Explosive synchronization of multi-layer complex networks based on star connection between layers with delay

Explosive synchronization (ES) is a kind of first-order jump phenomenon that exists in physical and biological systems. In recent years, researchers have focused on ES between single-layer and multi-layer networks. Most research on complex networks with delay has focused on single-layer or double-layer networks, multi-layer networks are seldom explored. In this paper, we propose a Kuramoto model of frequency weights in multi-layer complex networks with delay and star connections between layers. Through theoretical analysis and numerical verification, the factors affecting the backward critical coupling strength are analyzed. The results show that the interaction between layers and the average node degree has a direct effect on the backward critical coupling strength of each layer network. The location of the delay, the size of the delay, the number of network layers, the number of nodes, and the network topology are revealed to have no direct impact on the backward critical coupling strength of the network. Delay is introduced to explore the influence of delay and other related parameters on ES.