Research On Adaptive Synchronization Of Fractional-order Complex Networks

Nowadays, there are numerous natural or man-made complex networks. For examples, the World Wide Web, communication networks, social networks, food webs, and so on. All the above networks can be represented in terms of nodes and edges indicating connections between nodes. It is well known that complex networks pervade through almost all scientific and technological fields, including mathematics, physics, biological sciences, ecology, and social sciences etc. Due to its tremendous potentials in real applications, the research of complex networks has become a hot topic in modern scientific research.As is known to all, the fractional-order calculus is an old and new concept, which is a calculus model of any order. However, because there is no practical engineering application background, the fractional-order calculus theory has not attracted enough attention for a long time. But nowadays, it is found that many practical systems can be described by fractional-order differential equations, such as viscous system, dielectric polarization, electromagnetic wave etc. Compared with the integer-order calculus model, the fractional-order calculus has a better advantage in describing the model with memory and genetic characteristics. Therefore, he could describe the mathematical model more accurately.Because the fractional-order calculus has an incomparable advantage in describing the nonlinear system, we introduce the fractional-order calculus into complex networks. At present, the research on the fractional-order complex networks is very little. After referring to the previous research results, this paper has done the following work,(1) For two fractional-order complex networks, the hybrid synchronization problem is achieved by using linear feedback control and adaptive control to realize the hybrid synchronization of two fractional-order complex networks. Based on the fractional-order stability theory, the hybrid synchronization criterion and the response rate of the network are obtained.(2) According to the fractional-order complex networks with different topologies, we design an adaptive controller and an adaptive rate to achieve the synchronization between them. This controller is able to deal with the fractional-order complex networks with the same topology, and it can also deal with the fractional-order complex networks with different topologies.(3) At the end of the paper, it is proposed with a novel adaptive feedback control method which can identify with the unknown and uncertain system of parameters and topology structure with time delay in fractional-order complex network. And based on the stability theory of fractional order systems with time delay, a series of methods for identifying the parameters and topology of complex networks with time delay are obtained.