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

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 ${\mathrm{\Delta }}_c$: below ${\mathrm{\Delta }}_c$ the entire population locks to a collective phase velocity; above ${\mathrm{\Delta }}_c$ a desynchronous state emerges, characterized by strongly localized bursts on a slowly varying background. This minimal model thus establishes a new paradigm for complex synchronization landscapes arising from higher-order interactions.