2018/05/31 by Per Sebastian Skardal
Computer Science · Engineering · Environmental Science · Mathematics · Physics and Astronomy · #Cauchy distribution #Complex system #Computer science #Consistency (knowledge bases) #Curse of dimensionality #Ecosystem dynamics and resilience #Kuramoto model #Mathematical analysis #Mathematics #Nonlinear Dynamics and Pattern Formation #Physics #Scaling #Slime Mold and Myxomycetes Research #Stability (learning theory) #Statistical physics #Synchronization (alternating current) #Synchronization networks #Topology (electrical circuits) #nlin.AO
paper · pdf · doi:10.1103/physreve.98.022207
published as Phys. Rev. E 98, 022207 (2018)
arxiv created 2018/07/24 · openalex publication_date 2018/08/09 · arxiv updated 2018/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Kuramoto model is a paradigmatic tool for studying the dynamics of collective behavior in large ensembles of coupled dynamical systems. Over the past decade a great deal of progress has been made in analytical descriptions of the macroscopic dynamics of the Kuramoto model, facilitated by the discovery of Ott and Antonsen's dimensionality reduction method. However, the vast majority of these works relies on a critical assumption where the oscillators' natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which allows for a convenient closure of the evolution equations from the dimensionality reduction. In this paper we investigate the low-dimensional dynamics that emerge from a broader family of natural frequency distributions, in particular, a family of rational distribution functions. We show that, as the polynomials that characterize the frequency distribution increase in order, the low-dimensional evolution equations become more complicated, but nonetheless the system dynamics remain simple, displaying a transition from incoherence to partial synchronization at a critical coupling strength. Using the low-dimensional equations we analytically calculate the critical coupling strength corresponding to the onset of synchronization and investigate the scaling properties of the order parameter near the onset of synchronization. These results agree with calculations from Kuramoto's original self-consistency framework, but we emphasize that the low-dimensional equations approach used here allows for a true stability analysis categorizing the bifurcations.