Response solutions for degenerate reversible harmonic oscillators
Response solutions for degenerate reversible harmonic oscillators
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DOI：
10.3934/dcds.2021023
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发表时间：
2021
影响因子：
1.1
通讯作者：
Wen Si
中科院分区：
文献类型：
--
作者：
Wen Si
We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators \begin{document}$ \ddot{x}-\lambda x^n = \epsilon f(\omega t, x, \dot x, \epsilon), \; \; x\in \mathbb{R}, $\end{document} where \begin{document}$ \lambda = \pm 1 $\end{document} , \begin{document}$ n>1 $\end{document} is an integer and \begin{document}$ f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon) $\end{document} . With \begin{document}$ f $\end{document} satisfying certain non-degenerate conditions, we obtain the following results: (1) For \begin{document}$ \lambda = 1 $\end{document} and \begin{document}$ \epsilon $\end{document} sufficiently small, response solutions exist for each \begin{document}$ \omega $\end{document} satisfying a weak non-resonant condition; (2) For \begin{document}$ \lambda = -1 $\end{document} and \begin{document}$ \epsilon_* $\end{document} sufficiently small, there exists a Cantor set \begin{document}$ \mathcal{E}\in(0, \epsilon_*) $\end{document} with almost full Lebesgue measure such that response solutions exist for each \begin{document}$ \epsilon\in\mathcal{E} $\end{document} if \begin{document}$ \omega $\end{document} satisfies a Diophantine condition. Non-existence of response solutions is also discussed when \begin{document}$ f $\end{document} fails to satisfy the non-degenerate conditions.
We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators \begin{document}$ \ddot{x}-\lambda x^n = \epsilon f(\omega t, x, \dot x, \epsilon), \; \; x\in \mathbb{R}, $\end{document} where \begin{document}$ \lambda = \pm 1 $\end{document} , \begin{document}$ n>1 $\end{document} is an integer and \begin{document}$ f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon) $\end{document} . With \begin{document}$ f $\end{document} satisfying certain non-degenerate conditions, we obtain the following results: (1) For \begin{document}$ \lambda = 1 $\end{document} and \begin{document}$ \epsilon $\end{document} sufficiently small, response solutions exist for each \begin{document}$ \omega $\end{document} satisfying a weak non-resonant condition; (2) For \begin{document}$ \lambda = -1 $\end{document} and \begin{document}$ \epsilon_* $\end{document} sufficiently small, there exists a Cantor set \begin{document}$ \mathcal{E}\in(0, \epsilon_*) $\end{document} with almost full Lebesgue measure such that response solutions exist for each \begin{document}$ \epsilon\in\mathcal{E} $\end{document} if \begin{document}$ \omega $\end{document} satisfies a Diophantine condition. Non-existence of response solutions is also discussed when \begin{document}$ f $\end{document} fails to satisfy the non-degenerate conditions.