Global dynamical aspects of Abel equations of the third kind

• Grant number: 17/01410-9
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: August 15, 2017
• End date: September 04, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Regilene Delazari dos Santos Oliveira
• Visiting researcher: Claudia Valls Angles
• Visiting researcher institution: Instituto Superior Técnico (IST), Portugal
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 14/00304-2 - Singularities of differentiable mappings: theory and applications, AP.TEM

Abstract

This project belongs to the study of quadratic differential systems on the plane. There are more than one thousand papers published in quadratic polynomial differential systems (or simply quadratic systems). The difficulty of studying these differential systems is due to the fact that they depend on twelve parameters. Nowadays there are still many open questions regarding quadratic systems. In this project our objective is threefold for quadratic systems which come from Abel quadratic polynomial differential equations of the third kind. Abel differential equations appear in many text-books of ordinary differential equations as one of first non-trivial examples of non-linear differential equations. The Abel differential equations of the third and second kind have been studied intensively, either calculating their solutions or classifying their centers, and recently studied their polynomial solutions whenever they are reversible. However, nothing is known for the Abel differential equations known as of the third kind. An Abel differential equation of the third kind is of the formy^2 dy/dx= A(x)y+B(x),where A(x) and B(x) non-zero functions. In this project we propose to investigate the global dynamical of the Abel quadratic polynomial differential systems with some kind of symmetry (equivariance or reversibility). (AU)