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Generic Singularities of 3D Piecewise Smooth Dynamical Systems

In: Advances in Mathematics and Applications

Author

Listed:
  • Marco Antonio Teixeira

    (University of Campinas, Institute of Mathematics, Statistics and Scientific Computing)

  • Otávio M. L. Gomide

    (University of Campinas, Institute of Mathematics, Statistics and Scientific Computing)

Abstract

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.

Suggested Citation

  • Marco Antonio Teixeira & Otávio M. L. Gomide, 2018. "Generic Singularities of 3D Piecewise Smooth Dynamical Systems," Springer Books, in: Carlile Lavor & Francisco A. M. Gomes (ed.), Advances in Mathematics and Applications, pages 373-404, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-94015-1_15
    DOI: 10.1007/978-3-319-94015-1_15
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