Almost Periodic Solutions in Forced Harmonic Oscillators with Infinite Frequencies

摘要

In this paper, we consider a class of almost periodically forced harmonic oscillators @@@ sic + tau(2)x = epsilon f (t, x) @@@ where tau is an element of A with A being a closed interval not containing zero, the forcing term f is real analytic almost periodic functions in t with the infinite frequency omega = (... , omega, ... )(lambda is an element of Z). Using the modified Kolmogorov-Arnold-Moser (or KAM Arnold (Uspehi Mat. Nauk 18(5 (113)):13-40, 1963), Moser (Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II 1962:1-20 1962), Kolmogorov (Dokl. Akad. Nauk SSSR (N.S.) 98:527-530 1954)) theory about the lower dimensional tori, we show that there exists a positive Lebesgue measure set of tau contained in A such that the harmonic oscillators has almost periodic solutions with the same frequencies as f. The result extends the earlier research results with the forcing term f being quasi-periodic.

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