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Generic Singularities of 3D Piecewise Smooth Dynamical Systems

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

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.

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Copyright information

© 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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