Almost Periodic Solutions in Forced Harmonic Oscillators with Infinite Frequencies.

In this paper, we consider a class of almost periodically forced harmonic oscillators x ¨ + τ 2 x = ϵ f (t , x) <graphic href="12346_2022_635_Article_Equ41.gif"></graphic> where τ ∈ 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 ω = (⋯ , ω λ , ⋯) λ ∈ Z . Using the modified Kolmogorov–Arnold–Moser (or KAM Arnold (Uspehi Mat. Nauk 18(5 (113)):13–40, 1963), Moser (Nachr. Akad. Wiss. Göttingen 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 τ 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. [ABSTRACT FROM AUTHOR]