Analytic self-consistency for locked states in heterogeneous Stuart–Landau ensembles

Derive explicit analytical self-consistency equations for the collective frequency Ω and the mean-field coherence ℜ of the phase-locked state in an ensemble of globally coupled heterogeneous Stuart–Landau oscillators with mean-field coupling so that a linear stability analysis of the synchronized amplitude mode becomes feasible.

Background

In the discussion of globally coupled heterogeneous Stuart–Landau oscillators, the authors analyze phase-locked solutions characterized by a common frequency Ω and a mean-field coherence ℜ. They note that, unlike phase-only models, amplitude dynamics play a critical role in enabling frequency entrainment despite heterogeneity.

However, the authors explain that a linear stability analysis of the locked state is not feasible because Ω and ℜ are only implicitly defined and analytical expressions for the self-consistency equations could not be obtained. Establishing such closed-form expressions would enable rigorous stability analysis of the synchronized amplitude mode.