论文标题
filippov矢量场中单颗粒切向奇点的Lyapunov系数
Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields
论文作者
论文摘要
在平面分析矢量场中，可以通过Lyapunov系数区分焦点或中心之间的单个奇异性，这是根据围绕奇异性定义的第一回归图的功率序列系数给出的。在本文中，我们对分段分析矢量字段的单粒切向向量字段的类似问题感兴趣$ z =（z^+，z^ - ）$。首先，我们证明了在单个切向奇异性的邻域定义的第一回收图是分析性的，该图可以定义Lyapunov系数。然后，由于一般的一般属性，我们得到了第一个非散布lyapunov系数的索引始终是均匀的。此外，还获得了用于计算Lyapunov系数的数学算法的一般递归公式。我们还提供了有关从单粒子切向奇点分叉的极限周期的结果。分析了几个示例。
In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map defined around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields $Z=(Z^+ ,Z^-)$. First, we prove that the first-return map, defined in a neighborhood of a monodromic tangential singularity, is analytic, which allows the definition of the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a general recursive formula together with a Mathematica algorithm for computing the Lyapunov coefficients is obtained. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analyzed.
