Limit Tori in Three-Dimensional Vector Fields

• Grant number: 24/06926-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2024
• End date: August 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Lucas Queiroz Arakaki
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/13481-0 - Geometry of control, dynamical and stochastic systems, AP.TEM
• Associated scholarship(s): 25/08206-4 - Polycycles in planar vector fields: cyclicity and related problems, BE.EP.PD

Abstract

In this project, we tackle two problems concerning invariant tori for three-dimensional vector fields. The first one investigates the existence of invariant tori for vector fields having a 0-nilpotent singular point, i.e. vector fields whose linear term is yx. We propose the study of the invariant tori bifurcation for these vector fields using the Averaging theory to identify a Neimark-Sacker bifurcation at the associated return map, and importing some techniques from the study of nilpotent singular points to classify the tori in their regularity and stability. The second one consists in dealing with a broader problem: the asymptotic growth of the number N(m) of invariant tori with respect to the degree m of a polynomial three-dimensional differential system.