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

Abstract

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 $Δ_c$: below $Δ_c$ the entire population locks to a collective phase velocity; above $Δ_c$ 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.

Figures (4)

• Figure 1: Illustration of quantized frequency-locked states for increasing frequency heterogeneity: (a) $\Delta=0$, (b) $\Delta=0.4$, and (c) $\Delta=1$. Top panels: phase evolutions of four representative oscillators (color-coded); all rotate at the same constant collective velocity. Middle panels: space-time plot of the oscillator phases; the regular periodic pattern degrades as $\Delta$ grows. Bottom panels: the spatial distribution of local phase gradient $\Delta\theta$, which accumulates $8\pi$ (a), $4\pi$ (b), and $-2\pi$ (c) when traversing the ring from site 1 to 256. Other parameters: $K=1$ and $N=256$.
• Figure 2: Empirical probability density $P(\Omega)$ of the collective phase velocity for (a) $\Delta=0$, (b) $\Delta=0.2$, (c) $\Delta=0.4$, and (d) $\Delta=1$. As $\Delta$ increases, $P(\Omega)$ changes from sharp quantized lines (a) to broadened bands (b,c) and finally to a single continuous peak (d); the red curve in (d) is a Gaussian fit.
• Figure 3: Desynchronous states for (a) $\Delta=2.04$ and (b) $\Delta=3$. Top panels: instantaneous phase velocities $\dot{\theta}(t)$ of two active oscillators ($|\omega_i|>1$, black and red) and two passive ones ($|\omega_i|<1$, blue and magenta). Passive units remain close to the quantized frequency-locked background, whereas active units exhibit bursting dynamics. Bottom panels: spatial profile of the velocity fluctuation $F_i$. Bursting is strongly localized just above $\Delta_c$ and becomes more dispersed as $\Delta$ increases.
• Figure 4: Synchronization transitions. (a) Uniform frequency distribution where a continuous transition occurs at $\Delta_c=2$. The space averaged velocity fluctuation $\langle F\rangle$ (black) drops linearly toward zero, while the inverse participation ratio $\mathrm{IPR}$ (red) develops a sharp peak exactly at $\Delta_c$. (b) Lorentzian frequency distribution where no true transition occurs. $\langle F\rangle$ smoothly approaches zero only as $\Delta\rightarrow0$; however, the $\mathrm{IPR}$ reaches its maximum at $\Delta\simeq0.06$, beyond which $\langle F\rangle$ increases linearly with $\Delta$.