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Forced quasiperiodic and almost periodic solution for nonlinear systems. (English) Zbl 0796.34027

The existence of uniformly almost periodic solutions of systems of second order nonlinear ordinary differential equations with given almost periodic forcing is studied. The main result extends a result of the same authors [Nonlinear Anal., Theory Methods Appl. 19, 249-257 (1992; Zbl 0765.34031)] for forced almost periodic oscillations of a single second order equation of the Duffing type. Some precise results for the analogous quasiperiodic vibrations are obtained. In particular, if the forcing is quasiperiodic, generally it is proved that the displacement will also be quasiperiodic with the same frequencies.
The present paper is divided naturally into 3 parts. The first part (Section 2) describes almost periodic oscillations. The second part sharpens the result of Section 2 for quasiperiodic oscillations. The final part of the article (the Appendix) discusses the point of view (relative to known results on almost periodic and quasiperiodic functions) that enables the key results on forced oscillations to be obtained, in particular, the determination of the frequencies of almost periodic motions in terms of known frequencies of the forcing.
Reviewer: I.Ginchev (Varna)

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations

Citations:

Zbl 0765.34031
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References:

[1] Berger, M. S.; Chen, Y. Y., Forced quasiperiodic and almost periodic oscillations on nonlinear duffing equations, Nonlinear Analysis, 19, 3, 249-257 (1992) · Zbl 0765.34031
[2] Bohr, H., Almost Periodic Functions (1947), Chelsea, New York
[3] Besicovitch, A. S., Almost Periodic Functions (1954), Dover: Dover New York · Zbl 0065.07102
[4] Levitan, B. M.; Zhikov, V. V., Almost Periodic Functions and Differential Equations (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0499.43005
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