标题： 粗糙系统的强李雅普诺夫函数 显示英文标题

标题： Strong Lyapunov functions for rough systems

摘要： 我们开发了一种李雅普诺夫函数方法，以研究在具有霍尔德连续性的乘性噪声下自治常微分方程的长期行为，使用粗糙路径微积分和随机动力系统框架。通过引入{\it 强Lyapunov函数}的概念，我们得出结论，如果漂移项存在这样的函数，则扰动系统 admits 全局随机拉回吸引子，该吸引子关于噪声的二进制近似是上半连续的。研究了几种应用，例如单摆和 Fitzhugh Nagumo 神经系统。我们还证明了强李雅普诺夫函数在实践中可以通过李雅普诺夫神经网络进行近似。 显示英文摘要

摘要： We develop a Lyapunov function method 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. By introducing the concept of {\it strong Lyapunov functions}, 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 dyadic approximation of the noise. Several applications are studied, such as the pendulum and the Fitzhugh Nagumo neuro-system. We also prove that strong Lyapunov functions could be approximated in practice by Lyapunov neural networks.