Second-order Kuramoto model with adaptive simplicial complex

TL;DR

This paper studies how inertia and adaptive higher-order interactions influence synchronization in the second-order Kuramoto model on networks, revealing size-dependent transition behaviors and the role of adaptive feedback.

Contribution

It introduces an analysis of the second-order Kuramoto model with adaptive simplicial interactions, highlighting the impact of inertia and adaptation on synchronization transitions.

Findings

Backward transition remains controlled by adaptive feedback.

Forward transition becomes continuous in the thermodynamic limit.

Finite systems still exhibit abrupt synchronization transitions.

Abstract

We investigate the emergence of synchronization in the second-order Kuramoto model with adaptive simplicial interactions on a globally connected network. This inertial Kuramoto framework describes systems, where oscillator frequencies evolve over time. Unlike most previous work that ignores inertia, we examine how inertia combined with adaptive higher-order coupling alters synchronization transitions. Using self-consistency analysis, we derive the steady-state behavior and show that adaptation qualitatively reshapes the synchronization landscape. We find that the backward transition from synchronization to incoherence remains controlled by the adaptive feedback parameter, but the forward discontinuous jump to synchronization vanishes in the thermodynamic limit. In contrast, finite-size systems still display an abrupt transition to synchronization, with its onset precisely set by the adaptation control parameter. These results show how adaptive feedback and system size together govern the onset and robustness of synchronization in inertial oscillator networks with higher-order interactions.