Quantized Frequency-locking and Extreme Transitions in a Ring of Phase Oscillators with Three-Body Interactions

Synchronization phenomena in systems with higher-order interactions have attracted widespread attention. However, existing studies have mainly focused on simplicial interactions in globally coupled networks or complex topologies, which inherently conflate pairwise and higher-order contributions and lack a minimal model to isolate the distinct dynamical effects of higher-order interactions. Here we report a spectrum of exotic frequency-locked states in a ring of phase oscillators with pure three-body interactions. For identical oscillators, the system hosts a vast multiplicity of stable quantized frequency-locked states without phase coherence. Introducing frequency heterogeneity broadens each quantized level into a continuous band and drives an extreme second-order transition at a critical value: below which the entire population locks to a collective phase velocity; above which a desynchronous state emerges, characterized by strongly localized bursts on a slowly varying background. This work introduces a ring of oscillators with pure three-body interactions to explore synchronization beyond pairwise coupling. Quantized frequency-locked states exist without phase coherence for identical oscillators, and at a critical heterogeneity an extreme second-order transition from phase locking to a desynchronous state occurs.