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Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities


Authors: Hongbin Chen and Yi Li
Journal: Proc. Amer. Math. Soc. 135 (2007), 3925-3932
MSC (2000): Primary 34C10, 34C25
DOI: https://doi.org/10.1090/S0002-9939-07-09024-7
Published electronically: September 7, 2007
MathSciNet review: 2341942
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Abstract: We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities,

$\displaystyle x''+cx'+ax-x^{3}=h(t),\tag{$*$} $

where $ a$ and $ c>0$ are positive constants and $ h(t)$ is a positive $ T$-periodic function. We obtain sharp bounds for $ h$ such that $ (*)$ has exactly three ordered $ T$-periodic solutions. Moreover, when $ h$ is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.


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Additional Information

Hongbin Chen
Affiliation: Department of Mathematics, Xi’an Jiaotong University, Xi’an, People’s Republic of China
Email: hbchen@mail.xjtu.edu.cn

Yi Li
Affiliation: Department of Mathematics, Hunan Normal University, Changsha, Hunan, People’s Republic of China
Address at time of publication: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: yi-li@uiowa.edu

DOI: https://doi.org/10.1090/S0002-9939-07-09024-7
Keywords: Duffing equation, periodic solution, stability
Received by editor(s): September 27, 2006
Published electronically: September 7, 2007
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society

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