Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems
Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems
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准周期受迫系统的响应解和准周期简并分岔
DOI：
10.1088/1361-6544/aaa7b9
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发表时间：
2018-04
期刊：
影响因子：
1.7
通讯作者：
Jianguo Si
中科院分区：
文献类型：
--
作者：
Wen Si;Jianguo Si
本文共分两部分。在第一部分中，我们首先研究了具有退化平衡点的一维拟周期强迫系统的拟周期依赖于时间的扰动。我们研究了两种情况下的系统，其中一个系统承认一个响应的解决方案下的非共振条件下的频率向量比Brjuno-Rüssmann的弱，另一个系统也承认一个响应的解决方案，没有任何非共振条件。其次，利用赫尔曼方法研究了一类具有退化（包括完全退化）平衡点的拟周期扰动系统在Brjuno-Rüssmann非共振条件下响应解的存在性.在第二部分中，我们首先考虑一维退化向量场的普适开折的拟周期扰动。其次，我们考虑了具有完全退化平衡的正常二维Hamilton系统的普适开折的扰动。利用KAM理论和奇异性理论，证明了这两类泛开折在Brjuno-Rüssmann非共振条件下在大Cantor集上是持续的，这意味着可积部分的所有不变环面和所有分支情形都可以在大Cantor集上生存.哈密顿系统的结果可直接应用于拟周期受迫系统的响应。我们在本文中的结果可以被认为是相对于各种文献中的若干结果的改进（Broer等人2005 Nonlinearity 18 1735-69; Broer等人2006 J. Differ.当量ag 222 233-62; Wagener 2005 J. Differ.当量ag 216 216-81; Xu 2010 J. Differ.当量ag 250 551-71; Xu and Jiang 2010 Ergod. Theor.戴纳姆31 599-611; Lu and Xu 2014 Nonlinear Different.当量ag 21 361-70）。
This paper includes two parts. In the first part, we first focus on quasi-periodic time dependent perturbations of one-dimensional quasi-periodically forced systems with degenerate equilibrium. We study the system in two cases, for one of which system admits a response solution under a non-resonant condition on the frequency vector weaker than Brjuno–Rüssmann’s and for another of which system also admits a response solution without any non-resonant conditions. Next, we investigate the existence of response solutions of a quasi-periodic perturbed system with degenerate (including completely degenerate) equilibrium under Brjuno–Rüssmann’s non-resonant condition by using the Herman method. In the second part, we consider, firstly, the quasi-periodic perturbation of a universal unfolding of one-dimensional degenerate vector field . Secondly, we consider the perturbation of a universal unfolding of normal two-dimensional Hamiltonian system with completely degenerate equilibrium. With KAM theory and singularity theory, we show that these two classes of universal unfolding can persist on large Cantor sets under Brjuno–Rüssmann’s non-resonant condition, which implies all the invariant tori in the integrable part and all the bifurcation scenario can survive on large Cantor sets. The result for Hamiltonian system can apply directly to the response context for quasi-periodically forced systems. Our results in this paper can be regarded as an improvement with respect to several results in various literature (Broer et al 2005 Nonlinearity 18 1735–69; Broer et al 2006 J. Differ. Equ. 222 233–62; Wagener 2005 J. Differ. Equ. 216 216–81; Xu 2010 J. Differ. Equ. 250 551–71; Xu and Jiang 2010 Ergod. Theor. Dynam. Syst. 31 599–611; Lu and Xu 2014 Nonlinear Differ. Equ. Appl. 21 361–70).
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