Period function and criticality in planar differential systems

• Grant number: 23/07792-1
• Support Opportunities: Scholarships in Brazil - Master
• Start date: September 01, 2023
• End date: July 28, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Leandro de Jesus Nascimento Bacelar
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM
• Associated scholarship(s): 23/17662-8 - Integrability of centers in dynamical planar systems, BE.EP.MS

Abstract

This project deals with the periodic function of centers. A critical point p of a planar differential system is called a center if it has a neighborhood consisting entirely of periodic orbits. The largest neighborhood with this property is called the annular period of the center. The period function of the center assigns a period to each periodic orbit in the annular period. If the period function is constant, we say that the center is isochronous. Since the period function is defined on the set of periodic orbits at p, in order to study its qualitative properties, the first step is usually to parameterize this set. This can be done, for example, by taking a transversal section at p or, if the differential system has a global first integral, by using the energy level of the periodic orbits. If gamma_s is such a parameterization, the mapping T that associates the period of gamma_s to each gamma_s is a smooth mapping that provides the qualitative properties of the period function. In particular, the existence of critical periods, which are isolated critical points of this function, i.e., for each s_0 in the interval (0, 1) such that T'(s) = a(s - s_0)^k + o((s - s_0)^k) with a = 0 and k > 1. In this case, we say that gamma_s is a critical periodic orbit of multiplicity k of the center. It is worth noting that the number, maximum or minimum, and distribution of critical periods do not depend on the particular parameterization of the set of periodic orbits. A highly investigated question in this context is the maximum number of critical periods that a period function associated with a center can have. This project aims to introduce the student to the topic and investigate how to use the tools studied to study it in families of smooth and piecewise smooth centers.