Small-Twist Stability of Symplectic Integrators: A Nekhoroshev Approach

When a symplectic integrator is applied to an integrable Hamiltonian system, the step transition map can be viewed as a nearly integrable symplectic map with a step-size-dependent perturbation. In this paper, we establish a Nekhoroshev-type theorem for such maps, providing explicit exponential stability estimates that apply directly to small-twist cases. We explicitly emphasize that our analysis treats a small but strictly positive twist (m>0) with quantified m-dependence, rather than a uniform degenerate-twist result; accordingly, the admissible perturbation threshold explicitly deteriorates as the twist constant m approaches zero. Consequently, we prove that symplectic integrators exhibit exponential stability over exponentially long time scales. Our results, derived via resonant normal form construction and the geometric covering of the action space, furnish rigorous and computable bounds for the long-term preservation of action variables, thereby offering new insights into the nonlinear dynamical behavior of geometric discretizations.