Regularization of discontinuously foliated manifolds

• Grant number: 16/02031-9
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: September 03, 2016
• End date: October 01, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Paulo Ricardo da Silva
• Grantee: Paulo Ricardo da Silva
• Visiting researcher: Daniel Cantergiani Panazzolo
• Visiting researcher institution: Université de Haute-Alsace, France
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant: 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM

Abstract

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)

Scientific publications

(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, DEC 15 2017. (13/24541-0, 16/02031-9)