TY - JOUR
T1 - Boundedness in Asymmetric Quasi-Periodic Oscillations
AU - Xing , Xiumei
AU - Ma , Jing
AU - Jiao , Lei
JO - Communications in Mathematical Research 
VL - 2
SP - 121
EP - 128
PY - 2019
DA - 2019/11
SN - 33
DO - http://doi.org/10.13447/j.1674-5647.2017.02.03
UR - https://global-sci.org/intro/article_detail/cmr/13391.html
KW - boundedness, quasi-periodic, KAM theorem
AB - <p style="text-align: justify;">In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation $x′′+ax^+−bx^−+ ϕ(x) = p(t)$, where $a ≠ b$ are two positive constants and $ϕ(s)$, $p(t)$ are real analytic functions. Moreover, the $p(t)$ is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions, the quasi-periodic oscillator has the Lagrange stability.&nbsp;</p>