$3D$ Filippov系统中的链：一种混沌现象
Chains in $3D$ Filippov Systems: A Chaotic Phenomenon
摘要：
本文研究了分段光滑动力系统中的典型类属奇点（称为$T$ -奇点）之间的整体联系.这样的奇异性提出了所谓的非光滑空竹，这是由一对不变的锥从it. We分析的全球功能所产生的通信之间的分支的一个$T$ -奇异性的非光滑空竹，我们证明，在一般条件下，这种通信导致系统的混沌行为。更具体地说，我们涉及交叉轨道的Filippov系统提出了一定的交叉自连接到一个$T$ -奇点，与Smale马蹄铁的第一个返回地图相关联的系统。在这项工作中使用的技术依赖于检测的横截相交不变流形的双曲不动点的鞍型这样的第一个返回映射和Smale马蹄相关的分析。从我们的方法中讨论的具体情况下，我们提出了一个强大的混沌现象，其对应的光滑的情况下似乎只发生在高度退化的系统。
英文摘要：
This work is devoted to the study of global connections between typical generic singularities, named $T$-singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a pair of invariant cones emanating from it. We analyze global features arising from the communication between the branches of a nonsmooth diabolo of a $T$-singularity and we prove that, under generic conditions, such communication leads to a chaotic behavior of the system. More specifically, we relate crossing orbits of a Filippov system presenting certain crossing self-connections to a $T$-singularity, with a Smale horseshoe of a first return map associated to the system. The techniques used in this work rely on the detection of transverse intersections between invariant manifolds of a hyperbolic fixed point of saddle type of such a first return map and the analysis of the Smale horseshoe associated to it. From the specific case discussed in our approach, we present a robust chaotic phenomenon for which its counterpart in the smooth case seems to happen only for highly degenerate systems.
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