Pinning adaptive anti-synchronization between two general complex dynamical networks with non-delayed and delayed coupling

Dedicated to Professor Guanrong Chen on the occasion of his 65th birthday
https://doi.org/10.1016/j.amc.2012.01.007Get rights and content

Abstract

This paper investigates the anti-synchronization (AS) problem of two general complex dynamical networks with non-delayed and delayed coupling using pinning adaptive control method. Based on Lyapunov stability theory and Barbalat lemma, a sufficient condition is derived to guarantee the AS between two networks with non-delayed and delayed coupling. Numerical simulations are also presented to show the effectiveness of the proposed AS criterion.

Introduction

Over the past one decade, complex network behaviors have attracted a great deal of attention in variety of fields including biology, physics, mathematics, engineering and so on [1], [2], [3], [4]. A complex network is a large set of nodes and the connections between them, in which the nodes and connections can have different meanings in different situations. Some common examples of complex networks in the real world include the Internet, the World Wide Web (WWW), food webs, neural networks and social networks.

So far, various complex dynamical behaviors of complex networks have been extensively investigated, in which synchronization is a significative and interesting phenomenon [5], [6], [7], [8], [9]. The synchronization problems for small-world and scale-free networks have been studied in Refs. [10], [11], respectively. In Ref. [12], Zhou et al. investigated the locally and globally adaptive synchronization of an uncertain complex dynamical network. Recently, Sun et al. studied the global synchronization of a general complex network with non-delayed and delayed coupling [13]. In Ref. [14], Yu et al. investigate global synchronization of linearly hybrid coupled network with time-varying delay. Global synchronization in arrays of coupled networks with one single time-varying delay coupling is investigated in [15]. However, it is impossible to ensure synchronization by adding controllers to all nodes due to the complexity of the dynamical network. So researchers try to control a complex network by pinning part of nodes [16], [17], [18], [19]. The specifically pinning scheme and randomly pinning scheme were used to stabilize scale-free networks in [20]. In Ref. [21], Chen et al. pinned a complex network to a homogenous solution by a single controller. Yu et al. showed that the nodes with low degrees should be pinned first when the coupling strength is small [22]. Song et al. investigate the pinning synchronization problem of a hybrid-coupled complex network with mixed time-delays [23].

The above mentioned work focussed on the synchronization inside a network, which was called “inner synchronization” as in [24]. In 2007, Li et al. pioneered in studying “outer synchronization”, which refers to the synchronization phenomenon between two coupled networks regardless of “inner synchronization”, in Ref. [24]. An important example of outer synchronization is the spread of infectious diseases between different communities. Therefore, how to achieve synchronization between different networks is a very interesting and challenging work. Shortly after, Tang et al. realized the synchronization between two networks with identical and nonidentical topological structures by using adaptive controllers [25]. The problem of generalized outer synchronization of complex networks was considered in Refs. [26], [27].

Motivated by the above discussions, this paper focuses on the pinning adaptive outer anti-synchronization between two general complex dynamical networks with non-delayed and delayed coupling. As a special case of generalized synchronization, anti-synchronization (AS) phenomenon which can be characterized by vanishing of the sum of relevant state variables is a noticeable one in chaotic systems [28], [29], [30], [31]. In [32], AS of two complex networks using open-plus-closed-loop (OPCL) coupling was investigated. More recently, Sun et al. studied the hybrid synchronization problem of two coupled complex networks via the linear feedback and the adaptive linear feedback control methods [33].

In this paper, we study the AS between two general complex dynamical networks with non-delayed and delayed coupling based on the pinning controllers. A sufficient condition for the AS by adding adaptive feedback controllers to a fraction of network nodes is obtained. Numerical simulations are also provided to verify the effectiveness of our theoretical result.

The rest of this paper is organized as follows. In Section 2, the model of two general complex dynamical networks with non-delayed and delayed coupling is presented and some preliminaries are also provided. Pinning adaptive AS criterion is deduced in Section 3. Numerical simulations are given in Section 4. Finally, a conclusion is presented in Section 5.

Section snippets

Model description and preliminaries

Consider the following drive-response networks:x˙i(t)=f(xi(t),t)+c1j=1NaijΓxj(t)+c2j=1NbijΓxj(t-τ(t)),y˙i(t)=f(yi(t),t)+c1j=1NaijΓyj(t)+c2j=1NbijΓyj(t-τ(t))+uifor i=1,2,,N. Here xi,yiRn are respectively the state vector of the ith node in drive network X and response network Y,f:Rn×R+Rn is a continuously differentiable nonlinear vector-valued function, c1,c2>0 are the coupling strength, Γ is a inner-coupling matrix between nodes, τ(t) is the time-varying delay, and A=(aij)N×N,B=(bij)N×N

Pinning anti-synchronization criterion

Here, the pinning adaptive AS between the drive network X and the response network Y is investigated, and the main result is summarized in the following theorem.

Theorem 1

Suppose that Assumption 1, Assumption 2, Assumption 3 hold; Γ is a positive definite matrix and A is symmetric and irreducible. If:λmax(A2)<-L/c1,where P=(BTB)ΓT,k=λmax(P)/λmin(INΓ),L=L1+(c2k)/(2(1-ε))+c2/2, then the networks X and Y can realize AS.

Proof

From Assumption 1, Assumption 2, we get:(y+x)T(f(y,t)+f(x,t))=(y-(-x))T(f(y,t)-f(-x,t))

Numerical simulations

In this section, numerical simulations are presented to verify the effectiveness of the proposed AS criterion. In the numerical simulations throughout this paper, we assume that the coupling matrices A and B obey the scale-free distribution of the BA network model [2] with m0=m=8,N=100 and the small-world model [1] with the link probability p=0.1,m=4,N=100, respectively, the initial values are randomly chosen in the interval (0, 1) and the quantityE(t)=max{xi(t)+yi(t):i=1,2,,N},fort[0,+)is

Conclusion

In this paper, we have discussed the AS between two general complex dynamical networks with non-delayed and delayed coupling via pinning adaptive control. A rigorous AS theorem has been deduced through mathematical analysis in Section 3. Numerical simulations have verified the effectiveness and correctness of the theory.

Acknowledgements

The authors thank the referees and the editor for their valuable comments and suggestions on improvement of this paper. This work was supported in part by the National Natural Science Foundation of China (No. 10872119), National Basic Research Program of China, 973 Program (No. 2011CB706903), Key Disciplines of Shanghai Municipality under Grant No. S30104, and the Fundamental Research Funds for the Central Universities of Lanzhou University in China (lzujbky-2010-65 and lzujbky-2010-169).

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