The Dynamics Of Synchronous Pattern At Weak Coupling

With the rapid development of network technologies, such as the Internet, human society entered the complex networks age, and human’s life and production activities are relying more and more on the security and functioning of different kinds of complex networks. Consequent-ly, the complex system and network becomes a newly rising interdisciplinary field. The study of synchronization of complex networks is one of the central problems in interdisciplinary research. Synchronous behaviors are commonly observed in natural and man-made systems, and are widely recognized as important to the system functionality and operations. Therefore, studying the syn-chronization of complex networks is meaningful in both theory and reality. Here, we focus on some issues about network synchronous pattern at weak coupling.The first chapter is a brief review about complex networks, dynamical of networks, and syn-chronization patterns, including some basic concepts, related features, and the methods generally used in their investigations.In the second chapter, based on the stability of synchronization patterns, we study the consis-tency between functional and structural networks of coupled nonlinear oscillators. In data-based reconstruction of complex networks, dynamical information can be measured and exploited to gen-erate a functional network, but is it a true representation of the actual (structural) network? That is, when do the functional and structural networks match and is a perfect matching possible? To ad-dress these questions, we use coupled nonlinear oscillator networks and investigate the transition in the synchronization dynamics to identify the conditions under which the functional and structural networks are best matched. We find that, as the coupling strength is increased in the weak-coupling regime, the consistency between the two networks first increases and then decreases, reaching max-imum in an optimal coupling regime. Moreover, by changing the network structure, we find that both the optimal regime and the maximum consistency will be affected. In particular, the consis-tency for heterogeneous networks is generally weaker than that for homogeneous networks. Based on the stability of the functional network, we propose further an efficient method to identify the optimal coupling regime in realistic situations where the detailed information about the network structure, such as the network size and the number of edges, is not available. Two real-world exam-ples are given:corticocortical network of cat brain and the Nepal power grid. Our results provide new insights not only into the fundamental interplay between network structure and dynamics but also into the development of methodologies to reconstruct complex networks from data.In the third chapter, we investigate the controlling of synchronization patterns. Although the set of permutation symmetries of a complex network can be very large, few of the symmetries give rise to stable synchronous patterns, in this chapter, we present a new framework and develop tech-niques for controlling synchronization patterns in complex network of coupled chaotic oscillators. Specifically, according to the network permutation symmetry, we design a small-size and weight-ed network, namely the control network, and use it to control the large-size complex network by means of pinning coupling. We argue mathematically that for any of the network symmetries, there always exists a critical pinning strength beyond which the unstable synchronous pattern associated to this symmetry can be stabilized. The feasibility of the control method is verified by numeri-cal simulations of both artificial and real-work networks, and is demonstrated by experiment of coupled chaotic circuits. Our studies pave a way to the control of dynamical patterns in complex networks.In chapter four, we introduce a new control method which is able to induce chimera-synchron-ization in complex networks. In a recent study of chaos synchronization in symmetric complex networks, it is found that stable synchronous clusters may coexist with many non-synchronous nodes in the asynchronous regime, resembling the chimera state observed in regular networks of non-locally coupled periodic oscillators. Although of practical significance, this new type of state, namely the chimera-synchronization state, is hardly observed for the general complex networks, due to either the topological instabilities or the weak coupling strength. Here, based on the strategy of pinning coupling, we propose an effective method for inducing chimera-synchronization in sym-metric complex network of coupled chaotic oscillators. We are able to argue mathematically that, by pinning a group of nodes satisfying permutation symmetry in the network, there always exits a critical pinning strength beyond which the chimera-synchronization state can be stably generated. The feasibility and efficiency of the control method are verified by numerical simulations of both artificial and real-world complex networks.In the last chapter, we give our summary and perspectives.