Polycycles in planar vector fields: cyclicity and related problems

• Grant number: 25/08206-4
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: December 01, 2025
• End date: November 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Lucas Queiroz Arakaki
• Supervisor: Daniel Cantergiani Panazzolo
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Université de Haute-Alsace, France
• Associated to the scholarship: 24/06926-7 - Limit Tori in Three-Dimensional Vector Fields, BP.PD

Abstract

In this project, we tackle two problems related to the cyclicity of limit periodic sets: The first one is concerns families of vector fields $X_\mu$ having a hyperbolic polycycle $\Gamma$ for $\mu_0$. When the return map is well defined in the region enclosed by $\Gamma$, there exists a monodromic singularity in this region. Our goal is to study the simultaneous cyclicity of both the polycycle and the singularity for $\mu$ near $\mu_0$. The second problem deals with the finitude of the cyclicity of a class of non generic hyperbolic polycycles inside families of analytical vector fields.