Dulac's Problem and of the Focus Center on Two-Dimensional Manifolds

• Grant number: 18/19726-5
• Support Opportunities: Scholarships abroad - Research
• Start date: August 01, 2019
• End date: July 31, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Cláudio Gomes Pessoa
• Grantee: Cláudio Gomes Pessoa
• Host Investigator: Armengol Gasull
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain
• Associated research grant: 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM

Abstract

In this project our goal is the study of some current problems of the Qualitative Theory of the Ordinary Differentials Equations. We will study the Center-Focus Problem in central manifolds of vector fields in R3 and the Dulac's Problem for monodromic singular points of piecewise analytic vector fields in the plane.We will develop a method that allows us to computate the Lyapunov coefficients of a given system in R3 using the Lyapunov coefficients of simpler subsystems. We will use this method to study the cyclicity of centers and foci in central manifolds. We will then focus our efforts on the characterization of centers of some families of rigidi systems in R3. We hope to develop an algorithm to characterize them based on the existence of a comutator for these system.We will characterize the singular monodromic points of piecewise analytic vector fields and hope to solve the Dulac's Problem for these points.The problems will be developed in partnership with the group of dynamic systems of the Autonomous University of Barcelona, which has researchers of recognized scientific trajectory. We can mention the professors Armengol Gasull ("Host Researcher"), Jaume Llibre, and Joan Torregrosa.

Scientific publications

(9)

(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)

MAHDI, ADAM; PESSOA, CLAUDIO; RIBEIRO, JARNE D.. Rigid centres on the center manifold of tridimensional differential systems. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v. N/A, p. 23-pg., 2021-09-07. (18/19726-5, 19/10269-3)

PESSOA, CLAUDIO; RIBEIRO, RONISIO. Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian Differential System with Three Zones. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 32, n. 08, p. 16-pg., 2022-06-30. (19/10269-3, 18/19726-5)

LLIBRE, JAUME; PEREIRA, WEBER F.; PESSOA, CLAUDIO. PHASE PORTRAITS OF BERNOULLI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS. Electronic Journal of Differential Equations, p. 1-19, MAY 22 2020. (18/19726-5, 19/10269-3)

LLIBRE, JAUME; PESSOA, CLAUDIO; RIBEIRO, JARNE D.. Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations. Anais da Academia Brasileira de Ciências, v. 93, n. 4, 2021. (19/10269-3, 18/19726-5)

PESSOA, CLAUDIO; QUEIROZ, LUCAS. Monodromic Nilpotent Singular Points with Odd Andreev Number and the Center Problem. Qualitative Theory of Dynamical Systems, v. 21, n. 4, p. 24-pg., 2022-12-01. (19/13040-7, 19/10269-3, 18/19726-5)

BUZZI, CLAUDIO A.; PESSOA, CLAUDIO; MEDRADO, JOAO C.. Monodromy and Dulac's problem for piecewise analytical planar vector fields. BULLETIN DES SCIENCES MATHEMATIQUES, v. 188, p. 27-pg., 2023-11-01. (18/19726-5, 19/10269-3)

PESSOA, CLAUDIO; QUEIROZ, LUCAS; RIBEIRO, JARNE D.. Cyclicity of rigid centres on centre manifolds of three-dimensional systems. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v. N/A, p. 13-pg., 2023-02-09. (18/19726-5, 19/10269-3, 19/13040-7)

PESSOA, CLAUDIO; RIBEIRO, RONISIO. Limit cycles of planar piecewise linear Hamiltonian differential systems with two or three zones. Electronic Journal of Qualitative Theory of Differential Equations, v. N/A, n. 27, p. 19-pg., 2022-01-01. (18/19726-5, 19/10269-3)

LLIBRE, JAUME; PEREIRA, WEBER F.; PESSOA, CLAUDIO. PHASE PORTRAITS OF BERNOULLI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS. Electronic Journal of Differential Equations, v. N/A, p. 19-pg., 2020-05-22. (18/19726-5, 19/10269-3)