An introduction to the qualitative theory of ordinary differential equations

• Grant number: 23/05369-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: June 01, 2023
• End date: May 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Tiago de Carvalho
• Grantee: Bruno Henrique Garcia Malvestio
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• Associated research grant: 22/02819-6 - Intermittent vector fields: theoretical aspects and applications, AP.PNGP.PI

Abstract

We will study the qualitative theory of ordinary differential equations (ODEs) addressing a series of classical theorems that will alert the candidate to which propertiesare desirable (stability, convergence, hyperbolicity, etc) in ODE systems.We will approach the qualitative/geometric theory introduced in the 19th century by Henri Poincaré. To get to thatobjective, it will be necessary to understand the local theory for nonlinear systems, the Theorem of Existence and Uniqueness of solutions, dependenceof the solutions in relation to the initial conditions and parameters, the flow defined by a differential equation, the Tubular Flow Theorem, linearization of the differential equations, the Theorem ofGrobman-Hartman, limit sets and the Poincaré-Bendixson Theory in R2