通用一参数家族的三维Filippov系统
Generic one-parameter families of 3-dimensional Filippov Systems
AI总结：
本文研究了三维Filippov系统中一参数家族的结构稳定性及鲁棒性，基于接触理论和拓扑学分析了奇点的规范形式。
摘要：
本文探讨了一参数家族的三维分段光滑向量场（PSVFs）在典型奇点周围所具有的开放性、密度和结构稳定性条件。我们的分析是局部的，切换集M是一个嵌入在R³中的二维表面。简而言之，我们分析了PSVFs中出现的某些一维奇点的鲁棒性和规范形式。本文主要工具包括向量场与M的接触理论、分支理论和流形的拓扑学。我们的主要结果表明，我们建立的数学陈述类似于经典的Kupka-Smale定理，即我们证明了大量PSVFs呈现通用和近通用奇点的开放性和密度。由于PSVFs某些解的非唯一性，我们整个论文都采用Filippov理论作为基础。
英文摘要：
This paper addresses openness, density and structural stability conditions of one-parameter families of 3D piecewise smooth vector fields (PSVFs) defined around typical singularities. Our treatment is local and the switching set, $M$, is a $2D$ surface embedded in $\mathbb{R}^3$. In short, we analyze the robustness and normal forms of certain codimension one singularities that occur in PSVFs. The main machinery used in this paper involves the theory of contact between a vector field and $M$, Bifurcation Theory and the Topology of Manifolds. Our main result states robust mathematical statements resembling the classical Kupka-Smale Theorem in the sense that we establish the openness and density of a large class of PSVFs presenting generic and quasi-generic singularities. Due to the lack of uniqueness of certain solutions associated with PSVFs, we employ Filippov's theory as the basis of our approach throughout the paper.
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