Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Da Silva, Paulo R. (IBILCE-UNESP (Brazil). Departamento de Matemática)
Teixeira, Marco Antonio (IMECC-UNICAMP (Brazil))

Date:	2015
Abstract:	We consider a differential equation ˙p = X(p), p ϵ R3 with discontinuous right-hand side and discontinuities occurring on an algebraic variety ∑. We discuss the dynamics of the sliding mode which occurs when for any initial condition near p ϵ ∑ the corresponding solution trajectories are attracted to ∑. First we suppose that ∑ = H-1(0) where H is a polynomial function and 0 ϵ R is a regular value. In this case ∑ is locally di↵eomorphic to the set F = {(x, y, z) ϵ R3; z = 0} (Filippov). Second we suppose that ∑ is the inverse image of a non-regular value. We focus our attention to the equations defined around singularities as described in [8]. More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of R3 in terms of implicit functions and immersions are broken in a stable manner. In this case ∑ is locally diffeomorphic to one of the following sets D = {(x, y, z) ϵ R3; xy = 0} (double crossing); T = {(x, y, z) ϵ R3; xyz = 0} (triple crossing); C = {(x, y, z) ϵ R3; z2-x2-y2 = 0}(cone) or W = {(x, y, z) ϵ R3; zx2-y2 = 0} (Whitney's umbrella).
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-410 European Commission 316338 European Commission 318999
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Non-smooth vector fields ; Singular perturbation ; Sliding vector field ; Regularization ; Vector fields ; Manifolds with simple singularities
Published in:	Nonlinearity, Vol. 28 (2015) , p. 493-507, ISSN 1361-6544

DOI: 10.1088/0951-7715/28/2/493

Postprint 18 p, 1.2 MB

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