Filippov动力系统中的Shilnikov问题
Shilnikov problem in Filippov dynamical systems
摘要：
本文引入了$3$ D Filippov系统滑动Shilnikov轨道的概念。简而言之，这样的轨道是由Filippov轨迹组成的分段光滑闭曲线，其在开关曲面上滑动并且将Filippov平衡连接到其自身，即伪鞍焦点。一个版本的Shilnikov定理提供了这样的系统。特别是，我们表明，滑动Shilnikov轨道发生在一般的单参数家庭的Filippov系统，和任意接近滑动Shilnikov轨道存在可数无穷多个滑动周期轨道。这里，为了得到最后一个结果，不需要额外的类似Shilnikov的假设。此外，我们证明了不连续分段线性微分系统的滑动Shilnikov轨道的存在性。据我们所知，Fillippov系统的例子，本文提供的是第一次展示这样的滑动现象。
英文摘要：
In this paper we introduce the concept of sliding Shilnikov orbits for $3$D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon.
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