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

Xiumei Xing, Jing Ma & Lei Jiao

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

  • AMS Subject Headings

34C15, 70H08

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xingxm09@163.com (Xiumei Xing)

jiaolei0104@163.com (Lei Jiao)

  • BibTex
  • RIS
  • TXT
@Article{CMR-33-121, author = {Xing , Xiumei and Ma , Jing and Jiao , Lei}, 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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