论文标题
非平滑平衡附近的平面矢量场的动力学
Dynamics of planar vector fields near a non-smooth equilibrium
论文作者
论文摘要
在本文中，我们为平面分段光滑矢量场的定性和几何分析做出了贡献，这些平面平滑矢量场由两个平滑矢量场组成，这些平滑矢量场由直线$ y = 0 $隔开，并作为非分类平衡共享。从$σ$等效的意义上讲，我们为线性化提供了足够的条件，并为这些可线化的向量场提供了相位肖像和正常形式。由于存在不满足此条件时，这种情况很难被削弱，因为存在矢量场。关于扰动，当原点仍然是扰动下两个平滑矢量场的平衡时，建立了局部$σ$结构稳定性的必要条件。在反对这种情况下，我们证明，对于本文研究的任何分段平滑矢量字段，都有一个限制周期分叉，并且有一些分段平滑的矢量场，因此对于任何积极的整数$ m $，都有$ m $ $ $ $ $ $ $ $ $ $ $ $ $的限制周期。这里$ m $也许是无限。
In this paper we contribute to qualitative and geometric analysis of planar piecewise smooth vector fields, which consist of two smooth vector fields separated by the straight line $y=0$ and sharing the origin as a non-degenerate equilibrium. In the sense of $Σ$-equivalence, we provide a sufficient condition for linearization and give phase portraits and normal forms for these linearizable vector fields. This condition is hard to be weakened because there exist vector fields which are not linearizable when this condition is not satisfied. Regarding perturbations, a necessary and sufficient condition for local $Σ$-structural stability is established when the origin is still an equilibrium of both smooth vector fields under perturbations. In the opposition to this case, we prove that for any piecewise smooth vector field studied in this paper there is a limit cycle bifurcating from the origin, and there are some piecewise smooth vector fields such that for any positive integer $m$ there is a perturbation having exactly $m$ limit cycles bifurcating from the origin. Here $m$ maybe infinity.
