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Regularization of planar Filippov Systems near a codimension one singularity

Grant number: 14/13970-0
Support Opportunities:Scholarships abroad - Research Internship - Doctorate
Effective date (Start): September 22, 2014
Effective date (End): March 21, 2015
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Marco Antônio Teixeira
Grantee:Juliana Fernandes Larrosa
Supervisor: Maria Teresa Martínez-Seara Alonso
Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Research place: Universitat Politècnica de Catalunya (UPC), Spain  
Associated to the scholarship:11/22529-8 - Bifurcations of three-parameter families of non-smooth planar systems, BP.DR


The purpose of this project is to study the regularization of Planar Filippov Systems, that is, piecewise smooth systems for which the dynamics over the discontinuity set is given by the Filippov convention. A regularization process for piecewise dynamical systems has been introduced by Teixeira and Sotomayor. This method consists in creating an one parameter family of smooth vector fields which has, in a singular sense, the non-smooth system as a limit.Supposing that the discontinuity set is a regular curve, the first part of this project consists in studying the Teixeira-Sotomayor regularization near all codimension one singularities for these systems. We will study the relations between the topological features (such as sliding orbits, crossing orbits, homoclinic connections, etc) that appear in the unfoldings of the Filippov System and the correspondent regularized system, provided the last one also has a codimension one bifurcation.The second part of this project concerns to adapt the regularization method for systems which the discontinuity set is no longer a regular curve but an algebraic variety expressed by the zeros of a suitable polynomial. We intend to study the regularization near of all typical singularities of codimension zero and one for these systems. This study is important because it was noticed that even two piecewise vector fields are locally equivalent the regularization process can lead to non equivalent regularized systems.The main tools for the development of this project are basic elements of singularity theory of smooth dynamical systems, blowing-up and matching methods, the regularization of non-smooth systems and connection with techniques in singular perturbation theory.Since all the techniques that will be used can be applied to a n-dimensional space, as a future project, we expect to extend the obtained results to higher dimension systems, which have certainly a richer dynamics. (AU)

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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)
BONET-REVES, CARLES; LARROSA, JULIANA; M-SEARA, TERE. Regularization around a generic codimension one fold-fold singularity. Journal of Differential Equations, v. 265, n. 5, p. 1761-1838, . (11/22529-8, 14/13970-0)
LARROSA, JULIANA; M-SEARA, TERE; TEIXEIRA, MARCO ANTONIO. Local generic behavior of a planar Filippov system with non-smooth switching curve. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, . (14/13970-0, 11/22529-8)

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