Extent to which the spherical approximation captures key phenomena of randomly coupled Kuramoto models

Determine whether and to what extent the spherical mean-field model of globally coupled oscillators with a global spherical constraint and quenched random interactions reproduces phenomena reported in Kuramoto models with random interactions, specifically the volcano transition, chaotic dynamics, and the algebraic decay of the ferromagnetic order parameter.

Background

The paper introduces a solvable spherical model for coupled oscillators with fully random interactions and distributed natural frequencies, deriving dynamical mean-field equations and showing that any finite frequency dispersion suppresses a finite-temperature spin-glass transition. While the model captures certain benchmarks (e.g., the ferromagnetic synchronization transition) and provides analytic insight into the suppression of glassiness at finite temperature, the authors caution that residual zero-temperature glassiness may be an artifact of the spherical dynamics.

A key unresolved issue is whether the spherical approximation can faithfully capture complex dynamical phenomena known from Kuramoto models with random interactions, such as volcano transitions, chaotic behavior, and algebraic decay of the ferromagnetic order parameter. Assessing this requires comparing the spherical model’s predictions with those phenomena identified in prior studies of fully disordered Kuramoto-type systems.