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Regularization of discontinuously foliated manifolds


A finite set of vector fields, defines a piecewise-smooth vector field X on a manifold M and switching set E. A regularization is a 1-parameter family of smooth vector fields, converging to X, when the parameter goes to zero.We study regularization processes when the switching set is a regular surface and when its singular set is not empty. We introduce the concept of discontinuously foliated manifold and regularization of piecewise smooth oriented 1-foliation.The regularization processes studied are of the kinds Filippov, Nonlinear and the blow-up.We show that the regularization of the kind blow up gives a manifold on whichthe singular fiber has an invariant manifold S. The reduced flow on Sis equivalent to the sliding flow when S is normally hyperbolic. The case where S it is not normally hyperbolic appears when non Filippov regularizations are considered. (AU)

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(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
PANAZZOLO, DANIEL; DA SILVA, PAULO R.. Regularization of discontinuous foliations: Blowing up and sliding conditions via Fenichel theory. Journal of Differential Equations, v. 263, n. 12, p. 8362-8390, . (13/24541-0, 16/02031-9)

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