Boundedness of semilinear Duffing equations with Liouvillean frequency

Min Li; Xiong Li

doi:10.3934/dcds.2023127

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2024, Volume 44, Issue 3: 808-834. Doi: 10.3934/dcds.2023127

This issue Previous Article The connection between the dynamical properties of 3D systems and the image of the energy-Casimir mapping Next Article Decay of correlations for some non-uniformly hyperbolic attractors

Boundedness of semilinear Duffing equations with Liouvillean frequency

Min Li^1, and
Xiong Li^2, ,

1.
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
2.
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

^*Corresponding author: Xiong Li
^*Corresponding author: Xiong Li

Received: February 1, 2023

Revised: September 1, 2023

Early access: October 30, 2023

Published online: October 30, 2023

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Abstract

We are concerned with the quasi-periodic semilinear Duffing equation $ x''+\omega^2x+g(x,t) = 0, $ where $ \omega $ is a Diophantine number, $ g(x,t) $ is bounded, real analytic in $ x $ and $ t $, and is quasi-periodic in $ t $ with the frequency $ \tilde{\omega} = (1, \alpha) $, where $ \alpha $ is Liouvillean. Without assuming the twist condition and the polynomial-like condition on this equation, we will prove the boundedness of all solutions.

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Mathematics Subject Classification: Primary: 34D20, 37J40; Secondary: 34C15.

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References

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