Intralayer synchronization in a duplex network with noise

This paper concerns the impact of environmental noise on the intralayer synchronization of the duplex networks. A duplex network contains two layers. Different from the previous works (Tang et al., 2018; Shen & Tang, 2018), environmental noise is introduced into the dynamical system of the duplex network. We incorporate both the interlayer delay and the intralayer delay into the dynamical system. Both of the delays are time‐varying. However, Tang et al. (2018) only considered the intralayer delays, and they were assumed as constants, while Shen and Tang (2018) did not consider the interlayer delay or intralayer delay. When the system does not achieve automatic intralayer synchronization, we introduce two controllers: one is the state‐feedback controller and the other is the adaptive state‐feedback controller. Interestingly, we find that the intralayer synchronization will be achieved automatically if the interlayer coupling strength is large enough and the intralayer coupling strength is small enough under the situation that the time‐varying interlayer delays are absent. Finally, some interesting simulation results are obtained for the Chua–Chua chaotic system withapplication of our theoretic results, which show the feasibility and effectiveness of our control schemes.