Volume 33, Issue 2
Boundedness in Asymmetric Quasi-Periodic Oscillations

Commun. Math. Res., 33 (2017), pp. 121-128.

Published online: 2019-11

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• Abstract

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.

• Keywords

boundedness, quasi-periodic, KAM theorem

34C15, 70H08

xingxm09@163.com (Xiumei Xing)

jiaolei0104@163.com (Lei Jiao)

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@Article{CMR-33-121, author = {Xiumei and Xing and xingxm09@163.com and 5457 and School of Mathematics and Statistics, Yili Normal University, Yili, Xinjiang, 835000 and Xiumei Xing and Jing and Ma and and 5459 and School of Mathematics and Statistics, Yili Normal University, Yili, Xinjiang, 835000 and Jing Ma and Lei and Jiao and jiaolei0104@163.com and 5458 and School of Science, Nanjing University of Science and Technology, Nanjing, 210094 and Lei Jiao}, title = {Boundedness in Asymmetric Quasi-Periodic Oscillations}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {2}, pages = {121--128}, abstract = {

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.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.02.03}, url = {http://global-sci.org/intro/article_detail/cmr/13391.html} }
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 -

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.

Xiu-mei Xing, Jing Ma & Lei Jiao. (2019). Boundedness in Asymmetric Quasi-Periodic Oscillations. Communications in Mathematical Research . 33 (2). 121-128. doi:10.13447/j.1674-5647.2017.02.03
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