On structural stability of 3D Filippov systems

A semi-local approach


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.

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  1. 1.

    Bernardo, M.D., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems Theory and Applications. Springer, Berlin (2008)

    Google 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). https://doi.org/10.1016/j.cnsns.2017.02.014

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brogliato, B.: Nonsmooth Mechanics. Springer, New York (1999)

    Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Conley, C., Easton, R.: Isolated invariant sets and isolating blocks. Trans. Am. Math. Soc. 158(1), 35–61 (1971)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Alphen aan den Rijn (1988)

    Google 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). https://doi.org/10.1007/978-3-319-94015-1_15

    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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kozlova, V.S.: Roughness of a discontinuous system. Vestinik Moskovskogo Univ. Mat. 5, 16–20 (1984)

    MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Pallis, J., de Melo, W.: Geometric Theory of Dynamical Ssystems: An Introduction. Springer, Berlin (1982)

    Google Scholar 

  16. 16.

    Peixoto, M.C., Peixoto, M.: Structural stability in the plane with enlarged boundary conditions. An. Acad. Bras. Cie. 31, 135 (1959)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Teixeira, M.A.: Stability conditions for discontinuous vector fields. J. Differ. Equ. 88, 15 (1990)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Teixeira, M.A.: Generic bifurcation of sliding vector fields. J. Math. Anal. Appl. 176, 436 (1993)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Vishik, S.M.: Vector fields near the boundary of a manifold. Vestnik Moskovskogo Univ. Math. 27(1), 21–28 (1972)

    MATH  Google Scholar 

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Correspondence to Otávio M. L. Gomide.

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Otávio M. L. Gomide is supported by the FAPESP Grants 2015/22762-5 and 2016/23716-0. Marco A. Teixeira is supported by the FAPESP Grant 2012/18780-0 and by the CNPq Grant 300596/2009-0.

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Gomide, O.M.L., Teixeira, M.A. On structural stability of 3D Filippov systems. Math. Z. 294, 419–449 (2020). https://doi.org/10.1007/s00209-019-02252-6

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  • Structural stability
  • Filippov systems
  • 3-manifold
  • Nonsmooth dynamics

Mathematics Subject Classification

  • 37C10
  • 37C15
  • 37C20
  • 37C75