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

Abstract

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 show that the number of different stable solutions monotonically decreases as the coupling strength is decreased. Furthermore 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. In a cycle network with nearest-neighbor coupling we further show that phase-locked solutions with two or more angle differences larger than π/2 are all linearly unstable. We point out similarities between loop currents and vortices in superfluids and superconductors as well as persistent currents in superconducting rings and two-dimensional Josephson junction arrays.

Multistability of phase-locking in equal-frequency Kuramoto models on planar graphs

“…Refs. 12 and 28 on singlecycle networks were complemented by our recent work, 16 which showed that the bound N ≤ 2 · Int[n/4] + 1 is still valid for nonoriented coupling, even when one angle difference exceeds π/2, and that no stable fixed point exists with more than one angle difference exceeding π/2 for single-cycle networks. These results are summarized in Table I.…”

“…In this manuscript, we amplify on our earlier work 16 17 can only take discrete values, because angles are defined modulo 2π. As a matter of fact, single-valuedness of angle coordinates requires that summing over angle differences around any cycle in the network on which Eqs.…”

“…This latter observation eventually led to the characterization of circulating loop flows with discrete topological winding numbers. 6,16,26,27 Using winding numbers, Rogge and Aeyels 12 obtained an algebraic upper bound N ≤ 2 · Int[n/4] + 1 for the number of stable fixed point solutions with any angle difference in a Kuramoto model on a n-node ring with unidirectional nearest-neighbor couplings. The same upper bound has been calculated by Ochab and Góra 28 in a nonoriented n-node Kuramoto ring with nearest-neighbor coupling, under the condition that all angle differences are smaller than π/2.…”

“…Ref. 16 put this classification into perspective as it showed that, based on a theorem due to Taylor 31 (see also Ref. 32), only ℓ = 0, 1 is possible on a single-cycle network.…”

“…Third, in meshed networks, one has to revisit the one-toone relation between number of stable fixed points and number of possible winding numbers on which the strategy of Ref. 16 is based. Below, we present a method that overcomes these obstacles, which allows us to generalize the bound found in Refs.…”