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Revista Matemática Complutense

, Volume 29, Issue 1, pp 169–189 | Cite as

A geometric proof of the periodic averaging theorem on Riemannian manifolds

  • Misael Avendaño-CamachoEmail author
  • Guillermo Dávila-Rascón
Article

Abstract

We present a geometric proof of the averaging theorem for perturbed dynamical systems on a Riemannian manifold, in the case where the flow of the unperturbed vector field is periodic and the \({\mathbb {S}}^{1}\)-action associated to this vector field is not necessarily trivial. We generalize the averaging procedure [2, 3] defining a global averaging method based on a free coordinate approach which allow us to formulate our results on any open domain with compact closure.

Keywords

Averaging method Perturbation theory Periodic flows  Riemannian manifolds Horizontal lifts \({\mathbb {S}}^1\)-principal bundle 

Mathematics Subject Classification

37C10 37C55 37J40 53C20 55R10 58D17 

Notes

Acknowledgments

The authors are grateful to professor Yu. M. Vorobev for fruitful discussions on the preparation of this paper. They also would like to thank the referees for their useful suggestions that helped us to improve this work. This research was partially supported by the National Council of Science and Technology (CONACyT) under the Grant 219631, CB-2013-01.

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Copyright information

© Universidad Complutense de Madrid 2015

Authors and Affiliations

  1. 1.CONACYT Research Fellow-Departamento de MatemáticasUniversidad de SonoraHermosilloMexico
  2. 2.Departamento de MatemáticasUniversidad de SonoraHermosilloMexico

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