Strong Lyapunov functions for rough systems

Abstract

We extend the Lyapunov function technique, a fundamental tool for investigating asymptotic stability and existence of attractors for ordinary differential equations, by introducing the notion of a {\it strong Lyapunov function} for an autonomous drift under stochastic perturbation driven by general Hölder-continuous multiplicative noise, not necessarily Brownian. The mathematical setting within which our method proceeds consists of 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 a global random pullback attractor that 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 a numerical attractor for the discretization which is upper semi-continuous w.r.t. the noise intensity and converges to the continuous attractor as the step size tends to zero. Several applications, including dissipative systems, the pendulum, the FitzHugh-Nagumo neuro-system and the Lorenz system, demonstrate the power of our approach. We also prove that strong Lyapunov functions can be approximated in practice by Lyapunov neural networks.