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Determining the number of stable phase-locked solutions for locally coupledKuramoto models is a long-standing mathematical problem with importantimplications in biology, condensed matter physics and electrical engineeringamong others. We investigate Kuramoto models on networks with varioustopologies and show that different phase-locked solutions are related to oneanother by loop currents. The latter take only discrete values, as they arecharacterized by topological winding numbers. This result is generically validfor any network, and also applies beyond the Kuramoto model, as long as thecoupling between oscillators is antisymmetric in the oscillators' coordinates.Motivated by these results we further investigate loop currents inKuramoto-like models. We consider loop currents in nonoriented $n$-node cyclenetworks with nearest-neighbor coupling. Amplifying on earlier works, we givean algebraic upper bound $\mathcal{N} \le 2 \, {\rm Int}[n/4]+1$ for the number$\cal N$ of different, linearly stable phase-locked solutions. We show that thenumber of different stable solutions monotonically decreases as the couplingstrength is decreased. Furthermore stable solutions with a single angledifference exceeding $\pi/2$ emerge as the coupling constant $K$ is reduced, assmooth continuations of solutions with all angle differences smaller than$\pi/2$ at higher $K$. In a cycle network with nearest-neighbor coupling wefurther show that phase-locked solutions with two or more angle differenceslarger than $\pi/2$ are all linearly unstable. We point out similaritiesbetween loop currents and vortices in superfluids and superconductors as wellas persistent currents in superconducting rings and two-dimensional Josephsonjunction arrays.

Multistability of Phase-Locking and Topological Winding Numbers in Locally Coupled Kuramoto Models on Single-Loop Networks