Synchronization in Three Types of 2-Simplicial Ring Networks with Identical and Nonidentical Oscillators.

We explore the synchronization dynamics of Kuramoto oscillators in three 2-simplicial ring networks-node-coupled, edge-coupled, and ring-star-covering both identical and nonidentical systems. Notably, the 2-simplicial ring networks support much richer antiphase fully synchronized states than those in traditional simplicial complex models, with cluster sizes following a universal, finite-size-effect-free Gaussian distribution. Regarding in-phase fully synchronized states, we verify via the cellular dynamics method that they are linearly stable yet practically unreachable from random initial conditions due to their extremely small attraction basins. Under frequency heterogeneity, partial synchronization emerges, where oscillator frequencies cluster into two distinct groups. Among these networks, the ring-star 2-simplicial ring network exhibits the narrowest range of coupling strengths supporting partial synchronization states, while displaying the most significant disparity in cluster sizes. This research highlights the pivotal role of simplicial coupling and ring topology in regulating synchronization, offering new insights into higher-order complex networks.