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Global fixed-time synchronization of chaotic systems with different dimensions
Global fixed-time synchronization of chaotic systems with different dimensions
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Global fixed-time synchronization of chaotic systems with different dimensions Xiaozhen Guo, Guoguang Wen, Zhaoxia Peng, Yunlong Zhang PII: DOI: Reference:
S0016-0032(19)30867-1 https://doi.org/10.1016/j.jfranklin.2019.11.063 FI 4298
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Journal of the Franklin Institute
Received date: Revised date: Accepted date:
23 April 2019 26 September 2019 25 November 2019
Please cite this article as: Xiaozhen Guo, Guoguang Wen, Zhaoxia Peng, Yunlong Zhang, Global fixed-time synchronization of chaotic systems with different dimensions, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.063
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Global fixed-time synchronization of chaotic systems with different dimensions Xiaozhen Guoa , Guoguang Wena , Zhaoxia Pengb,∗, Yunlong Zhangc a Department
of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China of Transportation Science and Engineering, Beihang University, Beijing, P.R.China c CRIStAL, UMR CNRS 9189, Centrale Lille, Villeneuve d’Ascq 59651, France
b School
Abstract This paper studies the global fixed-time synchronization (GFTS) of chaotic systems with different dimensions. First, with the help of fixed-time stability theory of dynamic systems, a novel control protocol is put forward, which can achieve globally synchronization of two different dimensional chaotic systems (DDCS) in fixed time. Second, the GFTS of DDCS with uncertain parameters is also considered. The appropriate adaptive laws are designed to address the unknown parameters of the systems. Then, by the adaptive control, a new controller is presented to ensure the realization of DDCS within a given fixed time. Third, the GFTS is considered to networked DDCS, and the controller is also proposed accordingly. Different from conventional finite-time synchronization, the upper bound of settling time is independent of initial conditions of systems in GFTS. Finally, the effectiveness of the obtained results is demonstrated by corresponding numerical simulations. Keywords: Fixed-time synchronization, Chaotic systems, Different dimensions, Uncertain parameters, Networked chaotic systems.
1. Introduction Chaos is a kind of seemingly random phenomenon, but it is different from random phenomenon. Stochastic system is unpredictable in the short term, however, for the determined chaotic system, its short-term behavior is completely determined. And the sensitivity to the initial value makes its long-term behavior to be unpredictable. It is a kind of universal phenomenon in many nonlinear systems. Due to the works of Pecora and Carroll [1], the synchronization of chaotic systems has attracted a lot of attentions. There are many practical applications about chaos synchronization, for example, human brain [2], secure communication [3, 4], fluid mixing [5], medical images [6] and so on. In recent years, many types of control methods have been utilized, including feedback control [7–9], backstepping control [10], sliding model control [11, 12], fuzzy logic control [13, 14], pinning control [15, 16], impulsive control [17–19], adaptive control [20, 21] and so on. It’s worth noting that many existing literatures have considered the synchronization between two chaotic systems with the same dimensions [22, 23]. In fact, synchronization behavior also exists in different dimensional systems in practice, and it is also especially important. For example, the circulatory and respiratory systems with different dimensions exist synchronization behavior [24]. Because of the above reasons, more and more literatures have considered synchronization problems for different dimensional drive-response systems. Due to the active backstepping technique, the compound synchronization scheme was investigated between one third-dimension drive systems and one second-dimension response systems in [25]. And [26] realized the synchronization between a fourth-dimension system and a third-dimension system via adaptive control. They all have considered DDCS which reached the synchronization by reducing the dimension of master system because the dimension of master system is higher than slave system. Recently, there have been some researches on achieving the synchronization of chaotic systems by increasing dimension [27, 28], as well as some researches by reducing dimension and increasing dimension simultaneously [29, 30]. ∗ Corresponding
author. Email address: [email protected] (Zhaoxia Peng)
Preprint submitted to Elsevier
December 5, 2019
As we know, convergence rate plays a key role in the effect of synchronization. Finite-time control as a highspeed convergence technology, has been put forward to the synchronization of DDCS. In [31], the generalized finitetime synchronization of DDCS was first reached by using a finite-time control. In [32], the generalized finite-time synchronization was studied for DDCS with uncertain parameters. In [33], the global finite-time synchronization was investigated for the DDCS without unknown parameters and with unknown parameters, respectively. Although finite-time control schemes can achieve faster synchronization, the synchronization time is related to initial conditions [34]. However, in practical applications, the exact information of initial conditions is always unattainable which makes it hard to obtain estimated synchronization time. Besides, the synchronization time may also tend to infinity when the initial conditions approach to infinity. To overcome the drawbacks, the fixed-time control has been proposed [35], which can obtain more precise convergence time of error system independently of initial conditions. Recently, fixed-time synchronization has studied in [12, 36, 37], however the systems considered are the same dimensions. To the best of our knowledge, there is no research on the GFTS of DDCS up to now. Inspired by the above discussions, in this paper, the GFTS of two DDCS is investigated. The contributions of this paper are given as follows. A novel fixed-time control protocol is first put forward which can achieve globally synchronization of two DDCS in fixed time. Compared with existing results [31–33], this paper can achieve GFTS rather than finite-time synchronization. What’s more, the upper bound of settling time is independent of initial conditions of systems in GFTS but only dependent on the designed control parameters in this paper which is different from conventional finite-time synchronization. Then, the GFTS of DDCS with uncertain parameters is investigated. By employing the adaptive control technique, the adaptive update laws of parameters and new controller are proposed to achieve GFTS of two DDCS with uncertain parameters. Compared with the existing control methods in [38], the adaptive control approach is utilized to realize the GFTS of DDCS. Furthermore, the result of GFTS is extended to the networked chaotic systems with nonidentical dimensions. There exist some researches [39, 40] investigating the synchronization of networked systems with different dimensions, but the GFTS can not be achieved. However, this paper obtain the result of the GFTS for different dimensional networked chaotic systems. The structure of this paper is designed as follows. In Sec. 2, the preliminary definitions and lemmas for GFTS are given. Sec. 3 shows the main results including three parts. First, the GFTS of DDCS is derived. Second, the DDCS with uncertain parameters are considered, and the adaptive control is presented to reach GFTS. Third, the result is applied in networked DDCS. In Sec. 4, some examples are presented to illustrate the theoretical results. Finally, the conclusions are drawn in Sec. 5. Notations: In this paper, Rr denotes the real space with r dimensions. Rr×l denotes the set of all r × l real matrices. Let kxk = (xT x)1/2 , for x ∈ Rr , the superscript 0 T 0 denotes a matrix or vector transposition. For a square (nonsquare) matrix S ∈ Rm×m (∈ Rm×l ), S −1 is the inverse matrix of S (the left (m ≥ l, rank(S ) = l) or the right (m ≤ l, rank(S ) = m) inverse matrix of S ). 2. Preliminaries Consider the master system: x˙(t) = G(x(t)),
x(0) = x0 ,
(1)
where x(t) ∈ Rr is the state, G : Rr → Rr is a continuous function and G(0) = 0. Consider the slave system: y˙ (t) = H(y(t)) + u(t),
(2)
where y(t) ∈ Rl describes the state, H : Rl → Rl is a continuous function, and u(t) ∈ Rl is the control input. Remark 2.1. When r , l, the systems (1) and (2) have different dimensions. About the chaotic systems synchronization in finite time with different dimensions has been investigated in [31–33, 41], but to the best of our knowledge, there is no paper to study global synchronization for DDCS in fixed-time. To consider GFTS of DDCS, some indispensable definitions and lemmas are introduced below. 2
Definition 1. [33] Let ε(t) = ξ(x) − ζ(y) be the synchronization error between systems (1) and (2). Suppose there are two continuous differential functions ξ : Rr → Rm and ζ : Rl → Rm . If there exists a function τ : U\{0} → (0, +∞) where U ⊂ Rm is an open neighborhood of the origin, such that ε0 = ξ(x0 ) − ζ(y0 ) ∈ U and lim kε(t)k = lim kξ(x(t)) − ζ(y(t))k = 0,
t→τ(ε0 )
t→τ(ε0 )
kε(t)k ≡0, t > τ(ε0 ), then, the systems (1) and (2) can achieve finite-time synchronization. If U = Rm , then the systems (1) and (2) can achieve global finite-time synchronization. Definition 2. [35] The systems (1) and (2) are said to be GFTS if their error system reach global finite-time stable, and the setting time τ(ε0 ) is global bounded, that is to say, there exists a constant τmax > 0 such that τ(ε0 ) 6 τmax . Lemma 2.2. [42] Let χ1 , χ2 , · · · , χN be any nonnegative real numbers, then the following inequalities hold: !p N N P P χip > χi , when 0 < p 6 1, i=1 i=1 !p N N P p 1−p P χi , When p > 1. χi > N i=1
i=1
Lemma 2.3. [42] Given the following differential equation: m
p
y˙ = −αy n − βy q ,
y(0) = y0 ,
(3)
where α > 0, β > 0, and m, n, p, q are positive odd integers satisfying m > n and p < q. Then the equilibrium point of differential equation (4) is said to be fixed-time stable, and the upper bound of settling time satisfies τ<
1 q 1 n + . αm−n βq− p
(4)
3. Main results 3.1. GFTS of DDCS Let the error system of master and slave systems be ε(t) = ξ(x) − ζ(y),
(5)
where ξ : Rr → Rm and ζ : Rl → Rm are two continuous differential functions. Then, we have ε(t) ˙ =S ξ (x) x˙ − S ζ (y)˙y
=S ξ (x)G(x) − S ζ (y)(H(y) + u(t)),
(6)
where the Jacobian matrices of functions ξ(x) and ζ(y) are represented as S ξ (x) and S ζ (y), respectively. So they can be written as, ∂ξ1 (x) ∂ξ1 (x) ∂x1 · · · ∂ξ∂x1 (x) ∂x 2 r ∂ξ2 (x) ∂ξ2 (x) ∂ξ2 (x) ··· ∂x1 ∂x2 ∂xr , S ξ (x) = . . . .. .. .. .. . ∂ξm (x) ∂ξm (x) ∂ξm (x) · · · ∂x1 ∂x2 ∂xr ∂ζ1 (y) ∂ζ1 (y) ∂ζ1 (y) ∂y ··· ∂y2 ∂yl 1 ∂ζ2 (y) ∂ζ2 (y) ∂ζ2 (y) · · · ∂y y ∂y 2 l . S ζ (y) = . 1 . . .. .. .. .. . ∂ζ (y) ∂ζ (y) ∂ζm (y) m m · · · ∂y1 ∂y2 ∂yl 3
Assumption 3.1. m ≤ min{r, l} and Jacobian matrix S ζ (y) satisfies row full rank. Remark 3.2. There exists a right inverse of S ζ (y) defined as S ζ −1 (y) if Jacobian matrix S ζ (y) is row full rank and m ≤ l. In order to reach GFTS, the control input in (2) is designed, u(t) = −H(y) + S ζ −1 (y)(S ξ (x)G(x) + α1 sig(ε(t))m1 /n1 + β1 sig(ε(t)) p1 /q1 ),
(7)
where sig(ε(t))m1 /n1 = [sign(ε1 (t))|ε1 (t)|m1 /n1 , · · · , sign(εm (t))|εm (t)|m1 /n1 ]T , α1 , β1 are positive constants and m1 , n1 , p1 , q1 are positive odd integers satisfying m1 > n1 , p1 < q1 . Substituting (7) into (6), then the error dynamics can be rewritten as ε(t) ˙ = −α1 sig(ε(t))m1 /n1 − β1 sig(ε(t)) p1 /q1 .
(8)
Theorem 3.3. Suppose Assumption 3.1 holds. The error dynamics (8) of the systems (1) and (2) can be globally fixed-time stabilized at the origin if the control input is given by (7). That is, the systems (1) and (2) reach GFTS within the upper bound of settling time described by τ1 <
1 q1 m(m1 −n1 )/2n1 n1 + , α1 m1 − n1 β1 q1 − p1
(9)
where m is the dimension of vector ε(t), α1 , β1 are positive constants and m1 , n1 , p1 , q1 are positive odd integers satisfying m1 > n1 , p1 < q1 . Proof: Select the Lyapunov function as following form: V1 (t) = εT (t)ε(t).
(10)
Take the derivative of V1 (t) along (8) V˙ 1 (t) =2εT (t)ε(t) ˙ = − 2α1 εT (t)sig(ε(t))m1 /n1 − 2β1 εT (t)sig(ε(t)) p1 /q1
= − 2α1 εT (t)sign(ε(t))|ε(t)|m1 /n1 − 2β1 εT (t)sign(ε(t))|ε(t)| p1 /q1 m m X X (m1 +n1 )/n1 = − 2α1 − 2β1 |εk (t)| |εk (t)|(p1 +q1 )/q1 k=1
= − 2α1 By Lemma 2.2, (11) becomes
m X k=1
k=1
|εk (t)|2
(m1 +n1 )/2n1
− 2β1
m X k=1
|εk (t)|2
(p1 +q1 )/2q1
.
(p1 +q1 )/2q1 (m1 +n1 )/2n1 V˙ 1 (t) 6 − 2α1 m(n1 −m1 )/2n1 εT (t)ε(t) − 2β1 εT (t)ε(t) = − 2α1 m(n1 −m1 )/2n1 V1 (m1 +n1 )/2n1 − 2β1 V1 (p1 +q1 )/2q1 .
(11)
(12)
˙ ≤ 0, which indicates the system (8) is global asymptotical stable. Furthermore, It can be obtained that V(t) > 0, V(t) from Lemma 2.3, the system (8) is also global fixed-time stable, and the upper bound of settling time can be described by (9). The proof is completed. Remark 3.4. In [33], the global finite-time synchronization of DDCS has been investigated. However, the convergence time is not only related to the given designed parameters but also initial conditions. In this subsection, a new fixed-time control scheme is proposed in the Theorom3.3 to realize the GFTS and the strict proof is given. In addition, the convergence time is only dependent on the designed parameters, that is to say, the synchronization time can be estimated and given in advance. 4
3.2. GFTS of DDCS with uncertain parameters The chaotic systems’ parameters are unavoidably affected by external disturbances such as environment and measurement noises, that may be uncertain. Therefore, in this subsection, we will investigate the GFTS of DDCS with uncertain parameters. Consider the chaotic systems below, x˙(t) = G1 (x) + g(x)κ, y˙ (t) = H1 (y) + h(y)λ + u(t),
(13)
where x(t) ∈ Rr , y(t) ∈ Rl are state vectors of master system and slave system respectively, and G1 : Rr → Rr , H1 : Rl → Rl , g : Rr → Rr×p , h : Rl → Rl×q are all continuous functions, κ ∈ R p and λ ∈ Rq are the uncertain parameters. Assumption 3.5. In this paper, the vector of uncertain parameters κ and λ are assumed to be norm bounded as: kλk ≤ Θ,
(14)
κ˙ˆ =(S ξ (x)g(x))T ε(t),
(15)
λ˙ˆ = − (S ζ (y)h(y))T ε(t),
(16)
kκk ≤ Γ, where Γ and Θ are positive constants. The adaptive laws of parameters are proposed by
where κˆ and λˆ represent estimations of κ and λ, respectively. Then, by (15) and (16), the controller of slave system in (13) is proposed u(t) = − H1 (y) − h(y)λˆ + S ζ −1 (y)[S ξ (x)(G1 (x) + g(x)ˆκ) + α2 sig(ε(t))m2 /n2 + β2 sig(ε(t)) p2 /q2 sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 + β3 (kˆκk + Γ)(p2 +q2 )/q2 + α3 m(n2 −m2 )/2n2 2 kε(t)k kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ (kλk + Θ)(m2 +n2 )/n2 + β3 (kλk + Θ)(p2 +q2 )/q2 ], + α3 m(n2 −m2 )/2n2 2 kε(t)k kε(t)k2
(17)
where λ˜ = λˆ − λ, κ˜ = κˆ − κ, α2 , β2 , α3 , β3 are positive constants and m2 , n2 , p2 , q2 are positive odd integers satisfying m2 > n2 , p2 < q2 . Hence, the corresponding error system is obtained as ε(t) ˙ =S ξ (x)(G1 (x) + g(x)κ) − S ζ (y)(H1 (y) + h(y)λ + u(t)) = − S ξ (x)g(x)˜κ + S ζ (y)h(y)λ˜ − α2 sig(ε(t)m2 /n2 ) − β2 sig(ε(t) p2 /q2 )
sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 − β3 (kˆκk + Γ)(p2 +q2 )/q2 kε(t)k2 kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ − α3 m(n2 −m2 )/2n2 (kλk + Θ)(m2 +n2 )/n2 − β3 (kλk + Θ)(p2 +q2 )/q2 . 2 kε(t)k kε(t)k2
− α3 m(n2 −m2 )/2n2
(18)
Theorem 3.6. Suppose Assumption 3.1 and Assumption 3.5 hold. The error dynamics (18) of the master and slave systems (13) can be globally fixed-time stabilized at the origin if the control input is given by (17). That is to say, the systems (13) reach GFTS within the upper bound of settling time described by τ2 <
(3m)(m2 −n2 )/2n2 n2 1 q2 + , α m2 − n2 β q2 − p2
(19)
where α = min{α2 , α3 }, β = min{β2 , β3 }, m is the dimension of vector ε(t), m2 , n2 , p2 ,q2 are positive constants and satisfy m2 > n2 , p2 < q2 . 5
Proof: Select the Lyapunov function as follows ˜ V2 (t) = ε(t)T ε(t) + κ˜ T κ˜ + λ˜ T λ.
(20)
Take the derivative of V2 (t) along (15), (16) and (18) V˙ 2 (t) =2ε(t)T ε(t) ˙ + 2˜κT κ˙˜ + 2λ˜ T λ˙˜ =2ε(t)T ε(t) ˙ + 2˜κT κ˙ˆ + 2λ˜ T λ˙ˆ = − 2ε(t)T S ξ (x)g(x)˜κ + 2ε(t)T S ζ (y)h(y)λ˜ − 2α2 ε(t)T sig(ε(t))m2 /n2 − 2β2 ε(t)T sig(ε(t)) p2 /q2 sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 − 2β3 ε(t)T (kˆκk + Γ)(p2 +q2 )/q2 − 2α3 m(n2 −m2 )/2n2 ε(t)T kε(t)k2 kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ − 2α3 m(n2 −m2 )/2n2 ε(t)T (kλk + Θ)(m2 +n2 )/n2 − 2β3 ε(t)T (kλk + Θ)(p2 +q2 )/q2 kε(t)k2 kε(t)k2 T T + 2˜κT S ξ (x)g(x) ε(t) − 2λ˜ T S ζ (y)h(y) ε(t) sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 − 2β3 ε(t)T (kˆκk + Γ)(p2 +q2 )/q2 2 kε(t)k kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ − 2α3 m(n2 −m2 )/2n2 ε(t)T (kλk + Θ)(m2 +n2 )/n2 − 2β3 ε(t)T (kλk + Θ)(p2 +q2 )/q2 2 kε(t)k kε(t)k2 − 2α2 ε(t)T sign(ε(t))|ε(t)|m2 /n2 − 2β2 ε(t)T sign(ε(t))|ε(t)| p2 /q2 .
= − 2α3 m(n2 −m2 )/2n2 ε(t)T
(21)
ˆ + Θ ≥ kλk ˆ + kλk ≥ kλˆ − λk = λ˜ and According to Assumption 3.5 and kˆκk + Γ ≥ kˆκk + kκk ≥ kˆκ − κk = κ˜ and kλk (ε(t)T ε(t)) T sig(ε(t)) ε(t) kε(t)k2 = kε(t)k2 = 1 is satisfied. So, it has V˙ 2 (t) 6 − 2α2 ε(t)T sign(ε(t))|ε(t)|m2 /n2 − 2β2 ε(t)T sign(ε(t))|ε(t)| p2 /q2 (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2β3 κ˜ T κ˜ − 2α3 m(n2 −m2 )/2n2 κ˜ T κ˜ (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2α3 m(n2 −m2 )/2n2 λ˜ T λ˜ − 2β3 λ˜ T λ˜ , then, by Lemma 2.2 and equation (20), the (22) can be rewritten as the following (m2 +n2 )/2n2 (p2 +q2 )/2q2 V˙ 2 (t) 6 − 2α2 m(n2 −m2 )/2n2 εT (t)ε(t) − 2β2 εT (t)ε(t) (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2α3 m(n2 −m2 )/2n2 κ˜ T κ˜ − 2β3 κ˜ T κ˜ (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2β3 λ˜ T λ˜ − 2α3 m(n2 −m2 )/2n2 λ˜ T λ˜ (m2 +n2 )/2n2 6 − 2α(3m)(n2 −m2 )/2n2 εT (t)ε(t) + κ˜ T κ˜ + λ˜ T λ˜ (p2 +q2 )/2q2 − 2β εT (t)ε(t) + κ˜ T κ˜ + λ˜ T λ˜ 6 − 2α(3m)(n2 −m2 )/2n2 V2 (m2 +n2 )/2n2 − 2βV2 (p2 +q2 )/2q2 ,
(22)
(23)
˙ where α = min{α2 , α3 }, β = min{β2 , β3 } and m is the dimension of vector ε(t). Due to V(t) > 0, V(t) ≤ 0, it then follows from Lemma 2.3 that the error system (17) is global fixed-time stable. It means that the synchronization error converges to zero within the upper bound of settling time described by (19), i.e. τ2 < The proof is completed.
(3m)(m2 −n2 )/2n2 n2 1 q2 + . α m2 − n2 β q2 − p2 6
Remark 3.7. To achieve the GFTS of DDCS with uncertain parameters, the adaptive control methods is utilized. Finally, the estimations of uncertain parameters can converge to some constants in a fixed time. In [38], the finitetime synchronization of neural network systems with uncertain parameters is realized, but the GFTS of systems with different dimensions hasn’t been considered. 3.3. GFTS of networked DDCS In this subsection, we extend the GFTS from DDCS to networked chaotic systems with different dimensions. Consider a class of dynamical networks with M individuals indexed by k = 1, · · · , M, M X bk j x j (t), (24) x˙k (t) = gk (xk (t)) + h
j=1
iT
r
where xk (t) = xk1 (t), xk2 (t), · · · , xkr (t) ∈ R is a r-dimensional state vector of the kth node. Define x(t) = [x1 (t)T , x2 (t)T , · · · , x M (t)T ]T , g(x) := [g1 (x1 ), g2 (x2 ), · · · , g M (x M )]T where gk : Rr → Rr is a continuous function, and P − M b 1j j=2 .. B = . b M1
..
.
...
,
b1M
...
−
.. . M−1 P j=1
bM j
B = (bk j ) ∈ R M×M represents the coupling matrix. If node k is affected by node j( j , k), then bk j = b jk > 0; otherwise, bk j = b jk = 0. The vector form of system (24) is x˙(t) = g(x(t)) + Bx(t) = G(x).
(25)
y˙ (t) = h(y(t)) + By(t) + u(t) = H(y) + u(t),
(26)
The slave system also can be described as
where y(t) = [y1 (t)T , y2 (t)T , · · · , y M (t)T ]T represents the state vector of whole slave system. Here yk (t) = [yk1 (t), yk2 (t), · · · , ykl (t)]T ∈ Rl which is the state vector of the kth node, h(y) := [h1 (y1 ), h2 (y2 ), · · · , h M (y M )]T , hk : Rl → Rl is a continuous function and u(t) = [u1 (t)T , u2 (t)T , · · · , u M (t)T ]T denotes the vector of the control input, the control input of node k is denoted as uk (t) ∈ Rl . Then, the error system can be obtained by (25) and (26), ε(t) ˙ = S ξ (x) x˙ − S ζ (y)˙y
= S ξ (x)G(x) − S ζ (y)(H(y) + u(t)),
(27)
where ε(t) = [ε1 (t)T , ε2 (t)T , · · · , ε M (t)T ]T denotes the error vector, and εk (t) = ξ(xk (t)) − ζ(yk (t)) is the kth element of error vector. S ξ (x) = diag{S ξ (x1 ), S ξ (x2 ), · · · , S ξ (x M )} and S ζ (y) = diag{S ζ (y1 ), S ζ (y2 ), · · · , S ζ (y M )} are the Jacobian matrices of functions ξ(xk ) and ζ(yk ), and have the following forms S ξ (xk ) =
S ζ (yk ) =
∂ξ1 (xk ) ∂xk1 ∂ξ2 (xk ) ∂xk1
∂ξ1 (xk ) ∂xk2 ∂ξ2 (xk ) ∂xk2
∂ξm (xk ) ∂xk1
∂ξm (xk ) ∂xk2
∂ζ1 (yk ) ∂yk1 ∂ζ2 (yk ) ∂yk1
∂ζ1 (yk ) ∂yk2 ∂ζ2 (yk ) yk2
∂ζm (yk ) ∂yk1
∂ζm (yk ) ∂yk2
.. .
.. .
.. .
.. .
7
··· ··· .. . ··· ··· ··· .. . ···
∂ξ1 (xk ) ∂xkr ∂ξ2 (xk ) ∂xkr
.. .
∂ξm (xk ) ∂xkr ∂ζ1 (yk ) ∂ykl ∂ζ2 (yk ) ∂ykl
.. .
∂ζm (yk ) ∂yl
,
.
The objective is to guarantee that the error vector ε(t) tends to zero in a fixed time, then systems (25) and (26) reach synchronization. Thus the following control protocol is designed as: u(t) = − H(y) + S ζ −1 (y)(S ξ (x)G(x) + α4 sig(ε(t))m3 /n3 + β4 sig(ε(t)) p3 /q3 ),
(28)
where sig(ε(t))m3 /n3 = [(sig((ε1 (t))m3 /n3 ))T , · · · , (sig((ε M (t))m3 /n3 )T ]T , sig(εk (t))m3 /n3 = [sign(εk1 (t))|εk1 (t)|m3 /n3 , · · · , sign(εkm (t))|εkm (t)|m3 /n3 ]T , α4 , β4 are positive constants, m3 , n3 , p3 , q3 are positive odd integers and satisfy m3 > n3 , p3 < q3 . Combining (27) and (28), we can obtain the error dynamics as ε(t) ˙ = −α4 sig(ε(t))m3 /n3 − β4 sig(ε(t)) p3 /q3 .
(29)
Similarly, in the following theorem, the GFTS of systems (25) and (26) is proposed. Theorem 3.8. Suppose Assumption 3.1 holds. The error dynamics (29) of the master system (25) and slave system (26) can be globally fixed-time stabilized at the origin if the control input is given by (28). That is to say, the systems (25) and (26) reach GFTS within the upper bound of settling time described by τ3 <
1 q3 m(m3 −n3 )/2n3 n3 + , α4 m3 − n3 β4 q3 − p3
(30)
where m is the dimension of vector ε(t), α4 , β4 are positive constants, m3 , n3 , p3 , q3 are positive odd integers and satisfy m3 > n3 , p3 < q3 . Proof: The proof is omitted here because it is similar to that of Theorem 3.3. Remark 3.9. In this subsection, the results of DDCS are extended to the networked chaotic systems with different dimensions. It means that the GFTS of networked chaotic systems with different dimensions can be achieved. There exist some researches [39, 40] investigating the synchronization of networked systems with different dimensional nodes, but the GFTS can not be achieved. Furthermore, the investigation of GFTS for networked systems is necessary in practical applications, especially, neural networks and complex networks. 4. Simulation results In order to illustrate the effectiveness of the proposed control protocols for realizing GFTS of DDCS, in this section some examples are given. 4.1. GFTS between the hyperchaotic L¨u system and T system In order to demonstrate the proposed control protocol (7), the four dimensional hyperchaotic L¨u system [43] as master system and the three dimensional T system [44] as slave system are given below, respectively. The hyperchaotic L¨u system is given
and the T system is given
x˙1 (t) = −ax1 (t) + ax2 (t), x˙2 (t) = cx2 (t) − x1 (t)x3 (t), x˙3 (t) = −bx3 (t) + x1 (t)x2 (t), x˙4 (t) = x3 (t) − x4 (t),
y˙ 1 (t) = a1 (y2 (t) − y1 (t)) + u1 (t), y˙ 2 (t) = (c1 − a1 )y1 (t) − a1 y1 (t)y3 (t) + u2 (t), y˙ (t) = −b y (t) + y (t)y (t) + u (t), 3 1 3 1 2 3 8
(31)
(32)
Fig. 1: Trajectories of the errors εk (t)(k = 1, 2, 3): (a)α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9; (b)α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9.
Fig. 2: Trajectories of the errors εk (t)(k = 1, 2, 3): (c)α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9; (d)α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9.
with a = 15, b = 5, c = 10, a1 = 2.1, b1 = 0.6 and c1 = 30. Suppose ξ(x1 , x2 , x3 , x4 ) = (x1 + x3 , x2 + x4 , x2 x3 )T , ζ(y1 , y2 , y3 ) = (y1 , y2 , y3 )T . It then follows that the Jacobian matrices can be written as 1 0 0 1 0 1 0 S ξ (x) = 0 1 0 1 and S ζ (y) = 0 1 0 . 0 0 1 0 x3 x2 0 Hence, according to the system (6), we obtain the error system ε˙ 1 (t) = − 15x1 (t) + 15x2 (t) − 5x3 (t) − 2.1y2 (t) + 2.1y1 (t) − u1 (t), ε˙ 2 (t) =10x2 (t) − x1 (t)x3 (t) + x3 (t) − x4 (t) − 27.9y1 (t) + 2.1y1 (t)y3 (t) − u2 (t), ε˙ 3 (t) = − 50x2 (t)x3 (t) + 10x2 (t)(x2 (t))2 + 5x1 (t)(x3 (t))2 − (x1 (t))2 x2 (t)x3 (t) + 0.6y (t) + y (t)y (t) − u (t). 3
1
2
(33)
3
Fig. 1 and Fig. 2 show the trajectories of errors εk (t)(k = 1, 2, 3) with different initial conditions and the systems’ parameters. In Fig. 1(a) and (b), the same initial conditions of systems (31) and (32) are chosen as (x1 (0), x2 (0), x3 (0), x4 (0)) = (2, −2, 2, 1), (y1 (0), y2 (0), y3 (0)) = (1, 2, −2), but the parameters are selected as α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9 and α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9, respectively. In Fig. 2(c) and Fig. 2(d), the same initial conditions are chosen as (x1 (0), x2 (0), x3 (0), x4 (0)) = (2, 1, 1, 1.5), (y1 (0), y2 (0), y3 (0)) = (1, 1, 2), but the parameters are chosen as α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9 and α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9, respectively. We can see that Fig. 1(a), Fig. 2(c) have the same systems’ parameters but different initial conditions, as well as Fig. 1(b) and Fig. 2(d). According to Theorem 3.3, we can obtain the upper bound of convergence time are 0.6131 in Fig. 1(a) and Fig. 2(c), 7.4696 in Fig. 1(b) and Fig. 2(d), which are satisfied in Fig. 1 and Fig. 2. In addition, from Fig. 1(a) and Fig. 1(b) or from Fig. 2(c) and Fig. 2(d), the convergence time is dependent on system parameters. From Fig. 1(a) and Fig. 2(c) or from Fig. 1(b) and Fig. 2(d), the convergence time is independent of initial conditions of the systems. In addition, we suppose ξ(x1 , x2 , x3 , x4 ) = (x1 x4 , x2 + x3 )T , ζ(y1 , y2 , y3 ) = (y1 , 0.5y2 + 0.5y3 )T . It then follows that the Jacobian matrices can be written as " # " # x4 0 0 x1 1 0 0 S ξ (x) = and S ζ (y) = . 0 1 1 0 0 0.5 0.5 9
Fig. 3: Trajectories of the errors εk (t)(k = 1, 2)
The initial conditions of systems (31) and (32) in Fig. 3(a) and Fig. 3(b) are given as (x1 (0), x2 (0), x3 (0), x4 (0)) = (2, 1, 1, 1.5), (y1 (0), y2 (0), y3 (0)) = (1, 1, 2) and (x1 (0), x2 (0), x3 (0), x4 (0)) = (1, −0.5, −1.5, 2), (y1 (0), y2 (0), y3 (0)) = (1, −1, −1), respectively. The parameters in Fig. 3(a) and Fig. 3(b) are selected by α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9 and α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9, respectively. From Fig. 3, we can see that the synchronization error systems can also converge in a fixed time. Remark 4.1. In this subsection, Fig.1–Fig.3 show the GFTS can be achieved by the control protocol (7) which satisfy the Theorem 3.3. Fig.1 and Fig.2 verify the upper bound of setting time is independent of initial conditions but dependent on the designed parameters. In Fig.1 and Fig.2, the continuous functions are chosen as ξ : R4 → R3 and ζ : R3 → R3 . In Fig.3, the continuous functions are chosen as ξ : R4 → R2 and ζ : R3 → R2 . It means that the Fig.3 shows the lower dimensional GFTS can be achieved compared with Fig.1 and Fig.2. 4.2. GFTS between the hyperchaotic Lorenz system and Duffing system with uncertain parameters The following example is used to illustrate the adaptive laws for uncertain parameters and the corresponding proposed control scheme for the DDCS, where the hyperchaotic Lorenz system [45] is selected as the master system and Duffing system [46] is selected as the slave system. Consider the hyperchaotic Lorenz system x˙1 (t) = −κ1 x1 (t) + κ1 x2 (t), x˙2 (t) = κ2 x1 (t) − κ3 x2 (t) + x4 (t) − x1 (t)x3 (t), (34) x˙3 (t) = −κ4 x3 (t) + x1 (t)x2 (t), x˙4 (t) = −κ5 x2 (t) − κ6 x4 (t), and the Duffing system
y˙ 1 (t) = y2 (t) + u1 (t), y˙ 2 (t) = y1 (t) − y3 (t) − λ1 y2 (t) + λ2 cos(t) + u2 (t). 1
(35)
To show the hyperchaotic or chaotic behaviour for the systems (34) and (35), we choose κ1 = 12, κ2 = 23, κ3 = 1, κ4 = 2.1, κ5 = 6, κ6 = 0.2, λ1 = 1 and λ2 = 0.8. Suppose ξ(x1 , x2 , x3 , x4 ) = (x1 +x2 , x3 +x4 )T , ζ(y1 , y2 ) = (y1 , y2 )T . Then,
S ξ (x) =
"
1 0
1 0
0 1
0 1
#
and S ζ (y) =
From (15), we can get that the adaptive laws of parameters hold ˙ κˆ 1 (t) = (x2 (t) − x1 (t))ε1 (t), κ˙ˆ 2 (t) = x1 (t)ε1 (t), κ˙ˆ 3 (t) = −x2 (t)ε2 (t), κ˙ˆ 4 (t) = −x3 (t)ε2 (t), κˆ˙ 5 (t) = −x2 (t)ε2 (t), κ˙ˆ (t) = −x (t)ε (t), 6
4
2
10
"
1 0
0 1
#
.
(36)
Fig. 4: (a)Trajectories of the errors εk (t)(k = 1, 2); (b)trajectories of estimations κˆ k (k = 1, ..., 6); (c)trajectories of estimations λˆ k (k = 1, 2).
and
˙ λˆ 1 (t) = y2 (t)ε2 (t), λ˙ˆ (t) = − cos(t)ε (t). 2 2
(37)
According to (36) and (37), the control input of the slave system can be designed as follows by (17), u1 (t) = − y2 (t) − x1 (t)ˆκ1 + x2 (t)ˆκ1 + κˆ 2 x1 (t) − κˆ 3 x2 (t) + x4 (t) − x1 (t)x3 (t) m2 p2 + α2 sign(ε1 (t))|ε1 (t)| n2 + β2 sign(ε1 (t))|ε1 (t)| q2 sig(ε1 (t)) sig(ε1 (t)) + α3 m(n2 −m2 )/n2 (kˆκk + Γ)(m2 +n2 )/n2 + β3 (kˆκk + Γ)(p2 +q2 )/q2 2 2 kε(t)k kε(t)k sig(ε1 (t)) ˆ sig(ε1 (t)) ˆ (kλk + Θ)(m2 +n2 )/n2 + β3 (kλk + Θ)(p2 +q2 )/q2 , + α3 m(n2 −m2 )/n2 kε(t)k2 kε(t)k2 u2 (t) = − y1 (t) + y31 (t) + λˆ 1 y2 (t) − cos(t)λˆ 2 − x3 (t)ˆκ4 − x2 (t)ˆκ5 − x4 (t)ˆκ6 m2 p2 n2 q2 + x (t)x (t) + α sign(ε (t))|ε (t)| + β sign(ε (t))|ε (t)| 1 2 2 2 2 2 2 2 sig(ε2 (t)) sig(ε2 (t)) (kˆκk + Γ)(m2 +n2 )/n2 + β3 (kˆκk + Γ)(p2 +q2 )/q2 + α3 m(n2 −m2 )/n2 2 2 kε(t)k kε(t)k sig(ε2 (t)) ˆ sig(ε2 (t)) ˆ + α3 m(n2 −m2 )/n2 (kλk + Θ)(m2 +n2 )/n2 + β3 (kλk + Θ)(p2 +q2 )/q2 . 2 kε(t)k kε(t)k2
By the error dynamics (18), we can get ε˙ 1 (t) = − (x1 (t) − x2 (t))κ1 + κ2 x1 (t) − κ3 x2 (t) + x4 (t) − x1 (t)x3 (t) − y2 (t) − u1 (t), ε˙ 2 (t) = − κ4 x3 (t) + x1 (t)x2 (t) − κ5 x2 (t) − κ6 x4 (t)y1 (t) + y3 (t) + λ1 y2 (t) − λ2 cos(t) − u2 (t). 1
(38)
(39)
In this simulation we choose Γ = 27 and Θ = 1.71 where kκk k ≤ Γ (k = 1, ..., 6) and kλk k ≤ Θ (k = 1, 2). Fig.4(a) shows the trajectories of errors εk (t)(k = 1, 2), where the parameters are selected as α2 = 0.01, β2 = 10.5, α3 = 0.001, β3 = 0.001, m2 = 11, n2 = 9, p2 = 1, q2 = 11, the initial values of two systems are given as (x1 (0), x2 (0), x3 (0), x4 (0)) = (−1, −0.6, −1, −0.2), (y1 (0), y2 (0)) = (−2, −2). Fig.4(b) and Fig.4(c) show the trajectories of estimations of κˆ k (k = 1, ..., 6) and λˆ k (k = 1, 2) where the initial conditions of them are chosen as (ˆκ1 (0), κˆ 2 (0), κˆ 3 (0), κˆ 4 (0), κˆ 5 (0), κˆ 6 (0)) = (0, 0, 0, 0, 0, 0), (λ1 (0), λ2 (0)) = (−4.8, 1.5). According to Theorem 3.6, Fig.4(a) shows master and slave systems (13) can be synchronized in a fixed time via adaptive controller (17). Fig.4(b) and Fig.4(c) show the estimations of parameters can converge to some constants which satisfies the designed updated laws (15) and (16). 11
4.3. GFTS of networked Genesio system and Duffing system In this subsection, we consider the following networked Genesio [47] chaotic system with 10 nodes and networked Duffing chaotic system with 10 nodes as, x˙(t) = g(x(t)) + Bx(t), (40) y˙ (t) = h(y(t)) + By(t) + u(t), where in the networked Genesio master system, gk (xk (t)) holds 0 1 0 xk1 (t) 0 0 1 xk2 (t) + 0 gk (xk (t)) = 0 2 xk1 (t) xk3 (t) −6 −2.92 −1.2 and in the networked Duffing slave system, hk (yk (t)) holds ! ! 0 1 yk1 (t) hk (yk (t)) = + 1 −1 yk2 (t)
Then, we choose the symmetric coupling matrix B ∈ R10×10 as, 0 1 0 0 −3 0 0 −4 1 0 1 1 0 1 −3 0 1 0 1 0 0 −6 1 1 1 1 1 −4 0 0 B = 1 0 1 0 −3 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0
,
0 −y3k1 (t) + 0.8 cos(t) 1 0 0 1 0 1 −5 1 0 1
0 1 0 0 0 0 1 −4 1 1
1 0 0 1 1 0 0 1 −5 1
0 1 1 1 0 0 1 1 1 −6
!
.
.
Assuming the projective functions as ξ(x) = (xk1 + xk2 , xk3 )T , ζ(y) = (yk1 , yk2 )T , then, we get
S ξ (x) =
"
1 0
1 0
0 1
#
, S ζ (y) =
"
1 0
0 1
#
.
From (27), we have 10 X ε ˙ (t) =x (t) + x (t) − y (t) + bk j (xk1 (t) + xk2 (t) − yk1 (t)) − uk1 (t), k1 k1 k3 k2 j=1 2 ε˙ k2 (t) = − 6xk1 (t) − 2.92xk2 (t) − 1.2xk3 (t) + xk1 (t) − yk1 (t) + yk2 (t) + y3k1 (t) 10 X − 0.8 cos(t) + bk j (xk3 (t) − yk2 (t)) − uk2 (t).
(41)
j=1
According to (28), the controllers is designed as 10 X uk1 (t) =xk1 (t) + xk3 (t) − yk2 (t) + bk j (xk1 (t) + xk2 (t) − yk1 (t)) j=1 m3 p3 + α3 sign(εk1 (t))|εk1 (t)| n3 + β3 sign(εk1 (t))|εk1 (t)| q3 , 2 uk2 (t) = − 6xk1 (t) − 2.92xk2 (t) − 1.2xk3 (t) + xk1 (t) − yk1 (t) + yk2 (t) + y3k1 (t) − 0.8 cos(t) 10 X m3 p3 n3 q3 + b (x (t) − y (t)) + α sign(ε (t))|ε (t)| + β sign(ε (t))|ε (t)| . k j k3 k2 3 k2 k2 3 k2 k2 j=1 12
(42)
Fig. 5: (a)Trajectories of the errors εk1 (t)(k = 1, ..., 10); (b)trajectories of the errors εk2 (t)(k = 1, ..., 10)
Fig. 5 shows the trajectories of errors εk1 (t)(k = 1, ..., 10) and εk2 (t)(k = 1, ..., 10) where the parameters of systems are selected as α3 = 3.2, β3 = 5, m3 = 11, n3 = 5, p3 = 3, q3 = 7, and the initial conditions are chosen as x(0) = (1 + 5k, 2 + 4k, −1 − 8k)T , y(0) = (3 + 2k, 2 − 3k)T , (k = 1, ..., 10). From Theorem 3.8, the upper bound of settling time is obtained as τ3 < 0.7126 which is satisfied in Fig. 5. In addition, from Fig. 5(a) and Fig. 5(b), we can see that the convergence time of errors εk1 (t)(k = 1, ..., 10) and εk2 (t)(k = 1, ..., 10) is less than 0.7126. 5. Conclusions In this paper, the GFTS problems of DDCS with and without unknown parameters have been investigated, the corresponding controllers and adaptive control laws have been designed respectively. Then, GFTS problems have also been extended to networked DDCS. In addition, it is shown that the upper bound of settling time in fixed-time synchronization is independent of initial conditions of systems, which is different from finite-time synchronization. Certainly, more parameters in controller should be designed in advance which may more difficult to reach synchronization in other complex systems. So, our future work will consider the GFTS neural networks and complex networks with different dimensions. 6. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61977004 and 61503016, the Fundamental Research Funds for the Central Universities under Grants YWF-19-BJ-J-259 and 2017JBM067, The National Key Research and Development Program of China under Grant 2017YFB0103202. References [1] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems. Phys. Rev. Lett. 64 (8) (1990) 821-824. [2] S.J. Schiff, K. Jerger, D.H. Duong, T. Chang, M.L. Spano, W.L. Ditto, Controlling chaos in the brain. Nature 370 (6491) (1994) 615-620. [3] S. Bowong, Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys. Lett. A 326 (1) (2004) 102-113. [4] C.D. Li, X.F. Liao, K.W. Wong, Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication, Physica D 194 (3) (2004) 187-202. [5] J.M. Ottino, F.J. Muzzio, M. Tjahjadi, J.G. Franjione, S.C. Jana, H.A. Kusch, Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing processes, Science 257 (5071) (1992) 754-760. [6] Z.L. Li, M.H. Dong, S.P. Wen, X. Hu, P. Zhou, Z.G. Zeng, CLU-CNNs: Object detection for medical images, Neurocomputing 350 (2019) 53-59. [7] G. He, J.A. Fang, Z. Li, Finite-time synchronization of cyclic switched complex networks under feedback control, J. Frankl. Inst. 354 (9) (2017) 3780-3796. [8] J.M. Park, P.G. Park, H∞ sampled-state feedback control for synchronization of chaotic Lure systems with time delays, J. Frankl. Inst. 355 (16) (2018) 0016-0032. [9] Shengbo Wang, Yanyi Cao, Tingwen Huang, and Shiping Wen, Passivity and passification of memristive neural networks with leakage term and time-varying delays, Appl. Math. Comput. 361 (2019) 294-310. [10] Z.Y. Wu, X.C. Fu, Combination synchronization of three different order nonlinear systems using active backstepping design, Nonlinear Dyn. 73 (3) (2013) 1863-1872. [11] Z.B. Wang, H.Q.Wu, Projective synchronization in fixed time for complex dynamical networks with nonidentical nodes via second-order sliding mode control strategy, J. Frankl. Inst. 355 (15) (2018) 7306-7334. [12] A. Khanzadeh, M. Pourgholi, Fixed-time sliding mode controller design for synchronization of complex dynamical networks, Nonlinear Dyn. 88 (4) (2017) 2637-2649. [13] S.P. Wen, S.X. Xiao, Y. Yang, Z. Yan, Z.G. Zeng, T.W. Huang, Adjusting Learning Rate of Memristor-Based Multilayer Neural Networks via Fuzzy Method, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 38 (6) (2019) 1084-1094.
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Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
15
Journal Pre-proof
Global fixed-time synchronization of chaotic systems with different dimensions Xiaozhen Guo, Guoguang Wen, Zhaoxia Peng, Yunlong Zhang PII: DOI: Reference:
S0016-0032(19)30867-1 https://doi.org/10.1016/j.jfranklin.2019.11.063 FI 4298
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
23 April 2019 26 September 2019 25 November 2019
Please cite this article as: Xiaozhen Guo, Guoguang Wen, Zhaoxia Peng, Yunlong Zhang, Global fixed-time synchronization of chaotic systems with different dimensions, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.063
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Global fixed-time synchronization of chaotic systems with different dimensions Xiaozhen Guoa , Guoguang Wena , Zhaoxia Pengb,∗, Yunlong Zhangc a Department
of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China of Transportation Science and Engineering, Beihang University, Beijing, P.R.China c CRIStAL, UMR CNRS 9189, Centrale Lille, Villeneuve d’Ascq 59651, France
b School
Abstract This paper studies the global fixed-time synchronization (GFTS) of chaotic systems with different dimensions. First, with the help of fixed-time stability theory of dynamic systems, a novel control protocol is put forward, which can achieve globally synchronization of two different dimensional chaotic systems (DDCS) in fixed time. Second, the GFTS of DDCS with uncertain parameters is also considered. The appropriate adaptive laws are designed to address the unknown parameters of the systems. Then, by the adaptive control, a new controller is presented to ensure the realization of DDCS within a given fixed time. Third, the GFTS is considered to networked DDCS, and the controller is also proposed accordingly. Different from conventional finite-time synchronization, the upper bound of settling time is independent of initial conditions of systems in GFTS. Finally, the effectiveness of the obtained results is demonstrated by corresponding numerical simulations. Keywords: Fixed-time synchronization, Chaotic systems, Different dimensions, Uncertain parameters, Networked chaotic systems.
1. Introduction Chaos is a kind of seemingly random phenomenon, but it is different from random phenomenon. Stochastic system is unpredictable in the short term, however, for the determined chaotic system, its short-term behavior is completely determined. And the sensitivity to the initial value makes its long-term behavior to be unpredictable. It is a kind of universal phenomenon in many nonlinear systems. Due to the works of Pecora and Carroll [1], the synchronization of chaotic systems has attracted a lot of attentions. There are many practical applications about chaos synchronization, for example, human brain [2], secure communication [3, 4], fluid mixing [5], medical images [6] and so on. In recent years, many types of control methods have been utilized, including feedback control [7–9], backstepping control [10], sliding model control [11, 12], fuzzy logic control [13, 14], pinning control [15, 16], impulsive control [17–19], adaptive control [20, 21] and so on. It’s worth noting that many existing literatures have considered the synchronization between two chaotic systems with the same dimensions [22, 23]. In fact, synchronization behavior also exists in different dimensional systems in practice, and it is also especially important. For example, the circulatory and respiratory systems with different dimensions exist synchronization behavior [24]. Because of the above reasons, more and more literatures have considered synchronization problems for different dimensional drive-response systems. Due to the active backstepping technique, the compound synchronization scheme was investigated between one third-dimension drive systems and one second-dimension response systems in [25]. And [26] realized the synchronization between a fourth-dimension system and a third-dimension system via adaptive control. They all have considered DDCS which reached the synchronization by reducing the dimension of master system because the dimension of master system is higher than slave system. Recently, there have been some researches on achieving the synchronization of chaotic systems by increasing dimension [27, 28], as well as some researches by reducing dimension and increasing dimension simultaneously [29, 30]. ∗ Corresponding
author. Email address: [email protected] (Zhaoxia Peng)
Preprint submitted to Elsevier
December 5, 2019
As we know, convergence rate plays a key role in the effect of synchronization. Finite-time control as a highspeed convergence technology, has been put forward to the synchronization of DDCS. In [31], the generalized finitetime synchronization of DDCS was first reached by using a finite-time control. In [32], the generalized finite-time synchronization was studied for DDCS with uncertain parameters. In [33], the global finite-time synchronization was investigated for the DDCS without unknown parameters and with unknown parameters, respectively. Although finite-time control schemes can achieve faster synchronization, the synchronization time is related to initial conditions [34]. However, in practical applications, the exact information of initial conditions is always unattainable which makes it hard to obtain estimated synchronization time. Besides, the synchronization time may also tend to infinity when the initial conditions approach to infinity. To overcome the drawbacks, the fixed-time control has been proposed [35], which can obtain more precise convergence time of error system independently of initial conditions. Recently, fixed-time synchronization has studied in [12, 36, 37], however the systems considered are the same dimensions. To the best of our knowledge, there is no research on the GFTS of DDCS up to now. Inspired by the above discussions, in this paper, the GFTS of two DDCS is investigated. The contributions of this paper are given as follows. A novel fixed-time control protocol is first put forward which can achieve globally synchronization of two DDCS in fixed time. Compared with existing results [31–33], this paper can achieve GFTS rather than finite-time synchronization. What’s more, the upper bound of settling time is independent of initial conditions of systems in GFTS but only dependent on the designed control parameters in this paper which is different from conventional finite-time synchronization. Then, the GFTS of DDCS with uncertain parameters is investigated. By employing the adaptive control technique, the adaptive update laws of parameters and new controller are proposed to achieve GFTS of two DDCS with uncertain parameters. Compared with the existing control methods in [38], the adaptive control approach is utilized to realize the GFTS of DDCS. Furthermore, the result of GFTS is extended to the networked chaotic systems with nonidentical dimensions. There exist some researches [39, 40] investigating the synchronization of networked systems with different dimensions, but the GFTS can not be achieved. However, this paper obtain the result of the GFTS for different dimensional networked chaotic systems. The structure of this paper is designed as follows. In Sec. 2, the preliminary definitions and lemmas for GFTS are given. Sec. 3 shows the main results including three parts. First, the GFTS of DDCS is derived. Second, the DDCS with uncertain parameters are considered, and the adaptive control is presented to reach GFTS. Third, the result is applied in networked DDCS. In Sec. 4, some examples are presented to illustrate the theoretical results. Finally, the conclusions are drawn in Sec. 5. Notations: In this paper, Rr denotes the real space with r dimensions. Rr×l denotes the set of all r × l real matrices. Let kxk = (xT x)1/2 , for x ∈ Rr , the superscript 0 T 0 denotes a matrix or vector transposition. For a square (nonsquare) matrix S ∈ Rm×m (∈ Rm×l ), S −1 is the inverse matrix of S (the left (m ≥ l, rank(S ) = l) or the right (m ≤ l, rank(S ) = m) inverse matrix of S ). 2. Preliminaries Consider the master system: x˙(t) = G(x(t)),
x(0) = x0 ,
(1)
where x(t) ∈ Rr is the state, G : Rr → Rr is a continuous function and G(0) = 0. Consider the slave system: y˙ (t) = H(y(t)) + u(t),
(2)
where y(t) ∈ Rl describes the state, H : Rl → Rl is a continuous function, and u(t) ∈ Rl is the control input. Remark 2.1. When r , l, the systems (1) and (2) have different dimensions. About the chaotic systems synchronization in finite time with different dimensions has been investigated in [31–33, 41], but to the best of our knowledge, there is no paper to study global synchronization for DDCS in fixed-time. To consider GFTS of DDCS, some indispensable definitions and lemmas are introduced below. 2
Definition 1. [33] Let ε(t) = ξ(x) − ζ(y) be the synchronization error between systems (1) and (2). Suppose there are two continuous differential functions ξ : Rr → Rm and ζ : Rl → Rm . If there exists a function τ : U\{0} → (0, +∞) where U ⊂ Rm is an open neighborhood of the origin, such that ε0 = ξ(x0 ) − ζ(y0 ) ∈ U and lim kε(t)k = lim kξ(x(t)) − ζ(y(t))k = 0,
t→τ(ε0 )
t→τ(ε0 )
kε(t)k ≡0, t > τ(ε0 ), then, the systems (1) and (2) can achieve finite-time synchronization. If U = Rm , then the systems (1) and (2) can achieve global finite-time synchronization. Definition 2. [35] The systems (1) and (2) are said to be GFTS if their error system reach global finite-time stable, and the setting time τ(ε0 ) is global bounded, that is to say, there exists a constant τmax > 0 such that τ(ε0 ) 6 τmax . Lemma 2.2. [42] Let χ1 , χ2 , · · · , χN be any nonnegative real numbers, then the following inequalities hold: !p N N P P χip > χi , when 0 < p 6 1, i=1 i=1 !p N N P p 1−p P χi , When p > 1. χi > N i=1
i=1
Lemma 2.3. [42] Given the following differential equation: m
p
y˙ = −αy n − βy q ,
y(0) = y0 ,
(3)
where α > 0, β > 0, and m, n, p, q are positive odd integers satisfying m > n and p < q. Then the equilibrium point of differential equation (4) is said to be fixed-time stable, and the upper bound of settling time satisfies τ<
1 q 1 n + . αm−n βq− p
(4)
3. Main results 3.1. GFTS of DDCS Let the error system of master and slave systems be ε(t) = ξ(x) − ζ(y),
(5)
where ξ : Rr → Rm and ζ : Rl → Rm are two continuous differential functions. Then, we have ε(t) ˙ =S ξ (x) x˙ − S ζ (y)˙y
=S ξ (x)G(x) − S ζ (y)(H(y) + u(t)),
(6)
where the Jacobian matrices of functions ξ(x) and ζ(y) are represented as S ξ (x) and S ζ (y), respectively. So they can be written as, ∂ξ1 (x) ∂ξ1 (x) ∂x1 · · · ∂ξ∂x1 (x) ∂x 2 r ∂ξ2 (x) ∂ξ2 (x) ∂ξ2 (x) ··· ∂x1 ∂x2 ∂xr , S ξ (x) = . . . .. .. .. .. . ∂ξm (x) ∂ξm (x) ∂ξm (x) · · · ∂x1 ∂x2 ∂xr ∂ζ1 (y) ∂ζ1 (y) ∂ζ1 (y) ∂y ··· ∂y2 ∂yl 1 ∂ζ2 (y) ∂ζ2 (y) ∂ζ2 (y) · · · ∂y y ∂y 2 l . S ζ (y) = . 1 . . .. .. .. .. . ∂ζ (y) ∂ζ (y) ∂ζm (y) m m · · · ∂y1 ∂y2 ∂yl 3
Assumption 3.1. m ≤ min{r, l} and Jacobian matrix S ζ (y) satisfies row full rank. Remark 3.2. There exists a right inverse of S ζ (y) defined as S ζ −1 (y) if Jacobian matrix S ζ (y) is row full rank and m ≤ l. In order to reach GFTS, the control input in (2) is designed, u(t) = −H(y) + S ζ −1 (y)(S ξ (x)G(x) + α1 sig(ε(t))m1 /n1 + β1 sig(ε(t)) p1 /q1 ),
(7)
where sig(ε(t))m1 /n1 = [sign(ε1 (t))|ε1 (t)|m1 /n1 , · · · , sign(εm (t))|εm (t)|m1 /n1 ]T , α1 , β1 are positive constants and m1 , n1 , p1 , q1 are positive odd integers satisfying m1 > n1 , p1 < q1 . Substituting (7) into (6), then the error dynamics can be rewritten as ε(t) ˙ = −α1 sig(ε(t))m1 /n1 − β1 sig(ε(t)) p1 /q1 .
(8)
Theorem 3.3. Suppose Assumption 3.1 holds. The error dynamics (8) of the systems (1) and (2) can be globally fixed-time stabilized at the origin if the control input is given by (7). That is, the systems (1) and (2) reach GFTS within the upper bound of settling time described by τ1 <
1 q1 m(m1 −n1 )/2n1 n1 + , α1 m1 − n1 β1 q1 − p1
(9)
where m is the dimension of vector ε(t), α1 , β1 are positive constants and m1 , n1 , p1 , q1 are positive odd integers satisfying m1 > n1 , p1 < q1 . Proof: Select the Lyapunov function as following form: V1 (t) = εT (t)ε(t).
(10)
Take the derivative of V1 (t) along (8) V˙ 1 (t) =2εT (t)ε(t) ˙ = − 2α1 εT (t)sig(ε(t))m1 /n1 − 2β1 εT (t)sig(ε(t)) p1 /q1
= − 2α1 εT (t)sign(ε(t))|ε(t)|m1 /n1 − 2β1 εT (t)sign(ε(t))|ε(t)| p1 /q1 m m X X (m1 +n1 )/n1 = − 2α1 − 2β1 |εk (t)| |εk (t)|(p1 +q1 )/q1 k=1
= − 2α1 By Lemma 2.2, (11) becomes
m X k=1
k=1
|εk (t)|2
(m1 +n1 )/2n1
− 2β1
m X k=1
|εk (t)|2
(p1 +q1 )/2q1
.
(p1 +q1 )/2q1 (m1 +n1 )/2n1 V˙ 1 (t) 6 − 2α1 m(n1 −m1 )/2n1 εT (t)ε(t) − 2β1 εT (t)ε(t) = − 2α1 m(n1 −m1 )/2n1 V1 (m1 +n1 )/2n1 − 2β1 V1 (p1 +q1 )/2q1 .
(11)
(12)
˙ ≤ 0, which indicates the system (8) is global asymptotical stable. Furthermore, It can be obtained that V(t) > 0, V(t) from Lemma 2.3, the system (8) is also global fixed-time stable, and the upper bound of settling time can be described by (9). The proof is completed. Remark 3.4. In [33], the global finite-time synchronization of DDCS has been investigated. However, the convergence time is not only related to the given designed parameters but also initial conditions. In this subsection, a new fixed-time control scheme is proposed in the Theorom3.3 to realize the GFTS and the strict proof is given. In addition, the convergence time is only dependent on the designed parameters, that is to say, the synchronization time can be estimated and given in advance. 4
3.2. GFTS of DDCS with uncertain parameters The chaotic systems’ parameters are unavoidably affected by external disturbances such as environment and measurement noises, that may be uncertain. Therefore, in this subsection, we will investigate the GFTS of DDCS with uncertain parameters. Consider the chaotic systems below, x˙(t) = G1 (x) + g(x)κ, y˙ (t) = H1 (y) + h(y)λ + u(t),
(13)
where x(t) ∈ Rr , y(t) ∈ Rl are state vectors of master system and slave system respectively, and G1 : Rr → Rr , H1 : Rl → Rl , g : Rr → Rr×p , h : Rl → Rl×q are all continuous functions, κ ∈ R p and λ ∈ Rq are the uncertain parameters. Assumption 3.5. In this paper, the vector of uncertain parameters κ and λ are assumed to be norm bounded as: kλk ≤ Θ,
(14)
κ˙ˆ =(S ξ (x)g(x))T ε(t),
(15)
λ˙ˆ = − (S ζ (y)h(y))T ε(t),
(16)
kκk ≤ Γ, where Γ and Θ are positive constants. The adaptive laws of parameters are proposed by
where κˆ and λˆ represent estimations of κ and λ, respectively. Then, by (15) and (16), the controller of slave system in (13) is proposed u(t) = − H1 (y) − h(y)λˆ + S ζ −1 (y)[S ξ (x)(G1 (x) + g(x)ˆκ) + α2 sig(ε(t))m2 /n2 + β2 sig(ε(t)) p2 /q2 sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 + β3 (kˆκk + Γ)(p2 +q2 )/q2 + α3 m(n2 −m2 )/2n2 2 kε(t)k kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ (kλk + Θ)(m2 +n2 )/n2 + β3 (kλk + Θ)(p2 +q2 )/q2 ], + α3 m(n2 −m2 )/2n2 2 kε(t)k kε(t)k2
(17)
where λ˜ = λˆ − λ, κ˜ = κˆ − κ, α2 , β2 , α3 , β3 are positive constants and m2 , n2 , p2 , q2 are positive odd integers satisfying m2 > n2 , p2 < q2 . Hence, the corresponding error system is obtained as ε(t) ˙ =S ξ (x)(G1 (x) + g(x)κ) − S ζ (y)(H1 (y) + h(y)λ + u(t)) = − S ξ (x)g(x)˜κ + S ζ (y)h(y)λ˜ − α2 sig(ε(t)m2 /n2 ) − β2 sig(ε(t) p2 /q2 )
sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 − β3 (kˆκk + Γ)(p2 +q2 )/q2 kε(t)k2 kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ − α3 m(n2 −m2 )/2n2 (kλk + Θ)(m2 +n2 )/n2 − β3 (kλk + Θ)(p2 +q2 )/q2 . 2 kε(t)k kε(t)k2
− α3 m(n2 −m2 )/2n2
(18)
Theorem 3.6. Suppose Assumption 3.1 and Assumption 3.5 hold. The error dynamics (18) of the master and slave systems (13) can be globally fixed-time stabilized at the origin if the control input is given by (17). That is to say, the systems (13) reach GFTS within the upper bound of settling time described by τ2 <
(3m)(m2 −n2 )/2n2 n2 1 q2 + , α m2 − n2 β q2 − p2
(19)
where α = min{α2 , α3 }, β = min{β2 , β3 }, m is the dimension of vector ε(t), m2 , n2 , p2 ,q2 are positive constants and satisfy m2 > n2 , p2 < q2 . 5
Proof: Select the Lyapunov function as follows ˜ V2 (t) = ε(t)T ε(t) + κ˜ T κ˜ + λ˜ T λ.
(20)
Take the derivative of V2 (t) along (15), (16) and (18) V˙ 2 (t) =2ε(t)T ε(t) ˙ + 2˜κT κ˙˜ + 2λ˜ T λ˙˜ =2ε(t)T ε(t) ˙ + 2˜κT κ˙ˆ + 2λ˜ T λ˙ˆ = − 2ε(t)T S ξ (x)g(x)˜κ + 2ε(t)T S ζ (y)h(y)λ˜ − 2α2 ε(t)T sig(ε(t))m2 /n2 − 2β2 ε(t)T sig(ε(t)) p2 /q2 sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 − 2β3 ε(t)T (kˆκk + Γ)(p2 +q2 )/q2 − 2α3 m(n2 −m2 )/2n2 ε(t)T kε(t)k2 kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ − 2α3 m(n2 −m2 )/2n2 ε(t)T (kλk + Θ)(m2 +n2 )/n2 − 2β3 ε(t)T (kλk + Θ)(p2 +q2 )/q2 kε(t)k2 kε(t)k2 T T + 2˜κT S ξ (x)g(x) ε(t) − 2λ˜ T S ζ (y)h(y) ε(t) sig(ε(t)) sig(ε(t)) (kˆκk + Γ)(m2 +n2 )/n2 − 2β3 ε(t)T (kˆκk + Γ)(p2 +q2 )/q2 2 kε(t)k kε(t)k2 sig(ε(t)) ˆ sig(ε(t)) ˆ − 2α3 m(n2 −m2 )/2n2 ε(t)T (kλk + Θ)(m2 +n2 )/n2 − 2β3 ε(t)T (kλk + Θ)(p2 +q2 )/q2 2 kε(t)k kε(t)k2 − 2α2 ε(t)T sign(ε(t))|ε(t)|m2 /n2 − 2β2 ε(t)T sign(ε(t))|ε(t)| p2 /q2 .
= − 2α3 m(n2 −m2 )/2n2 ε(t)T
(21)
ˆ + Θ ≥ kλk ˆ + kλk ≥ kλˆ − λk = λ˜ and According to Assumption 3.5 and kˆκk + Γ ≥ kˆκk + kκk ≥ kˆκ − κk = κ˜ and kλk (ε(t)T ε(t)) T sig(ε(t)) ε(t) kε(t)k2 = kε(t)k2 = 1 is satisfied. So, it has V˙ 2 (t) 6 − 2α2 ε(t)T sign(ε(t))|ε(t)|m2 /n2 − 2β2 ε(t)T sign(ε(t))|ε(t)| p2 /q2 (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2β3 κ˜ T κ˜ − 2α3 m(n2 −m2 )/2n2 κ˜ T κ˜ (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2α3 m(n2 −m2 )/2n2 λ˜ T λ˜ − 2β3 λ˜ T λ˜ , then, by Lemma 2.2 and equation (20), the (22) can be rewritten as the following (m2 +n2 )/2n2 (p2 +q2 )/2q2 V˙ 2 (t) 6 − 2α2 m(n2 −m2 )/2n2 εT (t)ε(t) − 2β2 εT (t)ε(t) (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2α3 m(n2 −m2 )/2n2 κ˜ T κ˜ − 2β3 κ˜ T κ˜ (m2 +n2 )/2n2 (p2 +q2 )/2q2 − 2β3 λ˜ T λ˜ − 2α3 m(n2 −m2 )/2n2 λ˜ T λ˜ (m2 +n2 )/2n2 6 − 2α(3m)(n2 −m2 )/2n2 εT (t)ε(t) + κ˜ T κ˜ + λ˜ T λ˜ (p2 +q2 )/2q2 − 2β εT (t)ε(t) + κ˜ T κ˜ + λ˜ T λ˜ 6 − 2α(3m)(n2 −m2 )/2n2 V2 (m2 +n2 )/2n2 − 2βV2 (p2 +q2 )/2q2 ,
(22)
(23)
˙ where α = min{α2 , α3 }, β = min{β2 , β3 } and m is the dimension of vector ε(t). Due to V(t) > 0, V(t) ≤ 0, it then follows from Lemma 2.3 that the error system (17) is global fixed-time stable. It means that the synchronization error converges to zero within the upper bound of settling time described by (19), i.e. τ2 < The proof is completed.
(3m)(m2 −n2 )/2n2 n2 1 q2 + . α m2 − n2 β q2 − p2 6
Remark 3.7. To achieve the GFTS of DDCS with uncertain parameters, the adaptive control methods is utilized. Finally, the estimations of uncertain parameters can converge to some constants in a fixed time. In [38], the finitetime synchronization of neural network systems with uncertain parameters is realized, but the GFTS of systems with different dimensions hasn’t been considered. 3.3. GFTS of networked DDCS In this subsection, we extend the GFTS from DDCS to networked chaotic systems with different dimensions. Consider a class of dynamical networks with M individuals indexed by k = 1, · · · , M, M X bk j x j (t), (24) x˙k (t) = gk (xk (t)) + h
j=1
iT
r
where xk (t) = xk1 (t), xk2 (t), · · · , xkr (t) ∈ R is a r-dimensional state vector of the kth node. Define x(t) = [x1 (t)T , x2 (t)T , · · · , x M (t)T ]T , g(x) := [g1 (x1 ), g2 (x2 ), · · · , g M (x M )]T where gk : Rr → Rr is a continuous function, and P − M b 1j j=2 .. B = . b M1
..
.
...
,
b1M
...
−
.. . M−1 P j=1
bM j
B = (bk j ) ∈ R M×M represents the coupling matrix. If node k is affected by node j( j , k), then bk j = b jk > 0; otherwise, bk j = b jk = 0. The vector form of system (24) is x˙(t) = g(x(t)) + Bx(t) = G(x).
(25)
y˙ (t) = h(y(t)) + By(t) + u(t) = H(y) + u(t),
(26)
The slave system also can be described as
where y(t) = [y1 (t)T , y2 (t)T , · · · , y M (t)T ]T represents the state vector of whole slave system. Here yk (t) = [yk1 (t), yk2 (t), · · · , ykl (t)]T ∈ Rl which is the state vector of the kth node, h(y) := [h1 (y1 ), h2 (y2 ), · · · , h M (y M )]T , hk : Rl → Rl is a continuous function and u(t) = [u1 (t)T , u2 (t)T , · · · , u M (t)T ]T denotes the vector of the control input, the control input of node k is denoted as uk (t) ∈ Rl . Then, the error system can be obtained by (25) and (26), ε(t) ˙ = S ξ (x) x˙ − S ζ (y)˙y
= S ξ (x)G(x) − S ζ (y)(H(y) + u(t)),
(27)
where ε(t) = [ε1 (t)T , ε2 (t)T , · · · , ε M (t)T ]T denotes the error vector, and εk (t) = ξ(xk (t)) − ζ(yk (t)) is the kth element of error vector. S ξ (x) = diag{S ξ (x1 ), S ξ (x2 ), · · · , S ξ (x M )} and S ζ (y) = diag{S ζ (y1 ), S ζ (y2 ), · · · , S ζ (y M )} are the Jacobian matrices of functions ξ(xk ) and ζ(yk ), and have the following forms S ξ (xk ) =
S ζ (yk ) =
∂ξ1 (xk ) ∂xk1 ∂ξ2 (xk ) ∂xk1
∂ξ1 (xk ) ∂xk2 ∂ξ2 (xk ) ∂xk2
∂ξm (xk ) ∂xk1
∂ξm (xk ) ∂xk2
∂ζ1 (yk ) ∂yk1 ∂ζ2 (yk ) ∂yk1
∂ζ1 (yk ) ∂yk2 ∂ζ2 (yk ) yk2
∂ζm (yk ) ∂yk1
∂ζm (yk ) ∂yk2
.. .
.. .
.. .
.. .
7
··· ··· .. . ··· ··· ··· .. . ···
∂ξ1 (xk ) ∂xkr ∂ξ2 (xk ) ∂xkr
.. .
∂ξm (xk ) ∂xkr ∂ζ1 (yk ) ∂ykl ∂ζ2 (yk ) ∂ykl
.. .
∂ζm (yk ) ∂yl
,
.
The objective is to guarantee that the error vector ε(t) tends to zero in a fixed time, then systems (25) and (26) reach synchronization. Thus the following control protocol is designed as: u(t) = − H(y) + S ζ −1 (y)(S ξ (x)G(x) + α4 sig(ε(t))m3 /n3 + β4 sig(ε(t)) p3 /q3 ),
(28)
where sig(ε(t))m3 /n3 = [(sig((ε1 (t))m3 /n3 ))T , · · · , (sig((ε M (t))m3 /n3 )T ]T , sig(εk (t))m3 /n3 = [sign(εk1 (t))|εk1 (t)|m3 /n3 , · · · , sign(εkm (t))|εkm (t)|m3 /n3 ]T , α4 , β4 are positive constants, m3 , n3 , p3 , q3 are positive odd integers and satisfy m3 > n3 , p3 < q3 . Combining (27) and (28), we can obtain the error dynamics as ε(t) ˙ = −α4 sig(ε(t))m3 /n3 − β4 sig(ε(t)) p3 /q3 .
(29)
Similarly, in the following theorem, the GFTS of systems (25) and (26) is proposed. Theorem 3.8. Suppose Assumption 3.1 holds. The error dynamics (29) of the master system (25) and slave system (26) can be globally fixed-time stabilized at the origin if the control input is given by (28). That is to say, the systems (25) and (26) reach GFTS within the upper bound of settling time described by τ3 <
1 q3 m(m3 −n3 )/2n3 n3 + , α4 m3 − n3 β4 q3 − p3
(30)
where m is the dimension of vector ε(t), α4 , β4 are positive constants, m3 , n3 , p3 , q3 are positive odd integers and satisfy m3 > n3 , p3 < q3 . Proof: The proof is omitted here because it is similar to that of Theorem 3.3. Remark 3.9. In this subsection, the results of DDCS are extended to the networked chaotic systems with different dimensions. It means that the GFTS of networked chaotic systems with different dimensions can be achieved. There exist some researches [39, 40] investigating the synchronization of networked systems with different dimensional nodes, but the GFTS can not be achieved. Furthermore, the investigation of GFTS for networked systems is necessary in practical applications, especially, neural networks and complex networks. 4. Simulation results In order to illustrate the effectiveness of the proposed control protocols for realizing GFTS of DDCS, in this section some examples are given. 4.1. GFTS between the hyperchaotic L¨u system and T system In order to demonstrate the proposed control protocol (7), the four dimensional hyperchaotic L¨u system [43] as master system and the three dimensional T system [44] as slave system are given below, respectively. The hyperchaotic L¨u system is given
and the T system is given
x˙1 (t) = −ax1 (t) + ax2 (t), x˙2 (t) = cx2 (t) − x1 (t)x3 (t), x˙3 (t) = −bx3 (t) + x1 (t)x2 (t), x˙4 (t) = x3 (t) − x4 (t),
y˙ 1 (t) = a1 (y2 (t) − y1 (t)) + u1 (t), y˙ 2 (t) = (c1 − a1 )y1 (t) − a1 y1 (t)y3 (t) + u2 (t), y˙ (t) = −b y (t) + y (t)y (t) + u (t), 3 1 3 1 2 3 8
(31)
(32)
Fig. 1: Trajectories of the errors εk (t)(k = 1, 2, 3): (a)α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9; (b)α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9.
Fig. 2: Trajectories of the errors εk (t)(k = 1, 2, 3): (c)α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9; (d)α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9.
with a = 15, b = 5, c = 10, a1 = 2.1, b1 = 0.6 and c1 = 30. Suppose ξ(x1 , x2 , x3 , x4 ) = (x1 + x3 , x2 + x4 , x2 x3 )T , ζ(y1 , y2 , y3 ) = (y1 , y2 , y3 )T . It then follows that the Jacobian matrices can be written as 1 0 0 1 0 1 0 S ξ (x) = 0 1 0 1 and S ζ (y) = 0 1 0 . 0 0 1 0 x3 x2 0 Hence, according to the system (6), we obtain the error system ε˙ 1 (t) = − 15x1 (t) + 15x2 (t) − 5x3 (t) − 2.1y2 (t) + 2.1y1 (t) − u1 (t), ε˙ 2 (t) =10x2 (t) − x1 (t)x3 (t) + x3 (t) − x4 (t) − 27.9y1 (t) + 2.1y1 (t)y3 (t) − u2 (t), ε˙ 3 (t) = − 50x2 (t)x3 (t) + 10x2 (t)(x2 (t))2 + 5x1 (t)(x3 (t))2 − (x1 (t))2 x2 (t)x3 (t) + 0.6y (t) + y (t)y (t) − u (t). 3
1
2
(33)
3
Fig. 1 and Fig. 2 show the trajectories of errors εk (t)(k = 1, 2, 3) with different initial conditions and the systems’ parameters. In Fig. 1(a) and (b), the same initial conditions of systems (31) and (32) are chosen as (x1 (0), x2 (0), x3 (0), x4 (0)) = (2, −2, 2, 1), (y1 (0), y2 (0), y3 (0)) = (1, 2, −2), but the parameters are selected as α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9 and α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9, respectively. In Fig. 2(c) and Fig. 2(d), the same initial conditions are chosen as (x1 (0), x2 (0), x3 (0), x4 (0)) = (2, 1, 1, 1.5), (y1 (0), y2 (0), y3 (0)) = (1, 1, 2), but the parameters are chosen as α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9 and α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9, respectively. We can see that Fig. 1(a), Fig. 2(c) have the same systems’ parameters but different initial conditions, as well as Fig. 1(b) and Fig. 2(d). According to Theorem 3.3, we can obtain the upper bound of convergence time are 0.6131 in Fig. 1(a) and Fig. 2(c), 7.4696 in Fig. 1(b) and Fig. 2(d), which are satisfied in Fig. 1 and Fig. 2. In addition, from Fig. 1(a) and Fig. 1(b) or from Fig. 2(c) and Fig. 2(d), the convergence time is dependent on system parameters. From Fig. 1(a) and Fig. 2(c) or from Fig. 1(b) and Fig. 2(d), the convergence time is independent of initial conditions of the systems. In addition, we suppose ξ(x1 , x2 , x3 , x4 ) = (x1 x4 , x2 + x3 )T , ζ(y1 , y2 , y3 ) = (y1 , 0.5y2 + 0.5y3 )T . It then follows that the Jacobian matrices can be written as " # " # x4 0 0 x1 1 0 0 S ξ (x) = and S ζ (y) = . 0 1 1 0 0 0.5 0.5 9
Fig. 3: Trajectories of the errors εk (t)(k = 1, 2)
The initial conditions of systems (31) and (32) in Fig. 3(a) and Fig. 3(b) are given as (x1 (0), x2 (0), x3 (0), x4 (0)) = (2, 1, 1, 1.5), (y1 (0), y2 (0), y3 (0)) = (1, 1, 2) and (x1 (0), x2 (0), x3 (0), x4 (0)) = (1, −0.5, −1.5, 2), (y1 (0), y2 (0), y3 (0)) = (1, −1, −1), respectively. The parameters in Fig. 3(a) and Fig. 3(b) are selected by α1 = 5, β1 = 6, m1 = 7, n1 = 3, p1 = 5, q1 = 9 and α1 = 1/3, β1 = 0.4, m1 = 5, n1 = 3, p1 = 7, q1 = 9, respectively. From Fig. 3, we can see that the synchronization error systems can also converge in a fixed time. Remark 4.1. In this subsection, Fig.1–Fig.3 show the GFTS can be achieved by the control protocol (7) which satisfy the Theorem 3.3. Fig.1 and Fig.2 verify the upper bound of setting time is independent of initial conditions but dependent on the designed parameters. In Fig.1 and Fig.2, the continuous functions are chosen as ξ : R4 → R3 and ζ : R3 → R3 . In Fig.3, the continuous functions are chosen as ξ : R4 → R2 and ζ : R3 → R2 . It means that the Fig.3 shows the lower dimensional GFTS can be achieved compared with Fig.1 and Fig.2. 4.2. GFTS between the hyperchaotic Lorenz system and Duffing system with uncertain parameters The following example is used to illustrate the adaptive laws for uncertain parameters and the corresponding proposed control scheme for the DDCS, where the hyperchaotic Lorenz system [45] is selected as the master system and Duffing system [46] is selected as the slave system. Consider the hyperchaotic Lorenz system x˙1 (t) = −κ1 x1 (t) + κ1 x2 (t), x˙2 (t) = κ2 x1 (t) − κ3 x2 (t) + x4 (t) − x1 (t)x3 (t), (34) x˙3 (t) = −κ4 x3 (t) + x1 (t)x2 (t), x˙4 (t) = −κ5 x2 (t) − κ6 x4 (t), and the Duffing system
y˙ 1 (t) = y2 (t) + u1 (t), y˙ 2 (t) = y1 (t) − y3 (t) − λ1 y2 (t) + λ2 cos(t) + u2 (t). 1
(35)
To show the hyperchaotic or chaotic behaviour for the systems (34) and (35), we choose κ1 = 12, κ2 = 23, κ3 = 1, κ4 = 2.1, κ5 = 6, κ6 = 0.2, λ1 = 1 and λ2 = 0.8. Suppose ξ(x1 , x2 , x3 , x4 ) = (x1 +x2 , x3 +x4 )T , ζ(y1 , y2 ) = (y1 , y2 )T . Then,
S ξ (x) =
"
1 0
1 0
0 1
0 1
#
and S ζ (y) =
From (15), we can get that the adaptive laws of parameters hold ˙ κˆ 1 (t) = (x2 (t) − x1 (t))ε1 (t), κ˙ˆ 2 (t) = x1 (t)ε1 (t), κ˙ˆ 3 (t) = −x2 (t)ε2 (t), κ˙ˆ 4 (t) = −x3 (t)ε2 (t), κˆ˙ 5 (t) = −x2 (t)ε2 (t), κ˙ˆ (t) = −x (t)ε (t), 6
4
2
10
"
1 0
0 1
#
.
(36)
Fig. 4: (a)Trajectories of the errors εk (t)(k = 1, 2); (b)trajectories of estimations κˆ k (k = 1, ..., 6); (c)trajectories of estimations λˆ k (k = 1, 2).
and
˙ λˆ 1 (t) = y2 (t)ε2 (t), λ˙ˆ (t) = − cos(t)ε (t). 2 2
(37)
According to (36) and (37), the control input of the slave system can be designed as follows by (17), u1 (t) = − y2 (t) − x1 (t)ˆκ1 + x2 (t)ˆκ1 + κˆ 2 x1 (t) − κˆ 3 x2 (t) + x4 (t) − x1 (t)x3 (t) m2 p2 + α2 sign(ε1 (t))|ε1 (t)| n2 + β2 sign(ε1 (t))|ε1 (t)| q2 sig(ε1 (t)) sig(ε1 (t)) + α3 m(n2 −m2 )/n2 (kˆκk + Γ)(m2 +n2 )/n2 + β3 (kˆκk + Γ)(p2 +q2 )/q2 2 2 kε(t)k kε(t)k sig(ε1 (t)) ˆ sig(ε1 (t)) ˆ (kλk + Θ)(m2 +n2 )/n2 + β3 (kλk + Θ)(p2 +q2 )/q2 , + α3 m(n2 −m2 )/n2 kε(t)k2 kε(t)k2 u2 (t) = − y1 (t) + y31 (t) + λˆ 1 y2 (t) − cos(t)λˆ 2 − x3 (t)ˆκ4 − x2 (t)ˆκ5 − x4 (t)ˆκ6 m2 p2 n2 q2 + x (t)x (t) + α sign(ε (t))|ε (t)| + β sign(ε (t))|ε (t)| 1 2 2 2 2 2 2 2 sig(ε2 (t)) sig(ε2 (t)) (kˆκk + Γ)(m2 +n2 )/n2 + β3 (kˆκk + Γ)(p2 +q2 )/q2 + α3 m(n2 −m2 )/n2 2 2 kε(t)k kε(t)k sig(ε2 (t)) ˆ sig(ε2 (t)) ˆ + α3 m(n2 −m2 )/n2 (kλk + Θ)(m2 +n2 )/n2 + β3 (kλk + Θ)(p2 +q2 )/q2 . 2 kε(t)k kε(t)k2
By the error dynamics (18), we can get ε˙ 1 (t) = − (x1 (t) − x2 (t))κ1 + κ2 x1 (t) − κ3 x2 (t) + x4 (t) − x1 (t)x3 (t) − y2 (t) − u1 (t), ε˙ 2 (t) = − κ4 x3 (t) + x1 (t)x2 (t) − κ5 x2 (t) − κ6 x4 (t)y1 (t) + y3 (t) + λ1 y2 (t) − λ2 cos(t) − u2 (t). 1
(38)
(39)
In this simulation we choose Γ = 27 and Θ = 1.71 where kκk k ≤ Γ (k = 1, ..., 6) and kλk k ≤ Θ (k = 1, 2). Fig.4(a) shows the trajectories of errors εk (t)(k = 1, 2), where the parameters are selected as α2 = 0.01, β2 = 10.5, α3 = 0.001, β3 = 0.001, m2 = 11, n2 = 9, p2 = 1, q2 = 11, the initial values of two systems are given as (x1 (0), x2 (0), x3 (0), x4 (0)) = (−1, −0.6, −1, −0.2), (y1 (0), y2 (0)) = (−2, −2). Fig.4(b) and Fig.4(c) show the trajectories of estimations of κˆ k (k = 1, ..., 6) and λˆ k (k = 1, 2) where the initial conditions of them are chosen as (ˆκ1 (0), κˆ 2 (0), κˆ 3 (0), κˆ 4 (0), κˆ 5 (0), κˆ 6 (0)) = (0, 0, 0, 0, 0, 0), (λ1 (0), λ2 (0)) = (−4.8, 1.5). According to Theorem 3.6, Fig.4(a) shows master and slave systems (13) can be synchronized in a fixed time via adaptive controller (17). Fig.4(b) and Fig.4(c) show the estimations of parameters can converge to some constants which satisfies the designed updated laws (15) and (16). 11
4.3. GFTS of networked Genesio system and Duffing system In this subsection, we consider the following networked Genesio [47] chaotic system with 10 nodes and networked Duffing chaotic system with 10 nodes as, x˙(t) = g(x(t)) + Bx(t), (40) y˙ (t) = h(y(t)) + By(t) + u(t), where in the networked Genesio master system, gk (xk (t)) holds 0 1 0 xk1 (t) 0 0 1 xk2 (t) + 0 gk (xk (t)) = 0 2 xk1 (t) xk3 (t) −6 −2.92 −1.2 and in the networked Duffing slave system, hk (yk (t)) holds ! ! 0 1 yk1 (t) hk (yk (t)) = + 1 −1 yk2 (t)
Then, we choose the symmetric coupling matrix B ∈ R10×10 as, 0 1 0 0 −3 0 0 −4 1 0 1 1 0 1 −3 0 1 0 1 0 0 −6 1 1 1 1 1 −4 0 0 B = 1 0 1 0 −3 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0
,
0 −y3k1 (t) + 0.8 cos(t) 1 0 0 1 0 1 −5 1 0 1
0 1 0 0 0 0 1 −4 1 1
1 0 0 1 1 0 0 1 −5 1
0 1 1 1 0 0 1 1 1 −6
!
.
.
Assuming the projective functions as ξ(x) = (xk1 + xk2 , xk3 )T , ζ(y) = (yk1 , yk2 )T , then, we get
S ξ (x) =
"
1 0
1 0
0 1
#
, S ζ (y) =
"
1 0
0 1
#
.
From (27), we have 10 X ε ˙ (t) =x (t) + x (t) − y (t) + bk j (xk1 (t) + xk2 (t) − yk1 (t)) − uk1 (t), k1 k1 k3 k2 j=1 2 ε˙ k2 (t) = − 6xk1 (t) − 2.92xk2 (t) − 1.2xk3 (t) + xk1 (t) − yk1 (t) + yk2 (t) + y3k1 (t) 10 X − 0.8 cos(t) + bk j (xk3 (t) − yk2 (t)) − uk2 (t).
(41)
j=1
According to (28), the controllers is designed as 10 X uk1 (t) =xk1 (t) + xk3 (t) − yk2 (t) + bk j (xk1 (t) + xk2 (t) − yk1 (t)) j=1 m3 p3 + α3 sign(εk1 (t))|εk1 (t)| n3 + β3 sign(εk1 (t))|εk1 (t)| q3 , 2 uk2 (t) = − 6xk1 (t) − 2.92xk2 (t) − 1.2xk3 (t) + xk1 (t) − yk1 (t) + yk2 (t) + y3k1 (t) − 0.8 cos(t) 10 X m3 p3 n3 q3 + b (x (t) − y (t)) + α sign(ε (t))|ε (t)| + β sign(ε (t))|ε (t)| . k j k3 k2 3 k2 k2 3 k2 k2 j=1 12
(42)
Fig. 5: (a)Trajectories of the errors εk1 (t)(k = 1, ..., 10); (b)trajectories of the errors εk2 (t)(k = 1, ..., 10)
Fig. 5 shows the trajectories of errors εk1 (t)(k = 1, ..., 10) and εk2 (t)(k = 1, ..., 10) where the parameters of systems are selected as α3 = 3.2, β3 = 5, m3 = 11, n3 = 5, p3 = 3, q3 = 7, and the initial conditions are chosen as x(0) = (1 + 5k, 2 + 4k, −1 − 8k)T , y(0) = (3 + 2k, 2 − 3k)T , (k = 1, ..., 10). From Theorem 3.8, the upper bound of settling time is obtained as τ3 < 0.7126 which is satisfied in Fig. 5. In addition, from Fig. 5(a) and Fig. 5(b), we can see that the convergence time of errors εk1 (t)(k = 1, ..., 10) and εk2 (t)(k = 1, ..., 10) is less than 0.7126. 5. Conclusions In this paper, the GFTS problems of DDCS with and without unknown parameters have been investigated, the corresponding controllers and adaptive control laws have been designed respectively. Then, GFTS problems have also been extended to networked DDCS. In addition, it is shown that the upper bound of settling time in fixed-time synchronization is independent of initial conditions of systems, which is different from finite-time synchronization. Certainly, more parameters in controller should be designed in advance which may more difficult to reach synchronization in other complex systems. So, our future work will consider the GFTS neural networks and complex networks with different dimensions. 6. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61977004 and 61503016, the Fundamental Research Funds for the Central Universities under Grants YWF-19-BJ-J-259 and 2017JBM067, The National Key Research and Development Program of China under Grant 2017YFB0103202. References [1] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems. Phys. Rev. Lett. 64 (8) (1990) 821-824. [2] S.J. Schiff, K. Jerger, D.H. Duong, T. Chang, M.L. Spano, W.L. Ditto, Controlling chaos in the brain. Nature 370 (6491) (1994) 615-620. [3] S. Bowong, Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys. Lett. A 326 (1) (2004) 102-113. [4] C.D. Li, X.F. Liao, K.W. Wong, Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication, Physica D 194 (3) (2004) 187-202. [5] J.M. Ottino, F.J. Muzzio, M. Tjahjadi, J.G. Franjione, S.C. Jana, H.A. Kusch, Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing processes, Science 257 (5071) (1992) 754-760. [6] Z.L. Li, M.H. Dong, S.P. Wen, X. Hu, P. Zhou, Z.G. Zeng, CLU-CNNs: Object detection for medical images, Neurocomputing 350 (2019) 53-59. [7] G. He, J.A. Fang, Z. Li, Finite-time synchronization of cyclic switched complex networks under feedback control, J. Frankl. Inst. 354 (9) (2017) 3780-3796. [8] J.M. Park, P.G. Park, H∞ sampled-state feedback control for synchronization of chaotic Lure systems with time delays, J. Frankl. Inst. 355 (16) (2018) 0016-0032. [9] Shengbo Wang, Yanyi Cao, Tingwen Huang, and Shiping Wen, Passivity and passification of memristive neural networks with leakage term and time-varying delays, Appl. Math. Comput. 361 (2019) 294-310. [10] Z.Y. Wu, X.C. Fu, Combination synchronization of three different order nonlinear systems using active backstepping design, Nonlinear Dyn. 73 (3) (2013) 1863-1872. [11] Z.B. Wang, H.Q.Wu, Projective synchronization in fixed time for complex dynamical networks with nonidentical nodes via second-order sliding mode control strategy, J. Frankl. Inst. 355 (15) (2018) 7306-7334. [12] A. Khanzadeh, M. Pourgholi, Fixed-time sliding mode controller design for synchronization of complex dynamical networks, Nonlinear Dyn. 88 (4) (2017) 2637-2649. [13] S.P. Wen, S.X. Xiao, Y. Yang, Z. Yan, Z.G. Zeng, T.W. Huang, Adjusting Learning Rate of Memristor-Based Multilayer Neural Networks via Fuzzy Method, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 38 (6) (2019) 1084-1094.
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Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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