Skip to main content
SpringerLink
Log in

On the Inverse of Some Sign Matrices and on the Moments Sliding Vector Field on the Intersection of Several Manifolds: Nodally Attractive Case

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we consider selection of a sliding vector field of Filippov type on a discontinuity manifold \(\Sigma \) of co-dimension 3 (intersection of three co-dimension 1 manifolds). We propose an extension of the moments vector field to this case, and—under the assumption that \(\Sigma \) is nodally attractive—we prove that our extension delivers a uniquely defined Filippov vector field. As it turns out, the justification of our proposed extension requires establishing invertibility of certain sign matrices. Finally, we also propose the extension of the moments vector field to discontinuity manifolds of co-dimension 4 and higher.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The minimum value is \(\frac{-\sqrt{3}-3+3\sqrt{2}}{\sqrt{3}}\) and is attained at \(a=b=c=\frac{1}{\sqrt{3}}\).

References

  1. Alexander, J.C., Seidman, T.: Sliding modes in intersecting switching surfaces, I: blending. Houst. J. Math. 24, 545–569 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Barvinok, A.: A Course in Convexity. American Mathematical Society, Providence, RI (2007)

    MATH  Google Scholar 

  3. Brualdi, R., Shader, B.: Matrices of Sign-Solvable Linear Systems. Cambridge Tracts in Mathematics 116. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  4. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth Dynamical Systems. Theory and Applications. Applied Mathematical Sciences 163. Springer, Berlin (2008)

    MATH  Google Scholar 

  5. 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, 1800–1832 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dieci, L., Difonzo, F.: The moments sliding vector field on the intersection of two manifolds. J. Dyn. Differ. Equ. (2015). doi:10.1007/s10884-015-9439-9

  7. Dieci, L., Guglielmi, N.: Regularizing piecewise smooth differential systems: co-dimension 2 discontinuity surface. J. Dyn. Differ. Equ. 25(1), 71–94 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dieci, L., Lopez, L.: Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field. Numer. Math. 117, 779–811 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications. Kluwer Academic, Dordrecht (1988)

    Book  Google Scholar 

  10. Grunbaum, B.: Convex Polytopes. Springer, New York (2003)

    Book  MATH  Google Scholar 

  11. Jeffrey, M.: Dynamics at a switching intersection: hierarchy, isonomy, and multiple sliding. SIAM J. Appl. Dyn. Syst. 13, 1082–1105 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. Klee, V., Ladner, R.: Signsolvability revisited. Linear Algebr. Appl. 59, 131–157 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Klee, V., Ladner, R., Manber, R.: Qualitative matrices: strong sign-solvability and weak satisfiability. In: Greenberg, H., Maybee, J. (eds.) Computer-Assisted Analysis and Model Simplification, pp. 293–320. Academic Press, New York (1981)

    Chapter  Google Scholar 

  14. Llibre, J., Silva, P.R., Teixeira, M.A.: Regularization of discontinuous vector fields on \({\mathbb{R}}^3\) via singular perturbation. J. Dyn. Differ. Equ. 19, 309–331 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Thomassen, C.: When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse. Linear Algebr. Appl. 119, 27–34 (1989)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The first author gratefully acknowledges the support provided by a Tao Aoqing Visiting Professorship at Jilin University, Changchun (CHINA). The authors are very thankful to the anonymous referee for detecting a few mishaps in an early version of this paper

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Luca Dieci & Fabio Difonzo

Authors
  1. Luca Dieci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabio Difonzo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Dieci.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dieci, L., Difonzo, F. On the Inverse of Some Sign Matrices and on the Moments Sliding Vector Field on the Intersection of Several Manifolds: Nodally Attractive Case. J Dyn Diff Equat 29, 1355–1381 (2017). https://doi.org/10.1007/s10884-016-9527-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-016-9527-5

Keywords

Mathematics Subject Classification

Search

Navigation