The Framework of Mechanics for Dynamic Behaviors of Fractional-Order Stochastic Dynamic Systems

The paper derives first-moment synchronization conditions for a star-coupled fractional system with dichotomous multiplicative noise by closing the mean–noise correlator via a Shapiro–Loginov-type device and analyzing the resulting 2×2 fractional linear system. It obtains σ^2 < [ω+ε(N+1)]^2 + λ^α[ω+ε(N+1)] for the hub and σ^2 < (ω+ε)^2 + λ^α(ω+ε) for any leaf, and notes the latter implies the former (Eqs. (39)–(47) in the PDF) . The candidate solution independently diagonalizes the star’s deviation dynamics (hub and N−1 transversal modes), closes the same 2×2 first-moment system using a fractional Shapiro–Loginov identity, and applies Matignon’s criterion to recover exactly these inequalities and the implication. Minor sign conventions differ in the intermediate 2×2 system, but trace and determinant—and thus the stability domain—coincide. The paper’s closure for ⟨ξ D^αx⟩ uses the exponential-weighted Caputo operator (Eq. (19) and the Laplace-domain shift s→s+λ) , leading to the same λ^α dependence that the model writes explicitly. The approaches are therefore consistent and correct, albeit different in presentation.