Averaging Methods and Degenerate Bifurcations in Differential Systems

• Grant number: 25/15366-8
• Support Opportunities: Scholarships abroad - Research Internship - Master's degree
• Start date: November 01, 2025
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Murilo Rodolfo Cândido
• Grantee: Gean Franco Acuna de La Cruz
• Supervisor: Joan Torregrosa I Arus
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain
• Associated to the scholarship: 24/04433-3 - Detecting k-hyperbolic orbits and period-doubling bifurcations, BP.MS

Abstract

This project aims to study various averaging methods for the detection of periodic solutions in differential systems and to perform a comparison with other existing techniques. In the first stage, the classical first-order averaging method and its applications to zero-Hopf bifurcations in $\mathbb{R}^n$, Hamiltonian systems, and climate phenomena will be addressed. Then, higher-order averaging in arbitrary dimensions will be studied, including a proof of the general theorem of order~$n$. Applications will include the Hénon--Heiles Hamiltonian system, the existence of limit cycles in polynomial differential systems, and their relation to Hilbert's 16th problem. Moreover, extensions of the averaging theorem using the Lyapunov--Schmidt reduction method will be investigated. Computer algebra software will be employed to obtain analytical results on the existence of limit cycles in degenerate Hopf bifurcations, with emphasis on Kukles-type systems. (AU)