Cyclicity and local structural stability of piecewise vector fields

• Grant number: 21/12630-5
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): May 01, 2022
• Effective date (End): April 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Ana Livia Rodero
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM
• Associated scholarship(s): 23/05686-0 - Integrability, center, and cyclicity problems for planar analytical vector fields, BE.EP.PD

Abstract

The main goal of this project is the study of piecewise smooth vector fields defined in 2, 3 and 4-dimensional manifolds, in two distinct lines which will be addressed simultaneously. The goal of the first line of this project is to study the cyclicity problem for a system having multiple period rings, searching for an upper bound for the number of limit cycles that can bifurcate, simultaneously, of these rings. For the second line, we consider a piecewise smooth vector field Y=(X^+, X^-) defined in a 3-dimensional manifold. We would like to characterize the local structurally stable singularities for the system Y when we suppose that both X^+ and X^- are completely integrable and that Y is refractive. In the same context, we consider a piecewise-smooth refractive system Y defined in R^4 (under the hypoteses that X^+ and X^- are completely integrable and Y is refractive) and we would like to give initial answers about local stability and local bifurcations of codimension 1. (AU)