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Averaging and integral manifolds

Published online by Cambridge University Press:  17 April 2009

W. A. Coppel  and
K. J. Palmer
W. A. Coppel
Affiliation:
Australian National University, Canberra, ACT.
K. J. Palmer
Affiliation:
Australian National University, Canberra, ACT.
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Abstract

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An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 2 , April 1970 , pp. 197 - 222
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Bogolyubov, N.N. and Mitropolskii, Yu.A., Asymptotic methods in the theory of nonlinear oscillations (Russian), 3rd ed., (Gos. Izdat. Fiz.–Mat. Lit., Moscow, 1963). English transl. 2nd ed., (Gordon & Breach, New York, 1962).Google Scholar
[2]Coddington, Earl A. and Levinson, Norman, Theory of ordinary differential equations (McGraw-Hill, New York, Toronto, London, 1955).Google Scholar
[3]Coppel, W.A., “Dichotomies and reducibility (II)”, J. Differential Equations 4 (1968), 386–398.CrossRefGoogle Scholar