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, pr...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 294; no. 1-2; pp. 419 - 449
Main Authors Gomide, Otávio M. L., Teixeira, Marco A.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2020
Subjects
Mathematics
Mathematics and Statistics
Nonsmooth dynamics
Structural stability
37C20
37C75
37C10
3-manifold
Filippov systems
37C15
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-019-02252-6