Lyapunov Stability for nonautonomous systems on Manifolds
Dynamical Systems
2026-01-27 v1
Abstract
This paper studies the uniformly asymptotic stability of nonautonomous systems on Riemannian manifolds. We establish corresponding Lyapunov-type theorems (Theorems 2.1 and 2.2), extending classical Euclidean results (e.g., [9, Theorems 4.9 and 4.10]) to curved spaces. Our main contributions are: (i) an estimate for the domain of attraction linked to the equilibrium point's injectivity radius, where, under suitable conditions, this radius can be bounded using the sectional curvature (Proposition 2.1); (ii) a demonstration that this estimate depends on the choice of the Riemannian metric (Examples 2.1 and 2.2 and Remark 2.4); and (iii) a refined estimate compared to the Euclidean case, as detailed in item (6) of Remark 2.1 and in Remark 2.3.
Cite
@article{arxiv.2601.17700,
title = {Lyapunov Stability for nonautonomous systems on Manifolds},
author = {Li Deng and Xin Li},
journal= {arXiv preprint arXiv:2601.17700},
year = {2026}
}