Uniqueness of Filippov Sliding Vector Field on the Intersection of Two Surfaces in \(\mathbb

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• 作者：L. Dieci ; C. Elia ; L. Lopez
• 关键词：Filippov convexification ; Orbital equivalence ; Periodic orbit ; Stability ; 34A36
• 刊名：Journal of Nonlinear Science
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：25
• 期：6
• 页码：1453-1471
• 全文大小：772 KB
• 参考文献：Aizerman, M.A., Gantmacher, F.R.: On the stability of periodic motion. J. Appl. Math. 22, 1065-078 (1958)
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  Dieci, L.: Sliding motion on the intersection of two manifolds: spirally attractive case. Commun. Nonlinear Sci. Numer. Simul. 26, 65-4 (2015)MathSciNet CrossRef
  Dieci, L., Difonzo, F.: A comparison of Filippov sliding vector fields in co-dimension 2. J. Comput. Appl. Math. 262, 161-79. Corrigendum in J. Comput. Appl. Math. 272(2014), 273-73 (2014)
  Dieci, L., Difonzo, F.: The moments sliding vector field on the intersection of two manifolds. J. Dyn. Differ. Equ. (2015). doi:10.-007/?s10884-015-9439-9
  Dieci, L., Elia, C., Lopez, L.: A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis. J. Differ. Equ. 254, 1800-832 (2013)MATH MathSciNet CrossRef
  Dieci, L., Lopez, L.: Fundamental matrix solutions of Piecewise smooth differential systems. Math. Comput. Simul. 81, 932-53 (2011)MATH MathSciNet CrossRef
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  Jeffrey, M.R.: Dynamics at a switching intersection: hierarchy, isonomy, and multiple-sliding. SIAM J. Appl. Dyn. Syst. 13(3), 1082-105 (2014)MATH MathSciNet CrossRef
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• 作者单位：L. Dieci (1)
  C. Elia (2)
  L. Lopez (2)
  
  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA
  2. Dipartimento di Matematica, University of Bari, 70100, Bari, Italy
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

In this paper, we consider the class of smooth sliding Filippov vector fields in \(\mathbb {R}^3\) on the intersection \(\Sigma \) of two smooth surfaces: \(\Sigma =\Sigma _1\cap \Sigma _2\), where \(\Sigma _i=\{x:\ h_i(x)=0\}\), and \(h_i:\ \mathbb {R}^3\rightarrow \mathbb {R}\), \(i=1,2\), are smooth functions with linearly independent normals. Although, in general, there is no unique Filippov sliding vector field on \(\Sigma \), here we prove that—under natural conditions—all Filippov sliding vector fields are orbitally equivalent to \(\Sigma \). In other words, the aforementioned ambiguity has no meaningful dynamical impact. We also examine the implication of this result in the important case of a periodic orbit a portion of which slides on \(\Sigma \). Keywords Filippov convexification Orbital equivalence Periodic orbit Stability