Response solutions for quasi-periodically forced harmonic oscillators
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- by Jing Wang, Jiangong You and Qi Zhou PDF
- Trans. Amer. Math. Soc. 369 (2017), 4251-4274 Request permission
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.References
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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
- MR Author ID: 970244
- 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
- MR Author ID: 241618
- 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
- MR Author ID: 970275
- Email: qizhou628@gmail.com; qizhou@nju.edu.cn
- 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). - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4251-4274
- MSC (2010): Primary 34C15, 35B15, 37J40
- DOI: https://doi.org/10.1090/tran/6800
- MathSciNet review: 3624408