Synchronization in small-world networks

In the past decade, synchronization in complex networks, especially the question of synchronizability, attracts a lot of interest. The works on synchronizability in networks with a given topology can be divided into two classes according to the coupling matrix in networks. One is the static mechanism, where the coupling matrix remains fixed during the transition to synchronization. From the degree and load based weighted networks, the synchronizability becomes optimal when the intensities of all oscillators become uniform. The other one is the dynamical mechanism, where the coupling matrix evolves in time by introducing adaptive strengths between connected oscillators. The adaption process can enhance synchronization by modifying the coupling matrix in networks, but the synchronizability is still far from being optimal. This is because the resulting networks have nonuniform intensities even for networks with homogeneous degrees. In this paper we consider complete synchronization in smallworld networks of identical Rossler oscillators by applying a simple but effective dynamical optimization coupling scheme. We realize complete synchronization in networks with undelayed or delayed couplings, as well as ensure that all oscillators have uniform intensities during the transition to synchronization. Moreover, we obtain the coupling matrix with much better synchronizability in a certain range of the probability p for adding longrange connections.