Multistability of Phase-Locking and Topological Winding Numbers in Locally Coupled Kuramoto Models

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. Investigating Kuramoto models on networks with various topologies, it can be shown 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. Similarities between these loop currents and vortices in superfluids and superconductors as well as persistent currents in superconducting rings and two-dimensional Josephson junction arrays can be pointed out. In this talk, motivated by these results, we will further investigate loop currents in Kuramoto-like models.

To begin with, we will consider loop currents in nonoriented n-node cycle networks with nearest-neighbor coupling. Amplifying on earlier works, we will give an algebraic upper bound for the number of different, linearly stable phase-locked solutions. Stable solutions with a single angle difference exceeding π/2 emerge as the coupling constant K is reduced, as smooth continuations of solutions with all angle differences smaller than π/2 at higher K. Furthermore, we will show that in a cycle network with nearest-neighbor coupling, phase-locked solutions with two or more angle differences larger than π/2 are all linearly unstable. We will then investigate how the results for single-loop networks may be extended to multiple-loops networks and emphasize the issues arising from such a generalization.