Almost Periodic Solutions in Forced Harmonic Oscillators with Infinite Frequencies

• Shengqing Hu ^[1] ; Jing Zhang ^[2]
  1. [1] Chinese University of Hong Kong

     Chinese University of Hong Kong

     RAE de Hong Kong (China)

     
  2. [2] China Normal University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • In this paper, we consider a class of almost periodically forced harmonic oscillators x¨ + τ 2x = f (t, x) 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 = (··· , ωλ, ···)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.

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