# American Institute of Mathematical Sciences

August  2021, 41(8): 3951-3972. doi: 10.3934/dcds.2021023

## Response solutions for degenerate reversible harmonic oscillators

 School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author: Wen Si

Received  July 2020 Revised  December 2020 Published  January 2021

Fund Project: W. Si was partially supported by the National Natural Science Foundation of China (Grant Nos. 12001315); Shandong Provincial Natural Science Foundation, China (Grant Nos. ZR2020MA015); China Postdoctoral Science Foundation (Grant Nos. 2020M680089) and the Fundamental Research Funds of Shandong University (Grant Nos. 2019GN077). This paper is also supported by the National Natural Science Foundation of China (Grant Nos. 11971261, 11571201)

We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators
 $\ddot{x}-\lambda x^n = \epsilon f(\omega t, x, \dot x, \epsilon), \; \; x\in \mathbb{R},$
where
 $\lambda = \pm 1$
,
 $n>1$
is an integer and
 $f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon)$
. With
 $f$
satisfying certain non-degenerate conditions, we obtain the following results: (1) For
 $\lambda = 1$
and
 $\epsilon$
sufficiently small, response solutions exist for each
 $\omega$
satisfying a weak non-resonant condition; (2) For
 $\lambda = -1$
and
 $\epsilon_*$
sufficiently small, there exists a Cantor set
 $\mathcal{E}\in(0, \epsilon_*)$
with almost full Lebesgue measure such that response solutions exist for each
 $\epsilon\in\mathcal{E}$
if
 $\omega$
satisfies a Diophantine condition. Non-existence of response solutions is also discussed when
 $f$
fails to satisfy the non-degenerate conditions.
Citation: Wen Si. Response solutions for degenerate reversible harmonic oscillators. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3951-3972. doi: 10.3934/dcds.2021023
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