Bounded intra-layer synchronization of multilayer heterogeneous networks without external controllers

Multilayer network models with heterogeneous nodes characterize complex network in the reality to the best effect, and bounded synchronization has more important practical significance in engineering applications. This paper aims at the bounded intra-layer synchronization of multilayer heterogeneous networks where nodal dynamics is different each other. Based on a newly-introduced vector norm, we derive the sufficient conditions for the bounded intra-layer synchronization in multiplex networks with one-to-one connections between layers by using a new effective approach. The condition integrates the intra-layer and inter-layer structure information, and can be as well used to the accurate intra-layer synchronization in multiplex networks with homogeneous nodes each layer. More interestingly, the theoretical results show the negative/positive coupling between layers has negative/positive effect for intra-layer synchronization, and the partial cutting of connections between layers increases the bound of intra-layer synchronization. Finally, numerical examples are provided to verify the effectiveness of theoretical results.