Physic's differential equations

• Grant number: 21/05770-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2021
• End date: June 30, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Leonardo Lima da Silva
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM

Abstract

Differential equations have a remarkable in the description of phenomena around the world and specially to understand phenomena that involve rates. Some of these equations can be solved using differential calculus methods, but most of them require more sophisticated tools. Furthermore, differential equations can be found, for instance, in Medicine (to model the growth of cancer or the spread of a disease), in Engineering (to describe electricity), in Chemistry (to model chemical reactions), in Economics (to find better investment strategies), in Physics (to describe waves, pendulums or chaotic systems) among other areas.Then being able to solve differential equations could be an important skill for physicists and mathematicians. However, the techniques for solving ordinary differential equations only apply to very particular cases of equations.The qualitative theory of differential equations studies the behavior of differential equations, investigating qualitative aspects of their solutions, not seeking to find explicit solutions. This theory emerged from the work of Poincaré (1881) and Lyapunov (1892). There are relatively few differential equations that can be solved explicitly, but using topology and analysis tools, we can ``solve them'' in a qualitative sense, that is, describing information or properties of such solutions.This project aims to study differential equations that appear in physics using tools from the qualitative theory of odes.