Approximation methods for the stability analysis of complete synchronization on duplex networks

Recently, the synchronization on multi-layer networks has drawn a lot of attention. In this work, we study the stability of complete synchronization on duplex networks. We first numerically investigate the effects of coupling functions on complete synchronization on duplex networks. Then, we propose two approximation methods to deal with the stability of complete synchronization on duplex networks. In the first method, we introduce a modified master stability function and, in the second method, we only take into consideration the contributions of a few most unstable transverse modes to the stability of complete synchronization. We find that both methods work well for predicting the stability of complete synchronization for small networks. For large networks, the second method still works pretty well.