Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency

Hongyu Cheng; Shimin Wang

doi:10.3934/cpaa.2020222

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2021, Volume 20, Issue 2: 467-494. Doi: 10.3934/cpaa.2020222

This issue Previous Article Next Article On the Cahn-Hilliard equation with mass source for biological applications

Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency

Hongyu Cheng^1, and
Shimin Wang^2, ,

1.
School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
2.
School of Mathematics, Shandong University, Jinan 250100, China

^* Corresponding author
^* Corresponding author

Received: December 31, 2019

Revised: April 30, 2020

Early access: December 2020

Published: February 2021

The first author was supported by NSFC (No. 12001294). The second author was partially supported by CSC (No. 201706220147) and NSFC (No. 12001397)

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Abstract

This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators

$ \begin{equation*} \ddot{x}+\lambda^{2}x = \epsilon f(\omega t,x), \end{equation*} $

where $ \lambda \in \mathcal{O} $, a closed interval not containing zero, the forcing term $ f $ is real analytic, and the frequency vector $ \omega \in \mathbb{R}^d \, (d \geq 2) $ is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency $ \omega\in\mathbb{R}^{d}, $ which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi–periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite–dimensional harmonic oscillator systems with Liouvillean frequency.

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Mathematics Subject Classification: Primary: 70H08, 37J40; Secondary: 34C15.

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