Second-order Kuramoto model with feedback modulated simplicial interactions

Abstract

We investigate the emergence of synchronization in the second-order Kuramoto model with feedback modulated 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 feedback-modulated higher-order coupling, in which the feedback is governed by a measure of global phase synchronization, alters synchronization transitions. Using self-consistency analysis, we derive the steady-state behavior and show that the feedback mechanism qualitatively reshapes the synchronization landscape. We find that the backward transition from synchronization to incoherence is governed by the feedback strength, 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 feedback strength. These results show how the feedback control parameter and system size together govern the onset and robustness of synchronization in a coupled-oscillator system with higher-order interactions. © 2026 Elsevier Ltd.