Linear stability of the incoherent solution and the transition formula for the Kuramoto-Daido model

2009/11/26 by Hayato Chiba, Chiba, Hayato
Computer Science · Engineering · Mathematics · Medicine · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #Slime Mold and Myxomycetes Research #math.DS

paper · pdf · doi:10.48550/arxiv.0911.4993

arxiv created 2009/11/26 · openalex publication_date 2009/11/26 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Kuramoto-Daido model, which describes synchronization phenomena, is a system of ordinary differential equations on N-torus defined as coupled harmonic oscillators, whose natural frequencies are drawn from some distribution function. In this paper, the continuous model for the Kuramoto-Daido model is introduced and the linear stability of its trivial solution (incoherent solution) is investigated. Kuramoto's transition point K_c, at which the incoherent solution changes the stability, is derived for an arbitrary distribution function for natural frequencies. It is proved that if the coupling strength K is smaller than K_c, the incoherent solution is asymptotically stable, while if K is larger than K_c, it is unstable.

Citations

• Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function
• CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL

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