论文标题
$ 3D $ FILIPPOV系统的连锁系统：混乱现象
Chains in $3D$ Filippov Systems: A Chaotic Phenomenon
论文作者
论文摘要
这项工作致力于研究典型的通用奇点（称为$ t $ singularities）在分段平滑动力系统中的全球连接。这样的奇异性呈现出所谓的非平滑diabolo，它由一对不变的圆锥体组成。 我们分析了由$ t $ singularity的非平滑diobolo分支之间的通信产生的全球特征，我们证明，在一般条件下，这种通信会导致系统的混乱行为。更具体地说，我们将Filippov系统的跨轨道与$ t $ singularity展示某些交叉自相关的横向轨道，并与与系统相关的首次返回地图的Smale Horseshoe。这项工作中使用的技术依赖于检测该第一个返回图的鞍形类型的双曲线固定点之间的横向相交以及与之相关的Smale马蹄铁的分析。 从我们的方法中讨论的具体案例中，我们提出了一种强大的混沌现象，在这种情况下，它在平滑情况下似乎仅对高度退化的系统发生。
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.
