Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-12T16:54:05.397Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiphase averaging for generalized flows on manifolds

Published online by Cambridge University Press:  19 September 2008

H. S. Dumas ,
F. Golse  and
P. Lochak
H. S. Dumas
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati OH 45221–0025, USA
F. Golse
Affiliation:
UFR de Mathématiques, Université Paris VII, Tour 45-55, 5e étage, 4 place Jussieu, 75251 Paris Cedex 05, France
P. Lochak
Affiliation:
DMI, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a new proof of a strengthened version of Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in Rm and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPema-Lions theory of generalized flows for ODEs.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 1 , March 1994 , pp. 53 - 67
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adams, R.A.. Sobolev Spaces. Academic Press, New York, 1975.Google Scholar
[2]Anosov, D.V.. Averaging in systems of ODEs with rapidly oscillating solutions. Izv. Akad. Nauk. SSSR 24 (1960), 721–742 (untranslated).Google Scholar
[3]Amol'd, V.I. (ed). Dynamical Systems III Encyclopedia of Mathematical Sciences Vol. III, Springer, Berlin, 1988.Google Scholar
[4]Bakhtin, V.I.. Averaging in multifrequency systems. Funktional. Anal, i Prilozen. 20, No. 2 (1986), 1–7,Google Scholar
English translation: Functional Anal. Appl. 20, No. 2 (1986), 83–88.Google Scholar
[5]Brezis, H.. Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983.Google Scholar
[6]DiPerna, R. & Lions, P.-L.. Ordinary differential equations, transport theory and Sobolev spaces. Inv. Math. 98 (1989), 511–548.CrossRefGoogle Scholar
[7]Dodson, M.M., Rynne, B.P. & Vickers, J.A.G.. Averaging in multifrequency systems. Nonlinearity 2 (1989), 137–148.CrossRefGoogle Scholar
[8]Dumas, H.S.. A new proof of Anosov's averaging theorem, to appear in Hamiltonian Dynamical Systems: History, Theory, and Applications (Dumas, H.S., Meyer, K.R. and Schmidt, D. S., eds), Springer, 1994.Google Scholar
[9]Kasuga, T.. On the adiabatic theorem for the Hamiltonian system of differential equations in classical mechanics (parts I, II, and III). Proc. Acad. Japan 37 (1961), 366–382.Google Scholar
[10]Lochak, P. & Meunier, C.. Multiphase Averaging for Classical Systems, Applied Mathematical Sciences Vol. 72, Springer, New York, 1988.Google Scholar
[11]Neishtadt, A.I.. Averaging in multi-frequency systems, Dokl. Akad. Nauk. SSSR Mech. 223 (2) (1975), 314–317,Google Scholar
English translation: Sov. Phys. Dokl. 20 (7) (1975), 492–494;Google Scholar
Averaging in multi-frequency systems II, Dokl. Akad. Nauk. SSSR Mech. 226 (6) (1976), 1295–1298,Google Scholar
English translation: Sov. Phys. Dokl. 21(2) (1976), 80–82.Google Scholar