Existence and Stability of Dancing Equilibria in Asymmetric Kuramoto Networks

Abstract

We study nonzero-frequency phase-locked motions in asymmetrically coupled Kuramoto networks. Such motions are relative equilibria with fixed phase differences and a nonzero common angular velocity, and we call them dancing equilibria. Their existence requires all coupling sums to have the same nonzero value. We show that neither symmetric coupling nor an acyclic associated digraph can support a dancing equilibrium. We introduce structurally equitable and $q$-twisted state equitable partitions and prove a partition-based criterion for the resulting class-constant profiles, with standard labeled $q$-twisted profiles recovered from singleton partitions. For the forward $m$-neighbor model, we characterize existence by an exact indivisibility criterion. Stability is studied modulo the common phase-shift direction. For general directed networks, strong connectivity and edgewise phase differences in $\left(-π/2,π/2\right)$ imply local orbital exponential stability and yield an explicit positively invariant set contained in the local basin of attraction. For arbitrary twisted indices, this contraction argument gives a low-winding stability regime with explicit positively invariant neighborhoods. For each existing $q$-twisted branch of the forward model, a discrete Fourier transform criterion yields local orbital exponential stability when all nonzero Fourier-mode factors are positive and nonlinear instability when at least one is negative. In the unstable case, the proof constructs explicit escaping real Fourier perturbations. We further derive additional explicit stability and instability ranges for arbitrary twisted indices in terms of constants $N$, $m$, and $q$. For the first two twisted branches, sharper arguments yield a first-mode transition criterion for $q=1$ and a complete finite-size classification for $q=2$, with the degenerate case in each branch handled separately.