Phase Oscillator Networks with Nonlocal Higher-Order Interactions: Twisted States, Stability and Bifurcations

Abstract

The Kuramoto model provides a prototypical framework to synchronization phenomena in interacting particle systems. Apart from full phase synchrony where all oscillators behave identically, identical Kuramoto oscillators with ring-like nonlocal coupling can exhibit more elaborate patterns such as uniformly twisted states. It was discovered by Wiley, Strogatz and Girvan in 2006 that the stability of these twisted states depends on the coupling range of each oscillator. In this paper, we analyze twisted states and their bifurcations in the infinite particle limit of ring-like nonlocal coupling. We not only consider traditional pairwise interactions as in the Kuramoto model but also demonstrate the effects of higher-order nonpairwise interactions, which arise naturally in phase reductions. We elucidate how pairwise and nonpairwise interactions affect the stability of the twisted states, compute bifurcating branches, and show that higher-order interactions can stabilize twisted states that are unstable if the coupling is only pairwise.

Vlasov Equations on Directed Hypergraph Measures

“…More recently, studies have concentrated more on the complex dynamics caused by (i) the interaction types and (ii) the heterogeneity of the underlying network/graph [2]. In particular, certain IPS incorporating indirect interactions beyond pairwise interactions between two particles have been considered [6,7]. For instance, in a deterministic chemical reaction network, the concentration of a species A depends on all reactions where A appears either in the reactant or in the product [16].…”