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Bifurcations of twisted solutions in a continuum limit for the Kuramoto model on nearest neighbor graphs

We study bifurcations of twisted solutions in a continuum limit (CL) for the Kuramoto model (KM) of identical oscillators defined on nearest neighbor graphs, which may be deterministic dense, random dense or random sparse, when it may have phase-lag. We use the center manifold reduction, which is a standard technique in dynamical systems theory, and prove that the CL suffers bifurcations at which the one-parameter family of twisted solutions becomes unstable and a stable or unstable two-parameter family of modulated twisted solutions that oscillate or not depending on whether the phase-lag exists or not is born. We demonstrate the theoretical results by numerical simulations for the KM on deterministic dense, random dense and random sparse graphs.