Second-order Kuramoto model with adaptive simplicial complex

This study examines a second-order Kuramoto model, a widely recognized framework for understanding power grid dynamics, with a focus on adaptive simplicial interactions. Unlike previous research, our findings demonstrate that adaptive coupling eliminates an abrupt transition from an incoherent to a synchronized state in the thermodynamic limit. However, numerical simulations reveal a finite-size induced abrupt transition to coherence, with the adaptation control parameter governing the transition point. Our results highlight the impact of system size, inertia, adaptation control parameter, $2$-simplex coupling strength, and noise strength on this transition point. The results may offer valuable insights into the stability of dynamical systems characterized by adaptive interactions.