Borel Sensitivity and the Analyticity Limit: A New Foundation for Stability in Discrete-Time Dynamics

The faithful representation of continuous dynamical systems via discrete-time integrators is fundamentally limited by the emergence of artifactual bifurcations, instabilities induced by the discretization step ℎ that are absent in the original flow. While Backward Error Analysis (BEA) provides a framework for studying these effects through modified vector fields, the resulting formal power series typically exhibits a zero radius of convergence, obscuring the transition from stability to chaos. In this paper, we resolve the analyticity limit of the BEA by lifting the problem into the Borel plane. We demonstrate that the discrete evolution operator can be represented as a resurgent function whose singularities in the Borel plane 𝜁 correspond to the reciprocal of the system's Lyapunov exponents. We introduce a novel Borel Sensitivity Operator ∇μℒ̂(ζ), derived via the Duhamel formula, to quantify how variations in system parameters 𝜇 shift these singularities toward the origin. By analyzing the spectral flow of the operator ℒ = (μ𝑥 − 𝑥3)∂𝑥, we provide a precise criterion for the critical step size ℎ𝑐, identifying it as the point where the integration contour in the Laplace-Borel transform encounters a spectral singularity. Our results establish a theoretical foundation for "Spectral Step-Control" in AI-driven simulations, ensuring topological consistency between continuous models and their discrete implementations.