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This paper presents an averaging method for nonlinear systems defined onRiemannian manifolds. We extend closeness of solutions results for ordinarydifferential equations on $R^{n}$ to dynamical systems defined on Riemannianmanifolds by employing differential geometry. A generalization of closeness ofsolutions for periodic dynamical systems on compact time intervals is derivedfor dynamical systems evolving on compact Riemannian manifolds. Under localasymptotic (exponential) stability of the average vector field, we furtherrelax the compactness of the ambient Riemannian manifold and obtain thecloseness of solutions on the infinite time interval by employing the notion ofuniform normal neighborhoods of an equilibrium point of a vector field. Theseresults are also presented for time-varying dynamical systems where theiraveraged systems are almost globally asymptotically or exponentially stable oncompact manifolds. The main results of the paper are illustrated by severalexamples.

Averaging for nonlinear systems evolving on Riemannian manifolds