{"title":"Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation","authors":"Xin Yu Guan, Jian Guo Si, Wen Si","doi":"10.1007/s10114-023-1539-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider a class of normally degenerate quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators, </p><div><div><span>$$\\left\\{ {\\matrix{{\\dot x = y + {f_1}(\\omega t,x,y),} \\hfill \\cr {\\dot y = \\lambda {x^l} + {f_2}(\\omega t,x,y),} \\hfill \\cr } } \\right.$$</span></div></div><p> where 0 ≠ λ ∈ ℝ, <i>l</i> > 1 is an integer and the corresponding involution <i>G</i> is (−<i>θ, x</i>, −<i>y</i>) → (<i>θ, x, y</i>). The existence of response solutions of the above reversible systems has already been proved in [22] if [<i>f</i><sub>2</sub>(<i>ωt</i>, 0, 0)] satisfies some non-zero average conditions (See the condition (<b>H</b>) in [22]), here [ · ] denotes the average of a continuous function on <span>\\({\\mathbb{T}^d}\\)</span>. However, discussing the existence of response solutions for the above systems encounters difficulties when [<i>f</i><sub>2</sub>(<i>ωt</i>, 0, 0)] = 0, due to a degenerate implicit function must be solved. This article will be doing work in this direction. The purpose of this paper is to consider the case where [<i>f</i><sub>2</sub>(<i>ωt</i>, 0, 0)] = 0. More precisely, with 2<i>p</i> < <i>l</i>, if <i>f</i><sub>2</sub> satisfies <span>\\([{f_2}(\\omega t,0,0)] = [{{\\partial {f_2}(\\omega t,0,0)} \\over {\\partial x}}] = [{{{\\partial ^2}{f_2}(\\omega t,0,0)} \\over {\\partial {x^2}}}] = \\cdots = [{{{\\partial ^{p - 1}}{f_2}(\\omega t,0,0)} \\over {\\partial {x^{p - 1}}}}] = 0\\)</span>, either <span>\\({\\lambda ^{ - 1}}[{{{\\partial ^p}{f_2}(\\omega t,0,0)} \\over {\\partial {x^p}}}] < 0\\)</span> as <i>l</i> − <i>p</i> is even or <span>\\({\\lambda ^{ - 1}}[{{{\\partial ^p}{f_2}(\\omega t,0,0)} \\over {\\partial {x^p}}}] \\ne 0\\)</span> as <i>l</i> − <i>p</i> is odd, we obtain the following results: (1) For <span>\\(\\tilde \\lambda < 0\\)</span> (see <span>\\({\\tilde \\lambda }\\)</span> in (2.2)) and <i>ϵ</i> sufficiently small, response solutions exist for each <i>ω</i> satisfying a weak non-resonant condition; (2) For <span>\\(\\tilde \\lambda < 0\\)</span> and <i>ϵ</i><sub>*</sub> sufficiently small, there exists a Cantor set <span>\\({\\cal E} \\in (0,{_ * })\\)</span> with almost full Lebesgue measure such that response solutions exist for each <span>\\( \\in {\\cal E}\\)</span> if <i>ω</i> satisfies a Diophantine condition. In the remaining case where <span>\\({\\lambda ^{ - 1}}[{{{\\partial ^p}{f_2}(\\omega t,0,0)} \\over {\\partial {x^p}}}] > 0\\)</span> and <i>l</i> − <i>p</i> is even, we prove the system admits no response solutions in most regions.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-1539-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we consider a class of normally degenerate quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators,
$$\left\{ {\matrix{{\dot x = y + {f_1}(\omega t,x,y),} \hfill \cr {\dot y = \lambda {x^l} + {f_2}(\omega t,x,y),} \hfill \cr } } \right.$$
where 0 ≠ λ ∈ ℝ, l > 1 is an integer and the corresponding involution G is (−θ, x, −y) → (θ, x, y). The existence of response solutions of the above reversible systems has already been proved in [22] if [f2(ωt, 0, 0)] satisfies some non-zero average conditions (See the condition (H) in [22]), here [ · ] denotes the average of a continuous function on \({\mathbb{T}^d}\). However, discussing the existence of response solutions for the above systems encounters difficulties when [f2(ωt, 0, 0)] = 0, due to a degenerate implicit function must be solved. This article will be doing work in this direction. The purpose of this paper is to consider the case where [f2(ωt, 0, 0)] = 0. More precisely, with 2p < l, if f2 satisfies \([{f_2}(\omega t,0,0)] = [{{\partial {f_2}(\omega t,0,0)} \over {\partial x}}] = [{{{\partial ^2}{f_2}(\omega t,0,0)} \over {\partial {x^2}}}] = \cdots = [{{{\partial ^{p - 1}}{f_2}(\omega t,0,0)} \over {\partial {x^{p - 1}}}}] = 0\), either \({\lambda ^{ - 1}}[{{{\partial ^p}{f_2}(\omega t,0,0)} \over {\partial {x^p}}}] < 0\) as l − p is even or \({\lambda ^{ - 1}}[{{{\partial ^p}{f_2}(\omega t,0,0)} \over {\partial {x^p}}}] \ne 0\) as l − p is odd, we obtain the following results: (1) For \(\tilde \lambda < 0\) (see \({\tilde \lambda }\) in (2.2)) and ϵ sufficiently small, response solutions exist for each ω satisfying a weak non-resonant condition; (2) For \(\tilde \lambda < 0\) and ϵ* sufficiently small, there exists a Cantor set \({\cal E} \in (0,{_ * })\) with almost full Lebesgue measure such that response solutions exist for each \( \in {\cal E}\) if ω satisfies a Diophantine condition. In the remaining case where \({\lambda ^{ - 1}}[{{{\partial ^p}{f_2}(\omega t,0,0)} \over {\partial {x^p}}}] > 0\) and l − p is even, we prove the system admits no response solutions in most regions.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.