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On structural stability of 3D Filippov systems. A semi-local approach. (English) Zbl 1436.37029

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.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C75 Stability theory for smooth dynamical systems
37C10 Dynamics induced by flows and semiflows
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C20 Generic properties, structural stability of dynamical systems
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