Averaging theory, bifurcations and catastrophes

• Grant number: 23/11002-6
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: March 01, 2024
• End date: February 28, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Pedro Campos Christo Rodrigues Pereira
• Supervisor: Michael Raspady Jeffrey
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: University of Bristol, England
• Associated to the scholarship: 20/14232-4 - Bifurcation of invariant tori of differential systems via higher order averaging theory, BP.DR

Abstract

The Averaging Theory provides tools for studying perturbative systems of differential equations. In particular, by studying the so-called averaged system, qualitative properties of the original perturbed system can be inferred. Most importantly, a number of results relate the existence of compact invariant manifolds of the averaged system with the existence of integral manifolds of the original system. However, such results require generic conditions to be satisfied by the averaged equations. This can be an impediment to their application in the context of parameterized families of equations that undergo bifurcations. We propose to make use of the tools provided by Bifurcation Theory in order to explore which qualitative properties of a family of perturbative differential systems can be inferred from the family of averaged equations. More specifically, we will make use of the Catastrophe Theory in order to attempt a general description of the inheritance of qualitative properties from averaged systems. (AU)