Lyapunov Stability for nonautonomous systems on Manifolds

TL;DR

This paper extends Lyapunov stability theory to nonautonomous systems on Riemannian manifolds, providing new estimates for stability domains influenced by curvature and metric choices.

Contribution

It introduces Lyapunov-type theorems for manifold systems and derives bounds on attraction domains based on curvature and metric dependence.

Findings

Established Lyapunov theorems for manifold systems

Derived bounds on attraction domains linked to curvature

Showed metric choice affects stability estimates

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.