The Research On The Projective Synchronization Of Complex Dynamical Networks

As an essential phenomenon among dynamical behaviors of complex network, synchronizationhas been extensively investigated in various fields. Many kinds of synchronization manners arepresented to explain the complex dynamical behavior of physical systems, including projectivesynchronization.We mainly apply Lyapunov function method to study the projective synchronization of com-plex dynamical network, discussing the projective synchronization of two specific kinds of dynam-ical network respectively.Firstly, the adaptive projective synchronization of dynamical network with distributed timedelays is investigated. Network with unknown topology and network with both unknown topologyand system parameters of node dynamics are considered respectively. Based on Lyapunov sta-bility theory and LaSalle invariance principle, via designing appropriate adaptive controllers thesufcient conditions for achieving projective synchronization are obtained under the condition oflinearly independent. At the same time, the unknown topology of the network and system param-eters of node dynamics are well identified. The applicable Lyapunov functions play an importantrole in the proof of the theorem. In the numerical simulations, consider the projective synchro-nization of a complex network with distributed time delays consisting of5identical nodes in Lu¨chaotic system. Moreover, numerical results are illustrated to show the efectiveness of the proposedmethod.Secondly, cluster projective synchronization in community dynamical network with nonidenti-cal nodes is investigated. Networks consisting of coupled non-delayed systems and delayed systemsare both considered respectively. Based on Lyapunov stability theory via designing adaptivecontrollers which the linear parts of controllers are only pinned on those nodes which have directconnections between diferent communities several simple and useful synchronization criteria areobtained analytically. The cluster projective synchronization of the above networks are realized.Meanwhile, in order to realize the cluster projective synchronization more practically, adaptivecoupling strength method is adopted. In the numerical simulations, firstly consider the clusterprojective synchronization of the above two kinds of community network consisting of14nodeswith three communities. And then, consider the cluster projective synchronization of a communitynetwork consisting of three BA networks and a few connections. Numerical results are illustratedto show the efectiveness of the proposed method.