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Reducibility for real analytic quasi-periodic linear systems

Published online by Cambridge University Press:  09 April 2009

G. C. O'Brien
G. C. O'Brien
Affiliation:
Economics Department, La Trobe University, Bundoora Vic., 3083, Australia.
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Abstract

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In this paper we examine the linear differential equations x′ = Ax + P(φ)x, φ′ = ω where x ∈ Rn, φ ∈ Rm, A and ω are constant and P(φ) is real analytic and periodic in φ. We use the method of accelerated convergence to overcome the small divisors problem and reduce this system to the system y′ = By, φ′ = ω with constant coefficients.

This problem has already been examined by Mitropolśki and Samolenko but the calculations and details in their work are formidable and difficult to follow. Besides being simpler our method provides more precise estimates at all stages and can be extended to the differentiable case to provide a significant improvement over previous results.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 3 , May 1976 , pp. 289 - 298
Copyright
Copyright © Australian Mathematical Society 1976

References

Bogolyubov, N. N., Mitropolśki, Ju. A. and Samolenko, A. M. (1969), The method of accelerated convergence in non-linear mechanics (Russian) (Izd. “Naukova Dumka”, Kiev, 1969).Google Scholar
Gray, Alistair (to appear), A reducibility theorem for holomorphic linear systems via an implicit function theorem.Google Scholar
Mitropolśki, Ju. A. and Samolenko, A. M. (1965), ‘On constructing solutions of linear differential equations with quasiperiodic coefficients by the method of improved convergence’ (Russian), Ukrain, Mat. Z. 17, 42–59.Google Scholar