The Study On The Characteristics And Synchronization Of Fractional Chaotic System

Chaos is a seemingly random motion,which occurs in the determining system. There is a general phenomenon that chaos exists in the natural and social sciences. The study on the chaotic phenomenon of the system can explore the existence of ordered and practical value information behind the system that exists chaotic behavior, which is conducive to take advantage of the chaos. As the key issue of chaotic application, chaos synchronization has a great potential application in secret communication. And the study not only focuses on the synchronization between two chaos systems, but also expands into the synchronization of the complex networks. Fractional calculus is a generalization of integration, and the dynamic behaviors of fractional-order systems is more complex than that of integer-order systems. Therefor, that fractional calculus simulates the actual dynamic systems is more realistic. This paper studies on the chaos characteristics and synchronization of fractional-order systems and the synchronization of fractional-order complex networks.Firstly, take the fractional-order permanent magnet synchronous motor in example, based on the stability theory of fractional-order linear systems, the stability of equilibrium points is discussed. And how system’s order impacts its dynamic characteristics is studied by phase diagram and bifurcation diagram. System’s parameter characteristic is analyzed by Lyapunov exponent and bifurcation diagram.Secondly, for synchronization problem of fractional chaotic systems, on one hand, based on matrix theory, stability theory of fractional-order linear systems is simply applied in the synchronization controller design. On the other hand, considering the complex of actual problems, an universal method of designing controller is proposed, which not only synchronizes different chaotic systems with different orders and unknown parameters, but also identifies the unknown parameters of the respond system. Both the proposed methods’ effectiveness is certified by numerical simulation.Thirdly, based on WS small-world network, synchronization of fractional-order complex network with identical nodes is studied. And the nodes’ dynamics are fractional-order coupled dynamos systems. Without control, the synchronization of network is discussed under the conditions of the different rewiring probability and coupling strength. That is to say. that the network’s structure and the network’s coupling strength effect on its synchronization capabilities is studied, and the network is no external force.