Open AccessResponse solutions for quasi-periodically forced harmonic oscillators
Author(s) -
Jing Wang,
Jiangong You,
Qi Zhou
Publication year - 2016
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/6800
Subject(s) - algorithm , mathematics , computer science , artificial intelligence
We prove that the non-linear quasi-periodically forced harmonic oscillator with two frequencies ( 1 , α ) (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 α \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.