In this work we attempt to understand what behavior one should expect of a solution rajectory near Sigma when Sigma is attractive, what to expect when Sigma ceases to be attractive (at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature, whereby the piecewise smooth system is replaced by a smooth differential system. \ud\udThrough analysis and experiments in R-3 and R-4, we will confirm some known facts and provide some important insight: (i) when Sigma is attractive, a solution trajectory remains near Sigma, viz. sliding on Sigma is an appropriate idealization (though one cannot a priori decide which sliding vector field should be selected); (ii) when Sigma loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near while Sigma is attractive, and so that it will be leaving (a neighborhood of) Sigma when Sigma looses attractivity. \ud\udWe reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near Sigma should have been taking place.
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机译：在这项工作中，我们试图了解当Sigma有吸引力时，人们对Sigma附近解决方案轨迹的期望行为，当Sigma不再具有吸引力（在通用出口点）时，期望什么，最后，我们还对比和比较了一些行为。文献中提出的正则化方法，从而用平滑微分系统代替分段平滑系统。通过在R-3和R-4中进行分析和实验，我们将证实一些已知的事实并提供一些重要的见解：（i）当Sigma吸引人时，溶液轨迹仍保持在Sigma附近。在Sigma上滑动是一种合适的理想化方法（尽管不能先验地决定应选择哪个滑动矢量场）； （ii）当Sigma失去吸引力（在一阶退出条件下）时，典型的解决方案轨迹离开（iii）的邻域；没有明显的方法来规范化系统，以使规范化的轨迹保持接近，而Sigma具有吸引力，因此当Sigma失去吸引力时，它将离开Sigma（附近）。 \ ud \ ud我们仅考虑给定的分段光滑系统，而不对Sigma附近应该发生什么样的动力学做任何假设，从而得出上述结论。
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