New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.