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Response solutions for quasi-periodically forced harmonic oscillators


Authors: Jing Wang, Jiangong You and Qi Zhou
Journal: Trans. Amer. Math. Soc. 369 (2017), 4251-4274
MSC (2010): Primary 34C15, 35B15, 37J40
DOI: https://doi.org/10.1090/tran/6800
Published electronically: December 22, 2016
MathSciNet review: 3624408
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Abstract: We prove that the non-linear quasi-periodically forced harmonic oscillator with two frequencies $ (1,\alpha )$ has at least one response solution if the forcing is small. No arithmetic condition on the frequency is assumed and the smallness of the non-linear forcing does not depend on $ \alpha $. The result strengthens the existing results in the literature where the frequency is assumed to be Diophantine. The proof is based on a modified KAM theory for the lower dimensional tori.


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Additional Information

Jing Wang
Affiliation: Department of Mathematics, TU Dresden, 01062 Dresden, Germany
Address at time of publication: Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China
Email: jingwang018@gmail.com; jing.wang@njust.edu.cn

Jiangong You
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Address at time of publication: Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China
Email: jyou@nju.edu.cn

Qi Zhou
Affiliation: CNRS UMR 7586, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Batiment Sophie Germain, Case 7021, 75205 Paris Cedex 13, France
Address at time of publication: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: qizhou628@gmail.com; qizhou@nju.edu.cn

DOI: https://doi.org/10.1090/tran/6800
Received by editor(s): September 6, 2013
Received by editor(s) in revised form: February 28, 2015, and June 25, 2015
Published electronically: December 22, 2016
Additional Notes: The first author was supported by the German Research Council (DFG Emmy Noether Grant DE 1721/1-1), a Fellowship of the Humboldt Foundation, NNSF of China (11601230) and Natural Science Foundation of Jiangsu Province, China (BK20160816).
The second author was supported by NNSF of China (11471155) and 973 projects of China (2014CB340701)
The third author was partially supported by ERC Starting Grant “Quasiperiodic”, “Deng Feng Scholar Program B” of Nanjing University, Specially-appointed professor programme of Jiangsu Province and NNSF of China (11671192).
Article copyright: © Copyright 2016 American Mathematical Society