用正则化和爆破方法求解$\mathbb{R}^3$中的分段光滑可见-不可见二重奇异性
Resolution of the piecewise smooth visible-invisible two-fold singularity in $\mathbb{R}^3$ using regularization and blowup
摘要：
$\mathbb{R}^3$中分段光滑动力系统的二重奇异性一直是人们深入研究的课题。兴趣源于这样一个事实，即进入两重的轨迹与向前的非唯一性。关键问题是：我们如何在时间上继续前进？是否存在在所有候选者中区分开的轨道？我们解决这些问题，通过正规化的PWS动力系统的情况下，可见-不可见的双重。在这个框架内，我们考虑一个正则化函数的类外的索托马约尔和Teixera。然后，我们进行了严格的调查，使用几何奇异扰动理论和爆破。我们表明，确实有一个前进的轨道$U$，这是区别于所有可能的前进轨道离开两重。工作与正常形式的可见-不可见的两重，我们表明，吸引极限环可以获得（由于收缩向$U$），与全球回报机制组成后。我们提供了一些说明性的例子。
英文摘要：
Two-fold singularities in a piecewise smooth (PWS) dynamical system in $\mathbb{R}^3$ have long been the subject of intensive investigation. The interest stems from the fact that trajectories which enter the two-fold are associated with forward non-uniqueness. The key questions are: How do we continue orbits forward in time? Are there orbits that are distinguished among all the candidates? We address these questions by regularizing the PWS dynamical system for the case of the visible-invisible two-fold. Within this framework, we consider a regularization function outside the class of Sotomayor and Teixera. We then undertake a rigorous investigation, using geometric singular perturbation theory and blowup. We show that there is indeed a forward orbit $U$ that is distinguished amongst all the possible forward orbits leaving the two-fold. Working with a normal form of the visible-invisible two-fold, we show that attracting limit cycles can be obtained (due to the contraction towards $U$), upon composition with a global return mechanism. We provide some illustrative examples.
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