📝 SummarXiv

Abstract

We introduce the concept of {\it strong Lyapunov functions} to investigate the long term behavior of autonomous ordinary differential equations under a multiplicative noise of H\"older continuity, using rough path calculus and the framework of random dynamical systems. We conclude that if such a function exists for the drift then the perturbed system admits the global random pullback attractor which is upper semi-continuous w.r.t. the noise intensity coefficient and the dyadic approximation of the noise. Moreover, in case the drift is globally Lipschitz continuous, then there exists also a numerical attractor for the discritization which is also upper semi-continuous w.r.t. the noise intensity and also converges to the continuous attractor as the step size tends to zero. Several applications are studied, including the pendulum and the Fitzhugh Nagumo neuro-system. We also prove that strong Lyapunov functions could be approximated in practice by Lyapunov neural networks.