Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks

Determining the number of stable phase-locked solutions for locally coupled Kuramoto models is a long-standing mathematical problem with important implications in biology, condensed matter physics, and electrical engineering among others. We investigate Kuramoto models on networks with various topologies and show that different phase-locked solutions are related to one another by loop currents. The latter take only discrete values, as they are characterized by topological winding numbers. This result is generically valid for any network and also applies beyond the Kuramoto model, as long as the coupling between oscillators is antisymmetric in the oscillators’ coordinates. Motivated by these results, we further investigate loop currents in Kuramoto-like models. We consider loop currents in nonoriented n-node cycle networks with nearest-neighbor coupling. Amplifying on earlier works, we give an algebraic upper bound N≤2 Int[n/4]+1 for the number N of different, linearly stable phase-locked solutions. We sho...