Local generic behavior of a planar Filippov system with non-smooth switching curve

The version of record is available online at: http://dx.doi.org/10.1007/s40863-021-00270-zThis paper discusses the generic local classification of typical singularities of 2D piecewise smooth vector fields when the switching set is an non smooth curve. The main goal is to obtain classification results concerning structural stability and generic codimension one local bifurcations.Tere M. Seara the Catalan Grant 2014SGR504. Tere M-Seara is also supported by the Russian Scientic Foundation grant 14-41-00044 and the European Marie Curie Action FP7-PEOPLE-2012-IRSES: BREUDS. J. Larrosa has been supported by FAPESP grants 2011/22529-8 and 2014/13970-0 and the European Marie Curie Action FP7-PEOPLE-2012-IRSES: BREUDS.Peer ReviewedPostprint (author's final draft

Regularization around a generic codimension one fold-fold singularity

This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor–Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible–invisible case, the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding, specially the appearance and disappearance of periodic orbits that are not present in the Filippov vector field. In the case of a linear regularization, we prove that the regularized system is equivalent to a general slow-fast system studied by Krupa and SzmolyanPeer ReviewedPostprint (published version

Planar Filippov systems and generic bifurcations of low codimension

Orientador: Marco Antonio TeixeiraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação CientíficaResumo: Neste trabalho, abordamos aspectos geométricos e qualitativos da teoria de Sistemas Dinâmicos Suaves por Partes, mais especificamente a classe dos Sistemas Planares de Filippov. é feito um estudo sistemático das singularidades genéricas de um Sistema Planar de Filippov, bem como a noção de estabilidade estrutural local e uma classificação através de equivalências topológicas dos sistemas localmente estruturalmente estáveis. Estudamos ainda bifurcações genéricas locais e globais de codimensão um, apresentando seus desdobramentos genéricos. Além disso, damos uma classificação preliminar de todas as singularidades genéricas de codimensão dois e analisamos detalhadamente seus desdobramentos genéricos e a presença de curvas no espaço dos parâmetros onde ocorrem bifurcações globais de codimensão umAbstract: In this work some qualitative and geometric aspects of piecewise dynamical systems are discussed, specifically the class of Filippov Planar Systems. It is presented a systematic study of generic singularities of this class, as well as the notion of local structural stability and a classification by topological equivalences of the locally structurally stable systems. We also study the codimension-1 generic local and global bifurcations, showing their generic unfolding. Moreover, we give a preliminary classification of all codimension-2 generic singularities and analyze their generic unfolding and the appearance of curves on the parameter space where codimension-1 global bifurcations occursMestradoMatematicaMestre em Matemátic

Bifurcações genéricas em sistemas planares de Filippov

Orientadores: Marco Antonio Teixeira, Maria Tereza Martinez-Seara AlonsoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Considere um sistema planar de Filippov Z=(X,Y), onde X e Y são campos vetoriais suaves definidos em uma vizinhança da origem e cuja curva de descontinuidade é dada pelo conjunto de zeros da função f(x,y)=y. Neste trabalho apresentamos um estudo rigoroso das singularidades do tipo dobra-dobra de um sistema planar de Filippov. Mostramos que, sob algumas condições genéricas, o conjunto dos sistemas de Filippov que possuem uma singularidade dobra-dobra na origem é uma subvariedade mergulhada de codimensão um dentro do conjunto formado por todos os sistemas planares de Filippov definidos em torno da origem. Além disso, mostramos que para Z=(X,Y) pertencente à esta subvariedade todos os seus desdobramentos são equivalentes. Consideramos também sistemas suaves por partes Z=(X,Y) satisfazendo Z(x,y)=X(x,y) se xy> 0 e Z(x,y)=Y(x,y) se xy 0 and Z(x,y)=Y(x,y) if xy<0 . In this case, the discontinuity set is the set of zeros of f(x,y)=xy. For these systems, we present a classification of structurally stable and generic codimension one singularities. In addition, we present the bifurcation diagram of each codimension one singularity and we show that they are, in fact, universal unfoldings. In the sequel we study the Teixeira-Sotomayor regularization of planar Filippov systems having a fold-fold singularity and whose regularization has a critical point around the origin. In this context, we study the nature of this critical point and when the critical point presents a bifurcation, we study the relations between the bifurcation for the planar Filippov system and for the regularized systemDoutoradoMatematicaDoutora em Matemática2011/22529-8FAPES

Regularization around a generic codimension one fold-fold singularity

This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor–Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible–invisible case, the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding, specially the appearance and disappearance of periodic orbits that are not present in the Filippov vector field. In the case of a linear regularization, we prove that the regularized system is equivalent to a general slow-fast system studied by Krupa and SzmolyanPeer Reviewe