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

Wang, Zaihong

doi:10.1090/S0002-9939-02-06601-7

Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities
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Proc. Amer. Math. Soc. 131 (2003), 523-531 Request permission

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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