One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes

Abstract

We study synchronization in a system of phase-only oscillators residing on the sites of a one-dimensional periodic lattice. The oscillators interact with a strength that decays as a power law of the separation along the lattice length and is normalized by a size-dependent constant. The exponent $α$ of the power law is taken in the range $0 \le α<1$. The oscillator frequency distribution is symmetric about its mean (taken to be zero), and is non-increasing on $[0,\infty)$. In the continuum limit, the local density of oscillators evolves in time following the continuity equation that expresses the conservation of the number of oscillators of each frequency under the dynamics. This equation admits as a stationary solution the unsynchronized state uniform both in phase and over the space of the lattice. We perform a linear stability analysis of this state to show that when it is unstable, different spatial Fourier modes of fluctuations have different stability thresholds beyond which they grow exponentially in time with rates that depend on the Fourier modes. However, numerical simulations show that at long times, all the non-zero Fourier modes decay in time, while only the zero Fourier mode (i.e., the "mean-field" mode) grows in time, thereby dominating the instability process and driving the system to a synchronized state. Our theoretical analysis is supported by extensive numerical simulations.