On structural stability of 3D Filippov systems

A semi-local approach

Abstract

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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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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Keywords

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

Mathematics Subject Classification

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