基于奇异摄动的迟滞模型的爆破分析
Blowup analysis of a hysteresis model based upon singular perturbations
AI总结：
本文对基于奇异摄动的迟滞模型进行几何分析，利用爆破技术证明了不变圆柱的存在，发现慢漂移由 Filippov 滑动向量场主导，并在 grazing 情况下区分了平滑与迟滞的动力学差异。
摘要：
在本文中，我们对基于奇异摄动的一种新迟滞模型进行了几何分析。这里的迟滞是指分段光滑微分方程的一种正则化类型，其中轨迹在不连续集的小邻域内的过去状态决定了当前向量场。事实上，在不连续集的邻域消失的极限下，迟滞在适当的意义上收敛于 Filippov 滑动向量场。然而，最近（2022年），Bonet 和 Seara 表明，与通过平滑进行的正则化不同，迟滞在 grazing 分岔的正则化中会导致混沌，即使在二维情况下也是如此。我们在本文中分析的迟滞模型——由 Bonet 等人在 2017 年的一篇论文中提出，旨在统一分段光滑系统的不同正则化——涉及两个奇异摄动参数，并包含了慢-快和非光滑效应的组合。因此，从奇异摄动理论的角度来看，该模型的描述具有挑战性，即使在二维情况下也是如此。以爆破作为我们的主要技术工具，我们证明了一个不变圆柱的存在，该圆柱在方位角方向上承载快速动力学，在轴向方向上承载慢速漂移。我们发现，在首阶近似下，慢速漂移由 Filippov 滑动向量场给出。此外，在 grazing 的情况下，我们确定了两个重要的参数区间，它们将模型分别与平滑（通过极限环的鞍结分岔）和迟滞（通过折叠鞍点和新颖的返回机制导致的混沌动力学）联系起来。
英文摘要：
In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity {vanishes}, hysteresis converges in an appropriate sense to Filippov's sliding vector-field. Recently {(2022)}, however, {Bonet and Seara} showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper -- which was developed by {Bonet et al in a paper from 2017} as an attempt to unify different regularizations of piecewise smooth systems -- involves two singular perturbation parameters and includes a combination of slow-fast and nonsmooth effects. The description of this model is therefore -- from the perspective of singular perturbation theory -- challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov's sliding {vector-field} to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).
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