Collective synchronization in multilayer oscillator networks is a key topic in complex systems. However, most existing two-layer Kuramoto models mainly focus on first-order oscillators and pay limited attention to the combined effects of inertia, damping, and inter-layer interactions. In this work, a two-layer multi-rate Kuramoto model containing both first-order and second-order oscillators is developed within a unified framework. Numerical simulations are performed to explore the effects of coupling strength, topology, inertia, damping, and inter-layer coupling on synchronization behavior. The results show that stronger intra-layer and inter-layer coupling generally enhance synchronization. Compared with the first-order layer, the second-order layer reaches synchronization at a lower threshold but is more sensitive to parameter variations. In addition, asymmetric layer sizes and Erdős–Rényi (ER) topologies are found to promote global synchronization more easily. Larger inertia and weaker damping suppress synchronization, especially in the second-order layer. The study also reveals that multi-rate dynamics can produce asymmetric synchronization between layers. These findings help improve the understanding of synchronization mechanisms in heterogeneous multilayer networks.
Wang et al. (Fri,) studied this question.
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