Continuum Limits of Coupled Oscillator Networks Depending on Multiple Sparse Graphs

The continuum limit provides a useful tool for analyzing coupled oscillator networks. Recently, Medvedev (Commun Math Sci 17(4):883–898, 2019) gave mathematical foundation such an approach when the networks are defined on single graphs which may be dense or sparse, directed undirected, and deterministic random. In this paper, we consider depending multiple graphs, extend his results to show that is also valid in situation. Specifically, prove initial value problem (IVP) of corresponding has unique solution under general conditions becomes those IVP some adequate meaning. Moreover, if solutions stable asymptotically node number sufficiently large, then so limit, stable, weak meaning as tends infinity. These can applied with frequencies by regarding weight matrix another graph. We illustrate theory three variants Kuramoto model along numerical simulations.