论文标题
分段平滑矢量场引起的慢速正常形式
Slow-fast normal forms arising from piecewise smooth vector fields
论文作者
论文摘要
We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \ Mathbb {r}^{n} \ rightarrow \ Mathbb {r} $是一个平滑的地图，其0作为常规值。我们考虑$$ \ dot {z} = z _ {\ varepsilon}（z）= \ dfrac {1 +φ（f /\ varepsilon）} {2} {2} x（z） + dfrac {1- dfrac {f /v vareps y（z）， $φ$是过渡功能（不一定是单调）和其过渡功能单调的向量场$ z $的非线性正则化。众所周知的事实是，正规化系统是一个缓慢的系统。本文的主要贡献是研究由（线性或非线性）正规化产生的慢快速系统的典型奇异性。我们开发了一种算法来构建合适的过渡功能，并应用了这些想法，以从正常形式的平滑矢量场中创建慢速奇异性。我们介绍了过渡函数的示例，这些函数在正常形式正规化后，会产生正常双曲，折叠，跨批评和干草叉奇异点。
We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z) = \dfrac{1 + φ(F/\varepsilon)}{2}X(z) +\dfrac{1 - φ(F /\varepsilon)}{2}Y(z),$$where $φ$ is a transition function (not necessarily monotonic) and nonlinear regularizations of the vector field $Z$ whose transition function is monotonic. It is a well-known fact that the regularized system is a slow-fast system. The main contribution of this paper is the study of typical singularities of slow-fast systems that arise from (linear or nonlinear) regularizations. We developed an algorithm to construct suitable transition functions, and we apply these ideas in order to create slow-fast singularities from normal forms of piecewise smooth vector fields. We present examples of transition functions that, after regularization of a PSVF normal form, generate normally hyperbolic, fold, transcritical, and pitchfork singularities.
