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Existence and multiplicity of periodic solutions for the Duffing equation with singularity

Published online by Cambridge University Press:  21 May 2007

Jing Xia  and
Zaihong Wang
Jing Xia
Affiliation:
Department of Mathematics, Capital Normal University, Beijing 100037, People's Republic of China (zhwang@mail.cnu.edu.cn)
Zaihong Wang
Affiliation:
Department of Mathematics, Capital Normal University, Beijing 100037, People's Republic of China (zhwang@mail.cnu.edu.cn)
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Abstract

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In this paper, we consider the existence and multiplicity of periodic solutions for the Duffing equation $x''+g(x)=p(t)$ with a singularity. When the time map has oscillating properties, $g(x)$ possesses a singularity at the origin and tends to $+\infty$ as $x\to+\infty$ and other conditions hold. We obtain the existence of harmonic solutions and the multiplicity of subharmonic solutions of the given equation by using the phase-plane analysis methods and the generalized Poincaré–Birkhoff twist theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 3 , June 2007 , pp. 625 - 645
Copyright
2007 Royal Society of Edinburgh