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Almost Periodic Solutions in Forced Harmonic Oscillators with Infinite Frequencies

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Abstract

In this paper, we consider a class of almost periodically forced harmonic oscillators

$$\begin{aligned} \ddot{x}+\tau ^2 x=\epsilon f(t,x) \end{aligned}$$

where \(\tau \in {{\mathcal {A}}}\) with \({{\mathcal {A}}}\) being a closed interval not containing zero, the forcing term f is real analytic almost periodic functions in t with the infinite frequency \(\omega =(\cdots ,\omega _\lambda ,\cdots )_{\lambda \in {{\mathbb {Z}}}}\). 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 \(\tau \) contained in \({{\mathcal {A}}}\) 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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Notes

  1. See the definition 2.1 in Sect. 2.

  2. We call an equation \({\dot{x}}=f(x)\) is reversible, if its vector field satisfies that \(f(G(x))=-DG\cdot f(x)\), where \(G\circ G=id.\)

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen), Shenzhen, 518172, People’s Republic of China

    Shengqing Hu

  2. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, People’s Republic of China

    Jing Zhang

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Statements and declarations: The data used to support the findings of this study are available from the corresponding author upon request. This work was partially supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515111068), Shenzhen Science and Technology Innovation Program (Grant No. RCBS20210609103231040), National Natural Science Foundation of China (Grant No. 11871023) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

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Hu, S., Zhang, J. Almost Periodic Solutions in Forced Harmonic Oscillators with Infinite Frequencies. Qual. Theory Dyn. Syst. 21, 105 (2022). https://doi.org/10.1007/s12346-022-00635-5

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