| Synchronization is a very common phenomenon in nature,which has been paid more and more attention in nonlinear science including physics,optics,chemistry and biology.Synchronization in complex networks has been widely studied by researchers in various fields for its universal and profound practical significance.Since the seminal work of Pecora and Carroll on complete synchronization in 1990,synchronization of chaotic oscillators receives much more attention,and most research works are based on the invariant-manifold assumption,namely,the master stability function theory.Our work is focusing on pinning synchronization.Partial synchronization is usually generated by adding different forms of non-local edges to the network.For complex networks,it is very difficult to generate partial synchronization due to its random connection and large network size,but adding new controller to the original network can greatly increase the stability of generating partial synchronization.Based on the method of computational group theory,it is shown that pinning control as an effective way directly interferes with the target group to make partial synchronization happen,which is confirmed by theoretical analysis and numerical simulation.This provides a theoretical basis for the analysis of an abnormal phenomenon in pinning synchronization.In our present work,the pinning control of partial synchronization in networked chaotic oscillators is studied and the results show that in some circumstances the non-pinned nodes could be synchronized with each other and form a synchronous cluster.The trajectory of the cluster is different from the one formed by the pinned nodes and,interestingly,its stability can not be analyzed by the prevalent method of master stability function.By analyzing the statistical properties of the pinned and non-pinned nodes,it is revealed that the synchronization of non-pinned nodes is attributed to the significant modification of their synchronization manifold,which adjusts the critical coupling for generating partial synchronization.Our study points out the deformation of synchronization manifold in synchronization transition,which brings a caution to the application of the generalized master stability function method in synchronization analysis. |
