On structural stability of 3D Filippov systems

A semi-local approach
  • Otávio M. L. GomideEmail author
  • Marco A. Teixeira


The main purpose of this work is to provide a non-local approach to study aspects of structural stability of 3D Filippov systems. We introduce a notion of semi-local structural stability which detects when a piecewise smooth vector field is robust around the entire switching manifold, as well as, provides a complete characterization of such systems. In particular, we present some methods in the qualitative theory of piecewise smooth vector fields, which make use of geometrical analysis of the foliations generated by their orbits. Such approach displays surprisingly rich dynamical behavior which is studied in detail in this work. It is worth mentioning that this subject has not been treated in dimensions higher than two from a non-local point of view, and we hope that the approach adopted herein contributes to the understanding of structural stability for piecewise-smooth vector fields in its most global sense.


Structural stability Filippov systems 3-manifold Nonsmooth dynamics 

Mathematics Subject Classification

37C10 37C15 37C20 37C75 



  1. 1.
    Bernardo, M.D., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems Theory and Applications. Springer, Berlin (2008)zbMATHGoogle Scholar
  2. 2.
    Bonet, C., Seara, T.M., Fossas, E., Jeffrey, M.R.: A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems. Commun. Nonlinear Sci. Numer. Simul. 50, 142–168 (2017). MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brogliato, B.: Nonsmooth Mechanics. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Broucke, M.E., Pugh, C.C., Simić, S.N.: Structural stability of piecewise smooth systems. Geom. Differ. Equ. Dyn. Syst. Comput. Appl. Math. 20(1–2), 51–89 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Colombo, A., Jeffrey, M.R.: The two-fold singularity of discontinuous vector fields. SIAM J. Appl. Dyn. Syst. 8(2), 624–640 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colombo, A., Jeffrey, M.R.: Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows. SIAM J. Appl. Dyn. Syst. 10(2), 423–451 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colombo, A., Jeffrey, M.R.: The two-fold singularity of nonsmooth flows: leading order dynamics in n-dimensions. Phys. D Nonlinear Phenomena 263, 1–10 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Conley, C., Easton, R.: Isolated invariant sets and isolating blocks. Trans. Am. Math. Soc. 158(1), 35–61 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Alphen aan den Rijn (1988)CrossRefGoogle Scholar
  10. 10.
    Gomide, O.M.L., Teixeira, M.A.: Generic singularities of 3D piecewise smooth dynamical systems. In: Lavor, C., Gomes, F.A.M. (eds.) Advances in Mathematics and Applications. Springer, Berlin (2018). Google Scholar
  11. 11.
    Guardia, M., Seara, T., Teixeira, M.: Generic bifurcations of low codimension of planar filippov systems. J. Differ. Equ. 250(4), 1967–2023 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kozlova, V.S.: Roughness of a discontinuous system. Vestinik Moskovskogo Univ. Mat. 5, 16–20 (1984)zbMATHGoogle Scholar
  13. 13.
    Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar filippov systems. Int. J. Bifurc. Chaos 13(8), 2157–2188 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, C.M., Collins, P.J., Krauskopf, B., Osinga, H.M.: Tangency bifurcations of global poincaré maps. SIAM J. Appl. Dyn. Syst. 7(3), 712–754 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pallis, J., de Melo, W.: Geometric Theory of Dynamical Ssystems: An Introduction. Springer, Berlin (1982)CrossRefGoogle Scholar
  16. 16.
    Peixoto, M.C., Peixoto, M.: Structural stability in the plane with enlarged boundary conditions. An. Acad. Bras. Cie. 31, 135 (1959)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sotomayor, J., Teixeira, M.A.: Vector fields near the boundary of a 3-manifold. Dyn. Syst. 1331, 165 (1986)Google Scholar
  18. 18.
    Teixeira, M.A.: Generic bifurcation in manifolds with boundary. J. Differ. Equ. 25, 65 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Teixeira, M.A.: Stability conditions for discontinuous vector fields. J. Differ. Equ. 88, 15 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Teixeira, M.A.: Generic bifurcation of sliding vector fields. J. Math. Anal. Appl. 176, 436 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vishik, S.M.: Vector fields near the boundary of a manifold. Vestnik Moskovskogo Univ. Math. 27(1), 21–28 (1972)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUnicamp, IMECCCampinasBrazil

Personalised recommendations