A formal series approach to averaging: Exponentially small error estimates

Philippe Chartier; Ander Murua; Jesús María Sanz-Serna

doi:10.3934/dcds.2012.32.3009

Article Contents

2012, Volume 32, Issue 9: 3009-3027. Doi: 10.3934/dcds.2012.32.3009

This issue Previous Article Next Article The efficient approximation of coherent pairs in non-autonomous dynamical systems

A formal series approach to averaging: Exponentially small error estimates

1.
INRIA Rennes and ENS Cachan Bretagne, Campus Ker-Lann, av. Robert Schumann, F-35170 Bruz
2.
Konputazio Zientziak eta A. A. Saila, Informatika Fakultatea, UPV/EHU, E-20018 Donostia-San Sebastián
3.
Departamento de Matemática Aplicada e IMUVA, Facultad de Ciencias, Universidad de Valladolid, Valladolid

Received: November 30, 2011

Revised: February 29, 2012

Published: August 2012

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Abstract

The techniques, based on formal series and combinatorics, used nowadays to analyze numerical integrators may be applied to perform high-order averaging in oscillatory periodic or quasi-periodic dynamical systems. When this approach is employed, the averaged system may be written in terms of (i) scalar coefficients that are universal, i.e. independent of the system under consideration and (ii) basis functions that may be written in an explicit, systematic way in terms of the derivatives of the Fourier coefficients of the vector field being averaged. The coefficients may be recursively computed in a simple fashion. We show that this approach may be used to obtain exponentially small error estimates, as those first derived by Neishtadt. All the constants that feature in the estimates have a simple explicit expression.

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Mathematics Subject Classification: Primary: 34C29, 70K65; Secondary: 65L06.

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