Adaptive Pinning Synchronization in Complex Dynamical Networks with a Novel Adaptive Law

In this paper, we propose one adaptive pinning synchronization scheme in complex dynamical networks, whose adaptive law is simple in form. When a nonlinear vector field meets different conditions, we prove that the scheme is synchronous and that the synchronous solution is locally and globally asymptotically stabile. We analyze the process of using maximum eigenvalues of low matrix to judge pinning synchronization, and give one method to calculate the number of pinning nodes. At last, three numerical simulations are given to verify the effectiveness of the proposed scheme. The first two examples show relations between maximum eigenvalues of the principal submatriecs and the number of pinning nodes under the conditions of three pinning strategies in a scale-free network and under the conditions of the random pinning strategy in a nearest-neighbor network, respectively, and the last one shows effectiveness of the adaptive pinning synchronization by selecting pinning nodes randomly in a scale-free network.