Resonances and Phase Locking Phenomena for Foliation Preserving Torus Maps
Resonances and Phase Locking Phenomena for Foliation Preserving Torus Maps
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DOI：
10.1137/22m1485103
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
Xiaolong He;Rafael de la Llave
中科院分区：
文献类型：
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作者：
Xiaolong He;Rafael de la Llave
专家们众所周知，非线性系统中的共振会导致新的不变对象产生新的行为。本文的目的是研究在叶状保留环面图下由共振产生的不变集。这是一个保留无理线叶状结构的环面 $L_{\theta_{0}}=\{\theta_{0}+\Omega t | t\in\mathbb{R}\}\subset\mathbb{T}^{d}$。叶状保留图自然地表现为环面中线性流的重新参数化，并且在涉及耦合振荡器、延迟方程、具有移动壁的谐振器等的多种应用中也发挥着重要作用。我们在这里找到的不变对象导致了对这些模型行为的预测。由于本文的结果旨在应用于其他问题，因此我们开发了非常定量的结果，对现象和控制它们的不变对象给出了非常明确的描述。叶状保留图的锁相区域的结构与环面的通用图有很大不同。事实上，为了完整起见，我们对环面的通用贴图的情况进行了类似的分析，并表明叶状保留贴图中出现的对象在数量和质量上都与通用环面贴图不同。这会对应用产生影响。
It is well known for experts that resonances in nonlinear systems lead to new invariant objects that lead to new behaviors. The goal of this paper is to study the invariant sets generated by resonances under foliation preserving torus maps. That is torus which preserve a foliation of irrational lines $L_{\theta_{0}}=\{\theta_{0}+\Omega t | t\in\mathbb{R}\}\subset\mathbb{T}^{d}$. Foliation preserving maps appear naturally as reparametrization of linear flows in the torus and also play an important role in several applications involving coupled oscillators, delay equations, resonators with moving walls, etc. The invariant objects we find here, lead to predictions on the behavior of these models. Since the results of this paper are meant to be applied for other problems, we have developed very quantitative results giving very explicit descriptions of the phenomena and the invariant objects that control them. The structure of the phase locking regions for foliation preserving maps is very different than for generic maps of the torus. Indeed, for the sake of completeness, we have developed similar analysis for the case of generic maps of the torus and shown that the objects that appear in foliation preserving maps are quantitatively and qualitatively different from those of generic torus maps. This has consequences in applications.