Random attractor and SRB measure for stochastic Hopf bifurcation under discretization

The paper establishes (i) unique solvability of the backward Euler step and a discrete cocycle via a conjugacy transform, (ii) a random absorbing set hence a random attractor, and (iii) positivity of the numerical top Lyapunov exponent for large shear b by comparing a tractable observable Q̂ along continuous vs. discrete dynamics; this yields an SRB sample measure for the discrete RDS. The model’s solution proves the same end results but transfers positivity of Lyapunov exponents using general approximation theory for discretized variational flows (Grorud–Talay-type arguments) rather than the paper’s Q̂-based comparison. The model’s route requires additional regularity/integrability hypotheses it does not fully state, but under small step sizes and on the random attractor the conclusions remain consistent with the paper’s theorems.