Hysteresis in a Generalized Kuramoto Model with a Simplified Realistic Coupling Function and Inhomogeneous Coupling Strengths

The paper’s model definition, self-consistency equation, and linear stability boundary for the uniformly incoherent state are consistent and correctly used to map hysteresis regions across three coupling-strength distributions (truncated Gaussian, uniform, truncated power-law). In particular, the paper states and uses that the uniformly incoherent state is unstable below a critical threshold c0* and neutrally stable above it (cosβ and sinβ conditions in the μ→0+ limit) , and that fully locked states occur on the c0 = sinβ line with r = 1 . The candidate solution reverses the stability direction of incoherence (claiming stability for c0 < c0*), so its claimed coexistence interval (sinβ, c0*) is inverted relative to the paper’s analysis. This central mistake undermines the proposed hysteresis construction. Additional technical slips include: (i) the linearization at the fully locked state uses cos(θj−θi−β)=1 instead of cosβ; (ii) the symmetric-part Jacobian off-diagonal expression drops a term proportional to (Ki−Kj) sin(θj−θi) sinβ; (iii) the uniform-density sinβ equation is missing a 1/L factor. By contrast, the paper computes c0* from the correct boundary conditions (Eqs. (6–7)) and validates hysteresis regimes with stability checks (OA-based characteristic equation) and simulations across parameter sweeps in the (β,c0)-plane .