CENTER MANIFOLDS FOR RANDOM DYNAMICAL SYSTEMS WITH GENERALIZED TRICHOTOMIES

The paper’s theorem (existence and uniqueness of a Lipschitz invariant center manifold under a generalized α-trichotomy with σ+τ<1/2, plus the propagation bound) is proved via a carefully designed graph-transform on a product space and uses the precise definitions of σ and τ in (18)–(19); the contraction is established without requiring extra multiplicative properties of the rates. The model’s Lyapunov–Perron approach, however, assumes a contraction bound for a trajectory-space operator that, under the paper’s general α-trichotomy, is not justified: the stable/unstable tail terms involve time-dependent weights that do not reduce to τ as claimed. In addition, the model’s ‘sharp’ choice N = τ/(1−(σ+τ)) ignores the quadratic dependence that arises in the paper’s parameterization and is not generally correct.