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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
Published electronically: June 3, 2002
MathSciNet review: 1933343
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Abstract: In this paper, we study the dynamics of the mappings \begin{equation*} \begin {cases} \theta _1=\theta +2\alpha \pi +\frac {1}{r}\mu _1(\theta )+o(r^{-1}), r_1=r+\mu _2(\theta )+o(1),\quad \quad r\to +\infty , \end{cases} \end{equation*} 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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Zaihong Wang
Affiliation: Department of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China

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