Pinning Synchronization of Complex Dynamical Networks on Time Scales
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Pinning Synchronization of Complex Dynamical Networks on Time Scales Fang-Di Kong* and Jian-Ping Sun Abstract: The purpose of this paper is to investigate the synchronization problem of complex dynamical networks on time scales, which includes the synchronization problem of continuous-time and discrete-time complex dynamical networks as special cases. A pinning control strategy is designed to achieve synchronization of complex dynamical networks on time scales. Based on the theory of calculus on time scales and the Lyapunov method, pinning synchronization criteria for complex dynamical networks on time scales are established. Moreover, a numerical example is given to verify the effectiveness of theoretical results. Keywords: Complex dynamical network, pinning control, synchronization, time scale.
1.
INTRODUCTION
time complex dynamical network N
Many real-world systems can be modeled as complex dynamical networks with the nodes representing individuals in the system and the edges representing the interactions among them, such as the Internet [1], the World Wide Web, food webs and electrical power grids etc. (see, for example, [2] and references therein). Synchronization is one of the ubiquitous dynamical phenomena in nature. Synchronization of complex dynamical networks, which means that all nodes drive to a common state by utilizing information exchange among the nodes, has attracted considerable attention [3–9]. Control plays an indispensable role in forcing the network to achieve desired synchronization whenever a given network cannot realize synchronization by itself or the synchronized state is not the desired state. For a complex dynamical network, it is practically impossible or unnecessary to control all of its nodes to implement a desired objective. As an effective control scheme, pinning control strategy was introduced to synchronize a scale-free dynamical network to its equilibrium [10]. Since then, much effort has been devoted to synchronization of complex dynamical networks via pinning control scheme [11–18]. In [11–16], the authors investigated the pinning synchronization of continuous-time complex dynamical networks, and they concluded that continuous-time complex dynamical networks can be synchronized to a desired state by feedback pinning control scheme. In particular, Song and Cao [16] considered continuous-
x˙i (t) = f (xi (t)) + c ∑ gi j Γx j (t), j=1
t ∈ [0, +∞), i = 1, 2, · · · , N,
(1)
where xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T ∈ Rn is the state vector of the ith node at time t ∈ [0, +∞), f : Rn → Rn is a nonlinear vector function, the positive constant c > 0 is the coupling strength, the positive definite matrix Γ = diag(γ1 , γ2 , · · · , γn ) > 0 is the inner coupling matrix, and G = (gi j )N×N is called the coupling configuration matrix. Some low-dimensional pinning synchronization criteria were presented. In [17, 18], the pinning synchronization criteria of discrete-time complex dynamical networks were obtained.
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