Advanced search
Start date
Betweenand

On limit cycles in piecewise linear vector fields with algebraic discontinuity variety

Grant number: 21/10606-0
Support type:Scholarships abroad - Research
Effective date (Start): May 01, 2022
Effective date (End): July 31, 2022
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:Douglas Duarte Novaes
Grantee:Douglas Duarte Novaes
Host: Joan Torregrosa I Arus
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Research place: Universitat Autònoma de Barcelona (UAB), Spain  
Associated research grant:18/13481-0 - Geometry of control, dynamical and stochastic systems, AP.TEM

Abstract

The second part of the Hilbert's sixteenth problem consists in determining the upper bound H(n) for the number of limit cycles that planar polynomial vector fields of degree n can have. For n greater than or equal 2, it is still unknown whether H(n) is finite or not. The main achievements obtained so far establish lower bounds for H(n). Regarding asymptotic behavior, the best result says that H(n) grows as fast as n^2 log(n). Better lower bounds for small values of n are known in the research literature. In the recent paper "Some open problems in low dimensional dynamical systems" by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for L(n), which is defined as the maximum number of limit cycles that planar piecewise linear vector fields with two zones separated by a branch of an algebraic curve of degree n can have. So far, the best known general lower bound for L(n) is [n/2]. Again, better lower bounds for small values of n are known in the research literature. Providing upper bounds for L(n) has also been shown to be very challenging, even in the linear case, that is L(1). So far, it is still open whether L(1) is finite or not. Accordingly, the main goals of this project are: 1. To improve the lower bounds for L(n). Here, it is conjectured that L(n) grows as fast as n^2. The main techniques that is going to be used to approach this problem are: a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold; Chebyshev theory for ECT-systems; and Pseudo-Hopf Bifurcation.2. To obtain an upper bound for L(1). The main technique that is going to be used to approach this problem is an integral characterization of the half-return map for linear systems.

News published in Agência FAPESP Newsletter about the scholarship:
Articles published in other media outlets (0 total):
More itemsLess items
VEICULO: TITULO (DATA)
VEICULO: TITULO (DATA)

Please report errors in scientific publications list by writing to: cdi@fapesp.br.