Asymptotic behavior of the Hibert number in polynomial and piecewise polynomial planar vector fields

• Grant number: 23/06922-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: August 01, 2023
• End date: July 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Luana Ascoli
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/13481-0 - Geometry of control, dynamical and stochastic systems, AP.TEM

Abstract

Since the publication of Hilbert's list of problems, problem 16, concerning the maximum number H(n) (Hilbert number) of limit cycles of polynomial planar vector fields of degree n, has been one of the most important driving forces for new developments in the qualitative theory of vector fields. The finiteness of H(n), for n greater than or equal to 2, is still an open problem. The main advances in the problem consist in obtaining asymptotic behavior for H(n).Such a problem was also considered in the context of piecewise polynomial planar vector fields. In this context, Hc(n) denotes the maximum number of limit cycles that piecewise polynomial planar vector fields of degree n can have. Providing upper bounds for the maximum number of limit cycles in this context has also been shown to be very challenging, even in the linear case, n=1.In this project, we aim, firstly, a systematic study of classical theories that deal with planar vector fields, in particular, about limit cycles in polynomial systems. This study aims to provide a solid theoretical basis to tackle problems of great interest in the area involving systems with low regularity.