Regularizing Piecewise Smooth Differential Systems: Co-Dimension $2$ Disconti

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• 作者：Luca Dieci (1)
  Nicola Guglielmi (2)
  
• 关键词：Piecewise smooth systems ; Filippov solution ; Regularization ; Co ; dimension 2 ; 35L99 ; 65L15 ; 65L99.
• 刊名：Journal of Dynamics and Differential Equations
• 出版年：2013
• 出版时间：March 2013
• 年：2013
• 卷：25
• 期：1
• 页码：71-94
• 全文大小：622 KB
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• 作者单位：Luca Dieci (1)
  Nicola Guglielmi (2)
  
  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA
  2. Dipartimento di Matematica Pura e Applicata, Università di L’Aquila, via Vetoio, 67010, L’Aquila, Italy
  
• ISSN：1572-9222

文摘

In this paper, we are concerned with numerical solution of piecewise smooth initial value problems. Of specific interest is the case when the discontinuities occur on a smooth manifold of co-dimension 2, intersection of two co-dimension 1 singularity surfaces, and which is nodally attractive for nearby dynamics. In this case of a co-dimension 2 attracting sliding surface, we will give some results relative to two prototypical time and space regularizations. We will show that, unlike the case of co-dimension 1 discontinuity surface, in the case of co-dimension 2 discontinuity surface the behavior of the regularized problems is strikingly different. On the one hand, the time regularization approach will not select a unique sliding mode on the discontinuity surface, thus maintaining the general ambiguity of how to select a Filippov vector field in this case. On the other hand, the proposed space regularization approach is not ambiguous, and there will always be a unique solution associated to the regularized vector field, which will remain close to the original co-dimension 2 surface. We will further clarify the limiting behavior (as the regularization parameter goes to 0) of the proposed space regularization to the solution associated to the sliding vector field of Dieci and Lopez (Numer Math 117:779-11, 2011). Numerical examples will be given to illustrate the different cases and to provide some preliminary exploration in the case of co-dimension 3 discontinuity surface.