Synchronization and Explosive Synchronization of Two-Layer Second-Order Kuramoto Oscillator Networks

In the study of synchronization on complex networks, the second-order Kuramoto model has attracted extensive attention because the inclusion of inertia and damping enables a more realistic description of coupled systems with inertial and dissipative characteristics, such as power networks. However, existing studies on synchronization in second-order Kuramoto oscillator networks have mainly focused on single-layer networks, and the combined effects of inter-layer coupling, topological heterogeneity, and inertia in multilayer structures remain insufficiently explored. To address this issue, we extend the second-order Kuramoto model to two-layer oscillator networks with different architectures and allow flexible random inter-layer connections, and then systematically examine through numerical simulations the effects of damping/inertia, coupling strengths, topology variation, and layer size on synchronization. We find that when the inertia m is small, a moderate increase in the damping d can maintain a continuous synchronization transition while enhancing the overall synchronization level; in contrast, when m is large and d is small, explosive synchronization can be induced in the two-layer network. In addition, topologies with stronger degree heterogeneity are generally unfavorable for collective synchronization, and collective synchronization is more likely to be dominated by the larger layer. Layer-size mismatch can enhance the synchronization level of the two-layer system, and the global synchronization consistency of the whole system gradually increases as the size mismatch between the two layers becomes larger.