Adaptive synchronization in complex networks with non-delay and variable delay couplings via pinning control

The adaptive pinning synchronization in complex networks with non-delay and variable delay couplings is investigated in this letter. For both reducible and irreducible coupling matrixes, novel and practical synchronization criteria are obtained by Lasalle's invariance principle, and sufficient conditions for achieving the synchronization to require the minimum number of pinning nodes are derived. Moreover, the method for calculating the number of pinning nodes is given using the decreasing law of maximum eigenvalues of the principal submatrixes. At last, simulation examples are given to verify effectiveness of the proposed pinning synchronization scheme.