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Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities


Author: Zaihong Wang
Journal: Proc. Amer. Math. Soc. 131 (2003), 523-531
MSC (2000): Primary 34C25; Secondary 34B15
DOI: https://doi.org/10.1090/S0002-9939-02-06601-7
Published electronically: June 3, 2002
MathSciNet review: 1933343
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Abstract: In this paper, we study the dynamics of the mappings

\begin{displaymath}\begin{cases} \theta_1=\theta+2\alpha\pi+\frac{1}{r}\mu_1(\th... ...\ r_1=r+\mu_2(\theta)+o(1),\quad\quad r\to+\infty, \end{cases}\end{displaymath}

where $\alpha$ is a irrational rotation number. We prove the existence of orbits that go to infinity in the future or in the past by using the well-known Birkhoff Ergodic Theorem. Applying this conclusion, we deal with the unboundedness of solutions of Liénard equations with asymmetric nonlinearities.


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Additional Information

Zaihong Wang
Affiliation: Department of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China
Email: zhwang@mail.cnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-02-06601-7
Keywords: Unboundedness of solution, action-angle variable, asymmetric nonlinearity
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: September 22, 2001
Published electronically: June 3, 2002
Additional Notes: The author’s research was supported by the National Natural Science Foundation of China, No.10001025, and by the Natural Science Foundation of Beijing (1022003)
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society

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