Integrability, center, and cyclicity problems for planar analytical vector fields

• Grant number: 23/05686-0
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: August 28, 2023
• End date: August 27, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Ana Livia Rodero
• Supervisor: Jaume Gine Mesa
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Universidad de Lleida (UdL), Spain
• Associated to the scholarship: 21/12630-5 - Cyclicity and local structural stability of piecewise vector fields, BP.PD

Abstract

The integrability problem consists of explicitly finding a non-constant function that is constant on the orbits of a given system. For planar differential systems, the integrability theorypar excellence is Darboux's. Generalizations of this theory have been achieved during thelast few years, such as the integrability based on generalized cofactors and the Weierstrassintegrability, for example. Recently, a new integrability theory based on system solutionsthat could be written in the Puiseux series was developed and called Puiseux integrability. The main goal of this project is extending and applying recent integrability theories, as the Weierstrass and the Pueiseux ones in the study of the center and the cyclicity problems forplanar analytic vector fields. (AU)