) where A(t) = A ? K3 ? K′ 3 + Df(? x), F(x, ? x, e) = f(x) ? f(? x) ? Df(? x)e. By using Lemma 3 and e(t) = Z?1 (t)w(t), equation (19) is transformed into dw dt = Bw + Z(t)F(Z?1 (t)w(t)) + Z(t)(K3y + K′ 3z). (20) The corresponding GS manifold x(t) = ? x(t) + σ(y(t), z(t)) near ? x(t) is transformed into e(t) = Z?1 (t)w(t) = σ(y(t), z(t)), w(t) = Z(t)σ(y(t), z(t)) = Σ(y(t), z(t)). Obviously, the existence of the ?rst kind of GS manifold between the systems (1) and (2), and the system described by Eq. (21) means the existence of the second kind of GS manifold among the systems (1), (2), and (3), dw dt = Bw + Z(t)F(Z?1 (t)w(t)) + Z(t)K3y + Z(t)K′ 3z. (21) We derive the following theorem similar to Theorems 1 and 2. Theorem 3 Suppose the following conditions are satis?ed: in systems (1), (2), and (21) y ∈ Rn y , z ∈ Rn z and w ∈ Rn w, F : Rn x → Rn x, g : Rn y → Rn y , h : Rn z → Rn z are su?ciently smooth functions, and ZF(Z?1 (t)w(t)) + ZK3y + ZK′ 3z ? ZF(Z?1 (t)w′ (t)) ? ZK3y′ ? ZK′ 3z′ ≤ λ[|w ? w′ | + |y ? y′ | + |z ? z′ |], ZF(Z?1 (t)w(t)) + ZK3y + ZK′ 3z ≤ N, eBt ≤ M e?βt , when t ≤ τ, y(τ) = η ∈ Rn y ; t ≤ τ, z(τ) = χ ∈ Rn z . Based on Lemma 2, the solutions y(t) = y(t; τ, η, χ, x), z(t) = y(t; τ, η, χ, x) of the systems (1) and (2) exist on (?∞, τ). For any η, η′ , χ, χ′ , the inequality (8) holds. Additionally, |x| ≤ D, MN/β ≤ D, λM/β + λM(1 + ?)M1(∥K1∥ + ∥K2∥)/? β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K′ 12∥) < 1, and λM(1 + ?)M1 β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K12∥) ≤ ?, where λ, M, M1, N, β, ?, D, ? are all non-negative constants. Then there exists a GS manifold among systems (1), (2) and (21): S = {(w, y, z) |w = Σ(y, z), ?∞ < t < ∞, y ∈ Rn y , z ∈ Rn z }, 020511-6 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 satisfying |Σ(y, z)| ≤ D |Σ(y, z) ? Σ(y′ , z′ )| ≤ ?(|y ? y′ | + |z ? z′ |). Then there exists a second kind of GS manifold among systems (1), (2) and (3): x(t) = ? x(t) + σ(y(t), z(t)) = ? x(t) + Z?1 Σ(y(t), z(t)). 3. Numerical simulations In this section, we take the following uni?ed chaotic system[23] for illustration. ˙ u = (25α + 10)(v ? u), ˙ v = (28 ? 35α)u ? uw + (29α ? 1)v, ˙ w = uv ? α + 8 3 w. (22) When 0 ≤ α < 0.8, the system (22) is a Lorenz system;[24] when α = 0.8, it is a L¨ u system;[25] and when 0.8 < α ≤ 1, it is a Chen system.[26] We select L¨ u system and Chen system as the driving systems Y and Z, and take the Lorenz system as the response system X, which are described as Y system: ˙ y1 = (25α2 + 10)(y2 ? y1) + m11(x1 ? y1) + m12(z1 ? y1), ˙ y2 = (28 ? 35α2)y1 ? y1y3 + (29α2 ? 1)y2 + m21(x2 ? y2) + m22(z2 ? y2), ˙ y3 = y1y2 ? α2 + 8 3 y3 + m31(x3 ? y3) + m32(z3 ? y3); (23) Z system: ˙ z1 = (25α3 + 10)(z2 ? z1) + n11(x1 ? z1) + n12(y1 ? z1), ˙ z2 = (28 ? 35α3)z1 ? z1z3 + (29α3 ? 1)z2 + n21(x2 ? z2) + n22(y2 ? z2), ˙ z3 = z1z2 ? α3 + 8 3 z3 + n31(x3 ? z3) + n32(y3 ? z3), (24) when α2 = 0.8, α3 = 1, mij = nij = 0 (i = 1, 2, 3; j = 1, 2), systems Y and Z are chaotic. X system: ˙ x1 = (25α1 + 10)(x2 ? x1) + k11(y1 ? x1) + k12(z1 ? x1), ˙ x2 = (28 ? 35α1)x1 ? x1x3 + (29α1 ? 1)x2 + k21(y2 ? x2) + k22(z2 ? x2), ˙ x3 = x1x2 ? α1 + 8 3 x3 + k31(y3 ? x3) + k32(z3 ? x3), (25) when α1 = 0, kij = 0 (i = 1, 2, 3; j = 1, 2), system X is also chaotic. From the systems (23), (24) and (25), we have three modi?ed systems: The modi?ed Y system: ˙ y1 = (25α2 + 10)(y2 ? y1) + m11(?y1) + m12(?y1), ˙ y2 = (28 ? 35α2)y1 ? y1y3 + (29α2 ? 1)y2 + m21(?y2) + m22(?y2), ˙ y3 = y1y2 ? α2 + 8 3 y3 + m31(?y3) + m32(?y3). (26) The modi?ed Z system: ˙ z1 = (25α3 + 10)(z2 ? z1) + n11(?z1) + n12(?z1), ˙ z2 = (28 ? 35α3)z1 ? z1z3 + (29α3 ? 1)z2 + n21(?z2) + n22(?z2), 020511-7 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 ˙ z3 = z1z2 ? α3 + 8 3 z3 + n31(?z3) + n32(?z3). (27) The modi?ed X system: ˙ x1 = (25α1 + 10)(x2 ? x1) + k11(?x1) + k12(?x1), ˙ x2 = (28 ? 35α1)x1 ? x1x3 + (29α1 ? 1)x2 + k21(?x2) + k22(?x2), ˙ x3 = x1x2 ? α1 + 8 3 x3 + k31(?x3) + k32(?x3). (28) We use the auxiliary system approach to verify GS among systems (23), (24) and (25); as a result, the auxiliary system of X is introduced ˙ X1 = (25α1 + 10)(X2 ? X1) + k11(y1 ? X1) + k12(z1 ? X1), ˙ X2 = (28 ? 35α1)X1 ? X1X3 + (29α1 ? 1)X2 + k21(y2 ? X2) + k22(z2 ? X2), ˙ X3 = X1X2 ? α1 + 8 3 X3 + k31(y3 ? X3) + k32(z3 ? X3). (29) In the following simulations, the initial values of systems (23), (24), (25), and (29) are chosen to be (0.1, 0.2, 0.3), (0.4, 0.5, 0.6), (0.5, 1, 1.5) and (1.1, 1.2, 1.3), respectively. The ?rst kind of GS In this case, the system (28) collapses to an asymptotically stable equilibrium, and the systems (26) and (27) are chaotic. We select m11 = m12 = m21 = m22 = m31 = m32 = 0.1, n11 = n12 = n21 = n22 = n31 = n32 = 0.1, k11 = k12 = k21 = k22 = k31 = k32 = 0.5. Under the above conditions, we ?nd that the mod- i?ed X system (28) approaches the asymptotically sta- ble equilibrium (–9.2736, –10.2010, 25.8000), while the systems (26) and (27) are still chaotic. The response system (25) and the auxiliary system (29) carry out CS, which is shown in Fig. 1. Based on the auxiliary system method, we can have that the driving systems (23), (24), and the response system (25) realize GS. Fig. 1. Graphical representation of errors between the systems (25) and (29). The second kind of GS In this case, system (28) collapses to asymptotically stable periodic orbits, and the systems (26) and (27) are chaotic. We choose m11 = 2, m12 = 0.05, m21 = 1.5, m22 = 0.05, m31 = 2, m32 = 0.05, n11 = 0.74, n12 = 0.05, n21 = 1.1, n22 = 0.05, n31 = 1, n32 = 0.05, k11 = 0.19, k12 = 0, k21 = 0.01, k22 = 0.13, k31 = 0.24, k32 = 0.25. The results are shown in Figs. 2 and 3. Fig. 2. Graphical representation of the periodic orbit of the system (28). Fig. 3. Graphical representation of errors between the systems (25) and (29). 020511-8 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 From the above ?gures, we can see that the response system and the driving systems achieve two kinds of GS. It is worth while pointing out that the approach given in this paper can be easily extended to other complex systems, such as the multi-scroll chaotic systems.[27,28] 4. Conclusions and discussions The mathematical mechanism of the GS of chaotic systems has been unclear for many years. In the earlier works, we presented some theoretical results about the GS manifold of two coupled chaotic systems. This paper further studied the existence of GS manifold of three bidirectionally coupled chaotic systems, and theoretical proofs are proposed to the exponential attractive property of GS manifold. Future work will be focused on the analysis of GS of systems in the complex networks. References [1] Pecora L M and Carroll T L 1990 Phys. Rev. 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Sin. 55 590 (in Chinese) [17] Zhang R and Xu Z Y 2008 J. Sys. Sci. & Math. Scis. 28 1509 (in Chinese) [18] Guo L X and Xu Z Y 2008 Acta Phys. Sin. 57 6086 (in Chinese) [19] Guo L X and Xu Z Y 2008 Chaos 18 033134 [20] Hu A H, Xu Z Y and Guo L X 2009 Acta Phys. Sin. 58 6030 (in Chinese) [21] Wang R H and Wu Z Q 1979 Lectures on Ordinary Di?er- ential Equations (Beijing: People's Education Press) (in Chinese) [22] Ling Z S 1986 Almost Periodic Di?erential Equation and Integral Manifold (Shanghai: Shanghai Science and Tech- nology Press) (in Chinese) [23] Lu J A, Wu X Q and L¨ u J H 2002 Phys. Lett. A 305 365 [24] Lorenz 1963 J. Atmos. Sci. 20 130 [25] L¨ u J H and Chen G R 2002 Int. J. Bifur. Chaos 12 659 [26] Chen G R and Ueta T 1999 Int. J. Bifur. Chaos 9 1465 [27] L¨ u J H, Han F L, Yu X H and Chen G R 2004 Automatica 40 1677 [28] L¨ u J H, Yu S M, Leung H and Chen G R 2006 IEEE Trans. Circuits Syst. I 53 149 020511-9
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Chin. Phys. B Vol. 19, No. 2 (2010) 020511 The existence of generalized synchronisation of three bidirectionally coupled chaotic systems? Hu Ai-Hua(胡爱花)a)b)? , Xu Zhen-Yuan(徐振源)a) , and Guo Liu-Xiao(过榴晓)a) a)School of Science, Jiangnan University, Wuxi 214122, China b)School of Information Technology, Jiangnan University, Wuxi 214122, China (Received 18 February 2009; revised manuscript received 5 August 2009) The existence of two types of generalized synchronisation is studied. The model considered here includes three bidirectionally coupled chaotic systems, and two of them denote the driving systems, while the rest stands for the response system. Under certain conditions, the existence of generalised synchronisation can be turned to a problem of compression ?xed point in the family of Lipschitz functions. In addition, theoretical proofs are proposed to the exponential attractive property of generalised synchronisation manifold. Numerical simulations validate the theory. Keywords: generalised synchronisation manifold, compression ?xed point, exponential attractive property PACC: 0545 1. Introduction Since the pioneer work of Pecora and Carroll,[1] chaos synchronisation has attracted much attention due to its potential applications in various ?elds.[2?9] At present, several types of chaos synchronisation have been revealed, such as complete synchronisa- tion (CS), phase synchronisation (PS), lag synchro- nisation (LS), anticipated synchronisation (AS) and generalized synchronisation (GS). GS therein is an in- teresting and more important topic, which includes many synchronisation phenomena observed in labora- tory experiments.[10,11] In 1995, Rulkov et al.[12] ?rst described the GS phenomenon and presented the idea of mutual false nearest neighbours to detect the GS. In 1996, Abarbanel et al.[13] suggested the auxiliary system method to study GS in driving-response sys- tems, and the theory about this method was given in Ref. [14]. Recently, Hramov et al.[15] proposed a modi?ed system approach to study the GS. Based on the modi?ed system method, we have presented some theoretical results about GS in Refs. [16]–[20], however, the models in these references included only two chaotic systems. In this paper, we will further consider the existence of GS, and the model investigated here consists of three bidirection- ally coupled chaotic systems Y , Z, and X. Because this kind of model is close to the dynamical network, which exists everywhere in the real world, the research of the GS of three systems is more valuable. The pa- per proposes theoretical results about two types of GS, when the modi?ed systems Y and Z are chaotic, and system X collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbits. 2. The existence of the GS mani- fold Consider the following three bidirectionally cou- pled chaotic systems: dy dt = A1y + g(y) + K1(x ? y) + K′ 1(z ? y), (1) dz dt = A2z + h(z) + K2(x ? z) + K′ 2(y ? z), (2) dx dt = A3x + f(x) + K3(y ? x) + K′ 3(z ? x), (3) where y ∈ Rn y , z ∈ Rn z , x ∈ Rn x, g(y), h(z) and f(x) are smooth vector functions, Ai, Ki, K′ i, (i = 1, 2, 3) are n * n constant matrices. De?nition 1[13] Given two dynamical systems X and Y , if there exists a manifold S = {(X, Y ), X = Φ(Y )}, which includes at least one attractor, then X and Y carry out GS, and Φ is called the GS map. For three dynamical systems X, Y and Z, there are two expressions of GS manifold. One is X = ?Project supported by the National Natural Science Foundation of China (Grant No. 60575038), the Youth Foundation of Jiangnan University (Grant No. 314000-52210756) and the Program for Innovative Research Team of Jiangnan University. ?Corresponding author. E-mail: aihuahu@126.com c ? 2010 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/journals/cpb http://cpb.iphy.ac.cn 020511-1 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 Φ(Y, Z), in this case Y and Z are the driving systems, X is the response system; the other is X = Φ(Y ), and Z = Ψ(Y ), while Y is the driving system, and X and Z are the response systems. According to the dynamical property of the mod- i?ed Y and Z systems, one has dy dt = A1y + g(y) ? (K1 + K′ 1)y, (4) dz dt = A2z + h(z) ? (K2 + K′ 2)z, (5) and the modi?ed X system dx dt = A3x + f(x) ? (K3 + K′ 3)x. (6) The existence of two kinds of GS is theoretically proved, when the modi?ed Y and Z systems are chaotic, and the modi?ed X system collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbits. Based on the above, the GS manifold here is expressed as X = Φ(Y, Z). 2.1. The ?rst kind of GS Without loss of generality, suppose the system (6) has an equilibrium point x0 = 0 (If x0 ?= 0, we will use linear transformation to make 0 to be its equilibrium point), and the systems (4) and (5) are chaotic. Then the system (6) can be rewritten as dx dt = A3x ? (K3 + K′ 3)x + f′ (0)x + f(x) ? f′ (0)x = Bx + F(x), where B = A3 ? K3 ? K′ 3 + f′ (0), F(x) = f(x) ? f′ (0)x. In addition, all the eigenvalues of matrix B have the negative real part. Now the system (3) can be denoted as follows: dx dt = Bx + F(x) + K3y + K′ 3z. (7) Firstly, we introduce a useful lemma given in Ref. [21]. Lemma 1[21] Suppose ?(t), ψ(t), and ω(t) are continuous functions de?ned on [a, b], ω(t) > 0, while ψ(x) is a monotonous non-negative and non- decreasing function. If ?(x) ≤ ψ(x) + ∫ x a ω(s)?(s)ds, then ?(x) ≤ ψ(x) e ∫ x a ω(s)ds . Lemma 2 Suppose g(y) and h(z) in the systems (1) and (2) satisfy Lipschitz condition, i.e., |g(y1) ? g(y2)| ≤ L |y1 ? y2|, |h(z1) ? h(z2)| ≤ L |z1 ? z2|, where L is a positive constant, | ? | denotes vector norm; e(A1?K1?K′ 1)t ≤ M1 e?t , e(A2?K2?K′ 2)t ≤ M1 e?t , where ? > 0, t > 0, denotes matrix or operator norm. When t ≤ τ, y(τ) = η, z(τ) = χ, the solutions y(t) = y(t; τ, η, χ, x(.)), z(t) = z(t; τ, η, χ, x(.)) of the systems (1) and (2) exist for any continuous function x : (?∞, τ] → U ? Rn x on (?∞, τ], then for any η, η′ , χ, χ′ , x, x′ , the following formulae are satis?ed: |y(t; τ, η, χ, x(.)) ? y′ (t; τ, η′ , χ′ , x′ (.))| ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y ? y′ | + ∥K1∥ |x ? x′ | + ∥K′ 1∥ |z ? z′ |}ds, |z(t; τ, η, χ, x(.)) ? z′ (t; τ, η′ , χ′ , x′ (.))| ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |z ? z′ | + ∥K2∥ |x ? x′ | + ∥K′ 2∥ |y ? y′ |}ds. (8) Proof If dy dt = A1y + g(y) + K1(x ? y) + K′ 1(z ? y), y(τ) = η, dy′ dt = A1y′ + g(y′ ) + K1(x′ ? y′ ) + K′ 1(z′ ? y), y′ (τ) = η′ , then we have d(y ? y′ ) dt = (A1 ? K1 ? K′ 1)(y ? y′ ) + g(y) ? g(y′ ) + K1(x ? x′ ) + K′ 1(z ? z′ ), y(τ) ? y′ (τ) = η ? η′ . 020511-2 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 The solution of the equation can be described as follows: y ? y′ = e(A1?K1?K′ 1)(τ?t) (η ? η′ ) + ∫ τ t e(A?K1?K′ 1)(s?t) [(g(y) ? g(y′ )) + K1(x ? x′ ) + K′ 1(z ? z′ )]ds. Consequently, we have |y ? y′ | ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y ? y′ | + ∥K1∥ |x ? x′ | + ∥K′ 1∥ |z ? z′ |}ds. Another inequality in formulae (8) will be attained as the same. Based on the above, we ?rst present the theoretical result about the existence of the ?rst kind of GS manifold as Theorem 1. Theorem 1 Suppose the following conditions are satis?ed: in the systems (1), (2), and (7) y ∈ Rn y , z ∈ Rn z , and x ∈ Rn x, F : Rn x → Rn x, g : Rn y → Rn y , h : Rn z → Rn z are su?ciently smooth functions, and |F(x) + K3y + K′ 3z ? F(x′ ) ? K3y′ ? K′ 3z′ | ≤ λ[|x ? x′ | + |y ? y′ | + |z ? z′ |], (9) |F(x) + K3y + K′ 3z| ≤ N, (10) eBt ≤ M e?βt , (11) when t ≤ τ, y(τ) = η ∈ Rn y , we have t ≤ τ, z(τ) = χ ∈ Rn z . Based on Lemma 2, the solutions y(t) = y(t; τ, η, χ, x), z(t) = y(t; τ, η, χ, x) of the systems (1) and (2) exist on (?∞, τ). For any η, η′ , χ, χ′ , the inequality (8) holds. Additionally, |x| ≤ D, MN/β ≤ D, λM/β + λM(1 + ?)M1(∥K1∥ + ∥K2∥)/? β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K′ 12∥) < 1, λM(1 + ?)M1 β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K12∥) ≤ ?, (12) where λ, M, M1, N, β, ?, D, ? are all non-negative constants, ∥K′ 12∥ = max{∥K′ 1∥ , ∥K′ 2∥}. Then there exists a GS manifold among systems (1), (2), and (7): S = {(x, y, z) |x = σ(y, z), ?∞ < t < ∞, y ∈ Rn y , z ∈ Rn z }, (13) which satis?es |σ(y, z)| ≤ D, and |σ(y, z) ? σ(y′ , z′ )| ≤ ?(|y ? y′ | + |z ? z′ |). Proof Suppose FD,? is a family of Lipschitz function satisfying |σ(y, z)| ≤ D, |σ(y, z) ? σ(y′ , z′ )| ≤ ?(|y ? y′ | + |z ? z′ |). (14) De?ne the distance ∥σ1 ? σ2∥ = Supy∈Rn y ,z∈Rn z |σ1(y, z) ? σ2(y, z)|. It is easy to verify that FD,? is a complete metric space. Let t ≤ τ, and y(τ) = η ∈ Rn y , then the solutions y(t) = y(t; τ, η, χ) and z(t) = z(t; τ, η, χ) of the systems (1) and (2) exist on (?∞, τ). The operator G is de?ned as follows: G(σ(τ, η)) = ∫ τ ?∞ eB(τ?s) [F(σ(y(s), z(s))) + K3y(s) + K′ 3z(s)]ds, where y(s) = y(s; τ, η, χ) and z(s) = z(s; τ, η, χ). We will prove that the G maps FD,? on FD,?, and it is a contraction map: |G(σ(τ, η))| ≤ ∫ τ ?∞ e?β(τ?s) MN ds = MN β ≤ D. Suppose σ, σ′ are two maps satisfying formulae (14), η, η′ ∈ Rn y . Let y(t) = y(t; τ, η, χ), y′ (t) = y(t; τ, η′ , χ′ ); from expression (8), we have |y ? y′ | ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y ? y′ | + ∥K1∥ |x ? x′ | + ∥K′ 1∥ |z ? z′ |}ds 020511-3 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y1 ? y2| + ∥K1∥ |σ(y, z) ? σ′ (y′ , z′ )| + ∥K′ 1∥ |z ? z′ |}ds ≤ M1 e?(τ?t) |η ? η′ | + M1 ∫ τ t e?(s?t) {L |y1 ? y2| + ∥K1∥ [∥σ ? σ′ ∥ +?(|y ? y′ | + |z ? z′ |)] + ∥K′ 1∥ |z ? z′ |}ds ≤ M1 e?(τ?t) |η ? η′ | + M1 ∥K1∥ ∥σ ? σ′ ∥ e?(τ?t) /? + M1 ∫ τ t e?(s?t) {L |y1 ? y2| + ∥K1∥ ?(|y ? y′ | + |z ? z′ |) + ∥K′ 1∥ ∥ |z ? z′ |}ds. Then we obtain e?(t?τ) |y ? y′ | ≤ M1 |η ? η′ | + M1 ∥K1∥ ∥σ ? σ′ ∥ /? + M1 ∫ τ t e?(s?τ) {L |y ? y′ | + ∥K1∥ ?(|y ? y′ | + |z ? z′ |) + ∥K′ 1∥ |z ? z′ |}ds. (15) Similarly, we also have e?(t?τ) |z ? z′ | ≤ M1 |χ ? χ′ | + M1 ∥K2∥ ∥σ ? σ′ ∥ /? + M1 ∫ τ t e?(s?τ) {L |z ? z′ | + ∥K2∥ ?(|y ? y′ | + |z ? z′ |) + ∥K′ 2∥ |y ? y′ |}ds. (16) Adding formula (15) to formula (16) yields e?(t?τ) (|y ? y′ | + |z ? z′ |) ≤ M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /? + M1 ∫ τ t e?(s?τ) {L(|y ? y′ | + |z ? z′ |) + (∥K1∥ + ∥K2∥)?(|y ? y′ | + |z ? z′ |) + ∥K′ 12∥ |z ? z′ | + ∥K′ 12∥ |y ? y′ |}ds, where ∥K′ 12∥ = max{∥K′ 1∥ , ∥K′ 2∥}. Based on Lemma 1, the following formulae are satis?ed e?(t?τ) (|y ? y′ | + |z ? z′ |) ≤ {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?} e[M1(L+∥K1∥?+∥K2∥?+∥K′ 12∥](τ?t) , |y ? y′ | + |z ? z′ | ≤ {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?} e[M1(L+∥K1∥?+∥K2∥?+∥K′ 12∥)??](τ?t) . Consequently, the following expression can be obtained |G(σ(τ, η)) ? G(σ′ (τ, η′ ))| = ∫ τ ?∞ eB(τ?s) [F(σ(y(s), z(s))) + K3y(s) + K3z(s) ? F(σ′ (y′ (s), z′ (s))) ? K3y′ (s) ? K′ 3z(s)]ds ≤ ∫ τ ?∞ Mλ e?β(τ?s) [|σ(y(s), z(s)) ? σ′ (y′ (s), z′ (s))| + |y(s) ? y′ (s)| + |z(s) ? z′ (s)|]ds ≤ λM ∫ τ ?∞ e?β(τ?s) [∥σ ? σ′ ∥ + (1 + ?)(|y(s) ? y′ (s)| + |z(s) ? z′ (s)|]ds ≤ λM β ∥σ ? σ′ ∥ + λM(1 + ?) ∫ τ ?∞ e?β(τ?s) {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?} * e[M1(L+∥K1∥?+∥K2∥?+∥K′ 12∥)??](τ?s) ds ≤ λM β ∥σ ? σ′ ∥ + λM(1 + ?) β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K′ 12∥) {M1(|η ? η′ | + |χ ? χ′ |) + M1(∥K1∥ + ∥K2∥) ∥σ ? σ′ ∥ /?. Then the operator G is a Lipschitz contraction map on FD,?, and has a ?xed point, i.e., there exists a GS manifold S = {(σ(y, z), y, z)}. 020511-4 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 In addition, the manifold S is invariant. Suppose (x0, y0, z0) ∈ S, x0 = σ(y0, z0), for equations dy dt = A1y + g(y) + K1[σ(y, z) ? y] + K′ 1[z ? y], y(t0) = y0, dz dt = A2z + h(z) + K2[σ(y, z) ? z] + K′ 2[y ? z], z(t0) = z0. Let x(t) = σ(y(t), z(t)); there exists the unique solution of Eq. (7) xb(t) = ∫ t ?∞ eB(t?s) [F(σ(y(s), z(s))) + K2y(s) + K′ 2z(s)]ds. Moreover, xb(t) is bounded when t → ∞. It means that (x(t), y(t), z(t)) ∈ S, then S is invariant. Theorem 2 Under the conditions given in Theorem 1, GS manifold is exponentially attractive. Especially, when x(t), y(t), and z(t) are the solutions of the systems (1), (2) and (7) respectively, then |x(t) ? σ(y(t), z(t))| ≤ M |x(t0) ? σ(y(t0), z(t0))| e(?β+Mλ)(t?t0) , where Mλ < β. Proof The solution of Eq. (7) can be written as x(t) = eB(t?t0) x(t0) + ∫ t t0 eB(t?s) [F(x) + K3y + K′ 3z]ds, and on S, σ(y, z) = ∫ t ?∞ eB(t?s) [F(σ(y(s), z(s))) + K3y(s) + K′ 3z(s)]ds, then ξ(t) = x(t) ? σ(y, z) = eB(t?t0) x(t0) + ∫ t t0 eB(t?s) [F(x) + K3y + K′ 3z]ds ? ∫ t ?∞ eB(t?s) [F(σ(y(s), z(s))) + K3y(s) + K′ 3z(s)]ds, ξ(t) ? eB(t?t0) ξ(t0) = ∫ t t0 eB(t?s) [F(x(s)) ? F(σ(y(s), z(s)))]ds. Consequently, ξ(t) = x(t) ? σ(y(t), z(t)) = eB(t?t0) |ξ(t0)| + ∫ t t0 eB(t?s) [F(x(s)) ? F(σ(y(s), z(s)))]ds, |ξ(t)| ≤ e?β(t?t0) M |ξ(t0)| + λM ∫ t t0 e?β(t?s) |x(s) ? σ(y(s), z(s))| ds, eβ(t?t0) |ξ(t)| ≤ M |ξ(t0)| + λM ∫ t t0 e?β(t0?s) |ξ(s)| ds. Based on Lemma 1, we have eβ(t?t0) |ξ(t)| ≤ M |ξ(t0)| eMλ(t?t0) , then |ξ(t)| ≤ M |ξ(t0)| e(?β+Mλ)(t?t0) . 2.2. The second kind of GS In this subsection, consider that the modi?ed response system (6) has an asymptotically stable periodic solution ? x(t) = ? x(t + T), T > 0; and the systems (4) and (5) are chaotic. We introduce another two lemmas in Ref. [22] to prove the existence of this kind of GS manifold. Lemma 3[22] For a linear periodic di?erential equation described as dx dt = A(t)x, (17) 020511-5 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 where A(t) is a continuous function, A(t + T) = A(t), T > 0. Then there exists a continuous function Z(t) with period T, and equation (17) can be transformed into Eq. (18) using w(t) = Z(t)x, dw dt = Bw, (18) where B is a constant matrix. Lemma 4[22] Suppose the real parts of the eigenvalues of the matrix B in the equality (18) are β1, β2, ..., βn; the eigenvalues of (A(t)+A? (t))/2 are α1(t), α2(t), ..., αn(t); then there exists a unitary matrix S(t) = (Si,j(t)) such that βi = 1 T ∫ T 0 n ∑ j=1 |Si,j(t)| 2 αi,j(t)dt, i = 1, 2,n. Letting e(t) = x(t) ? ? x(t), we have de dt = A3x + f(x) ? K3x ? K′ 3x ? [A3 ? x + f(? x) ? K3 ? x ? K′ ? x3] + K3y + K′ 3z = A3e ? K3e ? K′ 3e + Df(? x)e + f(x) ? f(? x) ? Df(? x)e + K3y + K′ 3z = A(t)e + F(x, ? x, e) + K3y + K′ 3z, (19) where A(t) = A ? K3 ? K′ 3 + Df(? x), F(x, ? x, e) = f(x) ? f(? x) ? Df(? x)e. By using Lemma 3 and e(t) = Z?1 (t)w(t), equation (19) is transformed into dw dt = Bw + Z(t)F(Z?1 (t)w(t)) + Z(t)(K3y + K′ 3z). (20) The corresponding GS manifold x(t) = ? x(t) + σ(y(t), z(t)) near ? x(t) is transformed into e(t) = Z?1 (t)w(t) = σ(y(t), z(t)), w(t) = Z(t)σ(y(t), z(t)) = Σ(y(t), z(t)). Obviously, the existence of the ?rst kind of GS manifold between the systems (1) and (2), and the system described by Eq. (21) means the existence of the second kind of GS manifold among the systems (1), (2), and (3), dw dt = Bw + Z(t)F(Z?1 (t)w(t)) + Z(t)K3y + Z(t)K′ 3z. (21) We derive the following theorem similar to Theorems 1 and 2. Theorem 3 Suppose the following conditions are satis?ed: in systems (1), (2), and (21) y ∈ Rn y , z ∈ Rn z and w ∈ Rn w, F : Rn x → Rn x, g : Rn y → Rn y , h : Rn z → Rn z are su?ciently smooth functions, and ZF(Z?1 (t)w(t)) + ZK3y + ZK′ 3z ? ZF(Z?1 (t)w′ (t)) ? ZK3y′ ? ZK′ 3z′ ≤ λ[|w ? w′ | + |y ? y′ | + |z ? z′ |], ZF(Z?1 (t)w(t)) + ZK3y + ZK′ 3z ≤ N, eBt ≤ M e?βt , when t ≤ τ, y(τ) = η ∈ Rn y ; t ≤ τ, z(τ) = χ ∈ Rn z . Based on Lemma 2, the solutions y(t) = y(t; τ, η, χ, x), z(t) = y(t; τ, η, χ, x) of the systems (1) and (2) exist on (?∞, τ). For any η, η′ , χ, χ′ , the inequality (8) holds. Additionally, |x| ≤ D, MN/β ≤ D, λM/β + λM(1 + ?)M1(∥K1∥ + ∥K2∥)/? β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K′ 12∥) < 1, and λM(1 + ?)M1 β + ? ? M1(L + ∥K1∥ ? + ∥K2∥ ? + ∥K12∥) ≤ ?, where λ, M, M1, N, β, ?, D, ? are all non-negative constants. Then there exists a GS manifold among systems (1), (2) and (21): S = {(w, y, z) |w = Σ(y, z), ?∞ < t < ∞, y ∈ Rn y , z ∈ Rn z }, 020511-6 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 satisfying |Σ(y, z)| ≤ D |Σ(y, z) ? Σ(y′ , z′ )| ≤ ?(|y ? y′ | + |z ? z′ |). Then there exists a second kind of GS manifold among systems (1), (2) and (3): x(t) = ? x(t) + σ(y(t), z(t)) = ? x(t) + Z?1 Σ(y(t), z(t)). 3. Numerical simulations In this section, we take the following uni?ed chaotic system[23] for illustration. ˙ u = (25α + 10)(v ? u), ˙ v = (28 ? 35α)u ? uw + (29α ? 1)v, ˙ w = uv ? α + 8 3 w. (22) When 0 ≤ α < 0.8, the system (22) is a Lorenz system;[24] when α = 0.8, it is a L¨ u system;[25] and when 0.8 < α ≤ 1, it is a Chen system.[26] We select L¨ u system and Chen system as the driving systems Y and Z, and take the Lorenz system as the response system X, which are described as Y system: ˙ y1 = (25α2 + 10)(y2 ? y1) + m11(x1 ? y1) + m12(z1 ? y1), ˙ y2 = (28 ? 35α2)y1 ? y1y3 + (29α2 ? 1)y2 + m21(x2 ? y2) + m22(z2 ? y2), ˙ y3 = y1y2 ? α2 + 8 3 y3 + m31(x3 ? y3) + m32(z3 ? y3); (23) Z system: ˙ z1 = (25α3 + 10)(z2 ? z1) + n11(x1 ? z1) + n12(y1 ? z1), ˙ z2 = (28 ? 35α3)z1 ? z1z3 + (29α3 ? 1)z2 + n21(x2 ? z2) + n22(y2 ? z2), ˙ z3 = z1z2 ? α3 + 8 3 z3 + n31(x3 ? z3) + n32(y3 ? z3), (24) when α2 = 0.8, α3 = 1, mij = nij = 0 (i = 1, 2, 3; j = 1, 2), systems Y and Z are chaotic. X system: ˙ x1 = (25α1 + 10)(x2 ? x1) + k11(y1 ? x1) + k12(z1 ? x1), ˙ x2 = (28 ? 35α1)x1 ? x1x3 + (29α1 ? 1)x2 + k21(y2 ? x2) + k22(z2 ? x2), ˙ x3 = x1x2 ? α1 + 8 3 x3 + k31(y3 ? x3) + k32(z3 ? x3), (25) when α1 = 0, kij = 0 (i = 1, 2, 3; j = 1, 2), system X is also chaotic. From the systems (23), (24) and (25), we have three modi?ed systems: The modi?ed Y system: ˙ y1 = (25α2 + 10)(y2 ? y1) + m11(?y1) + m12(?y1), ˙ y2 = (28 ? 35α2)y1 ? y1y3 + (29α2 ? 1)y2 + m21(?y2) + m22(?y2), ˙ y3 = y1y2 ? α2 + 8 3 y3 + m31(?y3) + m32(?y3). (26) The modi?ed Z system: ˙ z1 = (25α3 + 10)(z2 ? z1) + n11(?z1) + n12(?z1), ˙ z2 = (28 ? 35α3)z1 ? z1z3 + (29α3 ? 1)z2 + n21(?z2) + n22(?z2), 020511-7 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 ˙ z3 = z1z2 ? α3 + 8 3 z3 + n31(?z3) + n32(?z3). (27) The modi?ed X system: ˙ x1 = (25α1 + 10)(x2 ? x1) + k11(?x1) + k12(?x1), ˙ x2 = (28 ? 35α1)x1 ? x1x3 + (29α1 ? 1)x2 + k21(?x2) + k22(?x2), ˙ x3 = x1x2 ? α1 + 8 3 x3 + k31(?x3) + k32(?x3). (28) We use the auxiliary system approach to verify GS among systems (23), (24) and (25); as a result, the auxiliary system of X is introduced ˙ X1 = (25α1 + 10)(X2 ? X1) + k11(y1 ? X1) + k12(z1 ? X1), ˙ X2 = (28 ? 35α1)X1 ? X1X3 + (29α1 ? 1)X2 + k21(y2 ? X2) + k22(z2 ? X2), ˙ X3 = X1X2 ? α1 + 8 3 X3 + k31(y3 ? X3) + k32(z3 ? X3). (29) In the following simulations, the initial values of systems (23), (24), (25), and (29) are chosen to be (0.1, 0.2, 0.3), (0.4, 0.5, 0.6), (0.5, 1, 1.5) and (1.1, 1.2, 1.3), respectively. The ?rst kind of GS In this case, the system (28) collapses to an asymptotically stable equilibrium, and the systems (26) and (27) are chaotic. We select m11 = m12 = m21 = m22 = m31 = m32 = 0.1, n11 = n12 = n21 = n22 = n31 = n32 = 0.1, k11 = k12 = k21 = k22 = k31 = k32 = 0.5. Under the above conditions, we ?nd that the mod- i?ed X system (28) approaches the asymptotically sta- ble equilibrium (–9.2736, –10.2010, 25.8000), while the systems (26) and (27) are still chaotic. The response system (25) and the auxiliary system (29) carry out CS, which is shown in Fig. 1. Based on the auxiliary system method, we can have that the driving systems (23), (24), and the response system (25) realize GS. Fig. 1. Graphical representation of errors between the systems (25) and (29). The second kind of GS In this case, system (28) collapses to asymptotically stable periodic orbits, and the systems (26) and (27) are chaotic. We choose m11 = 2, m12 = 0.05, m21 = 1.5, m22 = 0.05, m31 = 2, m32 = 0.05, n11 = 0.74, n12 = 0.05, n21 = 1.1, n22 = 0.05, n31 = 1, n32 = 0.05, k11 = 0.19, k12 = 0, k21 = 0.01, k22 = 0.13, k31 = 0.24, k32 = 0.25. The results are shown in Figs. 2 and 3. Fig. 2. Graphical representation of the periodic orbit of the system (28). Fig. 3. Graphical representation of errors between the systems (25) and (29). 020511-8 Chin. Phys. B Vol. 19, No. 2 (2010) 020511 From the above ?gures, we can see that the response system and the driving systems achieve two kinds of GS. It is worth while pointing out that the approach given in this paper can be easily extended to other complex systems, such as the multi-scroll chaotic systems.[27,28] 4. Conclusions and discussions The mathematical mechanism of the GS of chaotic systems has been unclear for many years. In the earlier works, we presented some theoretical results about the GS manifold of two coupled chaotic systems. This paper further studied the existence of GS manifold of three bidirectionally coupled chaotic systems, and theoretical proofs are proposed to the exponential attractive property of GS manifold. Future work will be focused on the analysis of GS of systems in the complex networks. References [1] Pecora L M and Carroll T L 1990 Phys. Rev. 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