P This paper studies how phase synchronization in complex networks depends on random shortcuts, using the uous chaotic Chua system as the nodes of the networks. It is found that for a given coupling strength when the number of random shortcuts is greater than a threshold the phase synchronization is induced. Phase synchronization becomes evident and reaches its maximum as the number of random shortcuts is further increased. These phenomena imply that random shortcuts can induce and enhance the phase synchronization in complex Chua systems. Furthermore, the paper also investigates the effects of the coupling strength and it is found that stronger coupling makes it easier to obtain the complete phase synchronization.
This paper studies pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions. To study a state-feedback pinning-controlled network with N nodes, it first converts the controlled network to an extended network of N＋1 nodes without controls. It is shown that the controlled synchronizability of the given network is determined by the real part of the smallest nonzero eigenvalue of the coupling matrix of its extended network when the synchronized region is unbounded; but it is determined by the ratio of the real parts of the largest and the smallest nonzero eigenvalues of the coupling matrix when the synchronized region is bounded. Both theoretical analysis and numerical simulation show that the portion of controlled nodes has no critical values when the synchronized region is unbounded, but it has a critical value when the synchronized region is bounded. In the former case, therefore, it is possible to control the network to achieve synchronization by pinning only one node. In the latter case, the network can achieve controlled synchronization only when the portion of controlled nodes is larger than the critical value.
