Generic Singularities of 3D Piecewise Smooth Dynamical Systems

  • Marco Antonio TeixeiraEmail author
  • Otávio M. L. Gomide


The aim of this paper is to provide a discussion on the current directions of research involving typical singularities of 3D nonsmooth vector fields. A brief survey of known results is also presented.

We describe the dynamical features of a fold–fold singularity in its most basic form and we give a complete and detailed proof of its local structural stability (or instability). In addition, classes of all topological types of a fold–fold singularity are intrinsically characterized. Such proof essentially follows from some lines laid out by Colombo, García, Jeffrey, Teixeira, and others and it offers a rigorous mathematical treatment under clear and crisp assumptions and solid arguments.

One should highlight that the geometric–topological methods employed lead us to the mathematical understanding of the dynamics around a T-singularity. This approach lends itself to applications in generic bifurcation theory. It is worth to say that such subject is still poorly understood in higher dimension.


  1. 1.
    A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of bifurcations of dynamic systems on a plane. John Wiley and Sons, 1971.Google Scholar
  2. 2.
    D. K. Arrowsmith, C. M. Place, An introduction to dynamical systems. Cambridge University Press, 1990.Google Scholar
  3. 3.
    W. Blaschke, G. Bol, Geometrie der gewebe: Topologische fragen der Differential Geometrie. Springer, 1938.Google Scholar
  4. 4.
    C. Bonatti, M. A. Teixeira, Topological equivalence of diffeomorphisms and curves. Journal of Differential Equations, vol. 118, 371–379, 1995.Google Scholar
  5. 5.
    C. A. Buzzi, J. C. R. Medrado, M. A. Teixeira, Generic bifurcation of refracted systems. Advances in Mathematics, vol. 234, 653–666, 2013.MathSciNetCrossRefGoogle Scholar
  6. 6.
    I. Ekeland, Discontinuité des champs Hamiltoniens er existence de solutions optimales en calcul des variations. Pub. IHES, 47, 5–32, 1977.CrossRefGoogle Scholar
  7. 7.
    A. Colombo, M. R. Jeffrey, The two-fold singularity of discontinuous vector fields. SIAM J. Applied Dynamical Systems, 8, 2, 624–640, 2009.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Colombo, M. R. Jeffrey, Nondeterministic Chaos, and the Two-fold Singularity in Piecewise Smooth Flows. SIAM J. Applied Dynamical Systems, 10, 2, 423–451, 2011.MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Colombo, M. R. Jeffrey, The two-fold singularity of nonsmooth flows: leading order dynamics in n-dimensions. Physica D: Nonlinear Phenomena, 263, 1–10, 2013.MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. F. Filippov, Differential equations with discontinuous righthand sides. Kluwer, 1988.Google Scholar
  11. 11.
    S. Fernández-García, D. Angulo García, G. Olivar Tost, M. di Bernardo, M. R. Jeffrey, Structural stability of the two-fold singularity, SIAM Journal on Applied Dynamical Systems, 2012, 11, 4, 1215–1230, 2012.Google Scholar
  12. 12.
    M. Guardia, T. M. Seara, M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems. J. Differential Equations, 250 (2011).Google Scholar
  13. 13.
    P. Hartman, On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana (2) 5, 1960.Google Scholar
  14. 14.
    V. S. Kozlova, Roughness of a discontinuous system. Vestinik Moskovskogo Universiteta, Matematika 5, 16–20, 1984.Google Scholar
  15. 15.
    Y. A. Kuznetsov, S. Rinaldi, A. Gragnani, One-parameter bifurcations in planar Filippov systems. International Journal of Bifurcation and Chaos, vol. 13, 8, 2157–2188, 2003.MathSciNetCrossRefGoogle Scholar
  16. 16.
    W. Melo, Moduli of stability of two-dimensional diffeomorphisms. Topology, 19 (1), 9–21, 1980.MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Palis, A differentiable invariant of topological conjugacies and moduli of stability. Astérisque, 51, 1978.Google Scholar
  18. 18.
    M. C. Peixoto, M. M. Peixoto, Structural stability in the plane with enlarged boundary conditions, An. Acad. Bras. Ciencias, 31, 1959.Google Scholar
  19. 19.
    P. B. Percell, Structural stability on manifolds with boundary, Topology, 12, 123–144, 1973.MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. V. Pereira, L. Pirio, An introduction to web geometry , IMPA Monographs, Springer, 2015.zbMATHGoogle Scholar
  21. 21.
    E. Ponce, R. Cristiano, D. Pagano, E. Freire, The Teixeira singularity degeneracy and its bifurcation in Piecewise Linear systems , Fourth Symposium on Planar Vector Fields, 2016.Google Scholar
  22. 22.
    J. Sotomayor, M. A. Teixeira, Vector fields near the boundary of a 3-manifold, Dynamical systems, vol 1331, 169–195, 1988.MathSciNetzbMATHGoogle Scholar
  23. 23.
    M. A. Teixeira, Generic bifurcation in manifolds with boundary. J. Differential Equations, 25 (1977).Google Scholar
  24. 24.
    M. A. Teixeira, Local and simultaneous structural stability of certain diffeomorphisms. Dynamical Systems and Turbulence, Warwick 1980, pp.382–390, 1981.MathSciNetGoogle Scholar
  25. 25.
    M. A. Teixeira, Generic singularities of discontinuous vector fields. Anais da Academia Brasileira de Ciências, 53(2), 1981.Google Scholar
  26. 26.
    M. A. Teixeira, On topological stability of divergent diagrams of folds. Mathematische Zeitschrift, 180 (2), 361–371, 1982.MathSciNetCrossRefGoogle Scholar
  27. 27.
    M. A. Teixeira, Stability conditions for discontinuous vector fields. J. Differential Equations, 88 (1990).Google Scholar
  28. 28.
    M. A. Teixeira, Generic bifurcation of sliding vector fields. Journal of Mathematical Analysis and Application, 176 (1993).Google Scholar
  29. 29.
    M. A. Teixeira, Divergent diagrams of folds and simultaneous conjugacy of involutions. Discrete and Continuous Dynamical Systems, 12 (4), 657–674, 2005.MathSciNetCrossRefGoogle Scholar
  30. 30.
    M. A. Teixeira, Perturbation Theory for Non-smooth Systems. Encyclopedia of Complexity and Systems Science, Springer New York, 6697–6709, 2009.Google Scholar
  31. 31.
    S. M. Vishik, Vector fields near the boundary of a manifold. Vestnik Moskovskogo Universiteta Mathematika, 27(1), 21–28, 1972.zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Marco Antonio Teixeira
    • 1
    Email author
  • Otávio M. L. Gomide
    • 1
  1. 1.Institute of Mathematics, Statistics and Scientific ComputingUniversity of CampinasCampinasBrazil

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