Qualitative theories and stability of Hill Equation solutions and generalizations

• Grant number: 25/22550-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: January 01, 2026
• End date: December 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: João Henrique Santos de Andrade
• Grantee: Athaide Belo da Silva Santos
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/15567-8 - Qualitative properties for geometric differential equations, AP.R

Abstract

This project aims to understand how patterns of stability and instability arise in differential equations that describe periodic phenomena such as vibrations and waves. The study is based on Hill's equation, a classical model originating in celestial mechanics with applications in physics, engineering, and optics. Periodic solutions and their stability will be analyzed using Floquet theory and qualitative methods from dynamical systems. A fourth-order version of Hill's equation will also be explored, related to beam vibrations and wave propagation. The project combines theoretical study and numerical simulations, contributing to the student's training in mathematical analysis and system stability. (AU)