Lyapunov Stability for nonautonomous systems on Manifolds

Li Deng; Xin Li

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arXiv · 2601.17700

Lyapunov Stability for nonautonomous systems on Manifolds

Li Deng·Xin Li

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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.

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Li Deng, Xin Li. 2026-01-25. Lyapunov Stability for nonautonomous systems on Manifolds. https://arxiv.org/abs/2601.17700

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