Exact solutions of the Kuramoto model with asymmetric higher order interactions of arbitrary order

• Published in: Chaos, solitons and fractals Vol. 195; p. 116243
• Main Authors: Costa, Guilherme S., Novaes, Marcel, de Aguiar, Marcus A.M.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.06.2025
• Subjects: Coupled oscillators; Higher-order interactions; Synchronization; Synchronization; Higher-order interactions; Coupled oscillators
• ISSN: 0960-0779
• DOI: 10.1016/j.chaos.2025.116243

Higher order interactions can lead to new equilibrium states and bifurcations in systems of coupled oscillators described by the Kuramoto model. However, even in the simplest case of 3-body interactions there are more than one possible functional forms, depending on how exactly the bodies are coupled. Which of these forms is better suited to describe the dynamics of the oscillators depends on the specific system under consideration. Here we show that, for a particular class of interactions, reduced equations for the Kuramoto order parameter can be derived for arbitrarily many bodies. Moreover, the contribution of a given term to the reduced equation does not depend on its order, but on a certain effective order, that we define. We give explicit examples where bi and tri-stability is found and discuss a few exotic cases where synchronization happens via a third order phase transition. •Reduced equations are derived for a class of asymmetric higher-order interactions.•The contribution of higher-order terms does not depend directly on their order.•We derive bifurcation diagrams for some particular cases, finding multi-stability.