Tiered synchronization in adaptive Kuramoto oscillators on simplicial complexes

TL;DR

This paper investigates how adaptation in higher-order interactions influences synchronization in Kuramoto oscillators on simplicial complexes, revealing new phenomena like tiered synchronization and second-order transitions.

Contribution

It introduces the concept that adaptation in higher-order coupling can lead to novel synchronization behaviors, supported by rigorous theoretical analysis and numerical simulations.

Findings

Adaptation causes tiered synchronization.

Second-order transition to synchronization observed.

Theoretical results match numerical simulations.

Abstract

An incorporation of higher-order interactions is known to lead an abrupt first-order transition to synchronization in otherwise smooth second-order one for pair-wise coupled systems. Here, we show that adaptation in higher-order coupling strength may yield completely different phenomena, notably, second-order transition to synchronization and tiered synchronization. Using the Ott-Antonsen approach, we perform rigorous theoretical calculations illustrating the origin of these emerging phenomena along with a complete description of all (un)stable states. Numerical simulations for limited-size networks are in agreement with the analytical predictions. These results would be important to comprehend dynamical behaviors of real-world complex systems with inherent higher-order interactions and adaptation through feedback coupling.