[Paper Review] Averaging for nonlinear systems evolving on Riemannian manifolds

Farzin Taringoo, Dragan Nešić+2 more|arXiv (Cornell University)|Nov 11, 2013

Stability and Controllability of Differential Equations28 references3 citations

TL;DR

This paper extends averaging theory for nonlinear dynamical systems to Riemannian manifolds by leveraging differential geometry, particularly the Levi-Civita connection and Riemannian distance. It establishes closeness of solutions between original and averaged systems on both finite and infinite time horizons, proving asymptotic and exponential stability results under compactness and local stability assumptions, with applications illustrated on a torus manifold.

ABSTRACT

This paper presents an averaging method for nonlinear systems defined on Riemannian manifolds. We extend closeness of solutions results for ordinary differential equations on $R^{n}$ to dynamical systems defined on Riemannian manifolds by employing differential geometry. A generalization of closeness of solutions for periodic dynamical systems on compact time intervals is derived for dynamical systems evolving on compact Riemannian manifolds. Under local asymptotic (exponential) stability of the average vector field, we further relax the compactness of the ambient Riemannian manifold and obtain the closeness of solutions on the infinite time interval by employing the notion of uniform normal neighborhoods of an equilibrium point of a vector field. These results are also presented for time-varying dynamical systems where their averaged systems are almost globally asymptotically or exponentially stable on compact manifolds. The main results of the paper are illustrated by several examples.

Motivation & Objective

• To generalize averaging methods for nonlinear systems from Euclidean spaces to Riemannian manifolds.
• To establish closeness of solutions between original time-varying systems and their averaged counterparts on compact Riemannian manifolds.
• To extend finite-time closeness results to infinite-time horizons by exploiting local asymptotic or exponential stability of the averaged system.
• To utilize geometric structures such as uniform normal neighborhoods and geodesic uniqueness to bridge Euclidean and Riemannian stability analysis.
• To validate the theoretical results through numerical examples on compact manifolds like the torus.

Proposed method

• Employment of the Levi-Civita connection to define covariant derivatives and geodesic flows on Riemannian manifolds.
• Use of Riemannian distance function as a metric to quantify solution closeness instead of Euclidean distance.
• Application of scaling techniques to bound the Riemannian metric by the Euclidean metric within precompact neighborhoods of equilibrium points.
• Utilization of uniform normal neighborhoods to ensure existence of unique minimizing geodesics and local injectivity of the exponential map.
• Adaptation of standard stability theory from [16] to Riemannian settings by leveraging Lyapunov functions with locally negative-definite Lie derivative.
• Numerical simulation of dynamical systems on the torus to demonstrate solution closeness under varying perturbation parameters (ε = 0.1 and ε = 0.03).

Experimental results

Research questions

• RQ1 Can averaging theory for nonlinear systems be extended from R^n to compact Riemannian manifolds while preserving solution closeness results?
• RQ2 How can the notion of stability be adapted to Riemannian manifolds to ensure infinite-time solution closeness in the averaging framework?
• RQ3 What geometric properties of Riemannian manifolds—such as geodesic uniqueness and normal neighborhoods—enable the extension of Euclidean averaging results?
• RQ4 Under what conditions does local asymptotic or exponential stability of the averaged system imply global or almost global stability on a compact manifold?
• RQ5 How can the Riemannian metric be locally compared to the Euclidean metric to facilitate stability and closeness analysis?

Key findings

• For compact Riemannian manifolds, if the averaged system is locally asymptotically or exponentially stable, then solutions of the original and averaged systems remain O(ε)-close on infinite time intervals.
• The closeness of solutions on infinite time horizons is established using uniform normal neighborhoods and local equivalence of Riemannian and Euclidean metrics via scaling.
• On compact manifolds, almost global asymptotic stability of the averaged system implies that all trajectories remain within an arbitrarily small δ-neighborhood of the equilibrium for all t ≥ t₀, provided ε is sufficiently small.
• Numerical results on the torus confirm that the nominal and averaged system trajectories remain O(ε)-close for ε = 0.1 and ε = 0.03 over t ∈ [0, ∞).
• The existence of unique minimizing geodesics in normal neighborhoods allows for a rigorous comparison of solution trajectories using Riemannian distance, enabling the extension of classical averaging results.
• The Lyapunov function with locally negative-definite Lie derivative ensures local stability of the averaged system, which is sufficient to guarantee long-term solution closeness under the proposed framework.