Averaging theory for finding periodic solutions of Carathéodory differential equations via topological degree theory

• Grant number: 22/04184-8
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: August 01, 2022
• End date: July 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Francisco Bruno Gomes da Silva
• Supervisor: Armengol Gasull
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain
• Associated to the scholarship: 18/22689-4 - Topological methods in the study of periodic solutions of non-smooth differential equations, BP.DR

Abstract

Motivated by recent works generalizing the averaging theory for low regularity differential equations, this project proposes a higher order analysis for a family of perturbed Carathéodory differential equations aiming at finding sufficient conditions to guarantee the existence of periodic solutions of these equations that bifurcate from the non-perturbed system. It is also proposed to investigate how these periodic solutions converge as the perturbation parameter tends to zero. It is hoped that further ahead the results obtained in this research can be applied in problems of optimal control theory, where Carathéodory differential equations appear naturally. (AU)