Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia

We discuss the uniqueness and well-ordering property of phase-locked states emerged from some admissible class of initial configurations for the Kuramoto model under the effect of frustration and inertia. Our results rely on the nonlinear stability and structure of phase-locked states for the Kuramoto model. When the coupling strength is sufficiently large and the diameter of initial phase configuration is sufficiently small, we show that the emergent phase configurations are stable in [Formula: see text]-norm with respect to initial configurations and they tend to the unique collision-free phase-locked state up to rotation. Moreover, we verify that the geometric shape of the emergent phase-locked state is invariant under the effect of inertia. We provide several numerical examples and compare them with our analytical results.