Invariant manifolds and limit periodic sets of discontinuous foliations

• Grant number: 21/10198-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): January 01, 2023
• Status: Discontinued
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Otavio Henrique Perez
• 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): 24/00392-0 - Sliding cycles of regularized piecewise smooth vector fields having higher order tangency points, BE.EP.PD

Abstract

Slow-fast systems (also known as singular perturbation problems) are a special class of ordinary dierential equations characterized by the existence of solutions with multiple time scales. Besides its wide applicability in many branches of science and engeneering, this class of systems draws attention for its rich and challenging dynamics. This research project aims to answer two questions concerning slow-fast systems. The first one is a conjecture stated in the recent paper [Meza-Sarmiento, Oliveira e Silva em Nonlinear Analysis: Real World Applications 60:103283, 2021] that considers global dynamics of planar quadratic slow-fast systems. The authors characterized the global phase portraits of this class of slow-fast systems in the Poincaré sphere and they conjectured a global version of the Fenichel Theorem. The Fenichel Theorem is a result of major importance in Geometric Singular Perturbation Theory that assures the persistence of compact normally hyperbolic locally invariant manifolds under singular perturbations, as well as the slow dynamics on such manifolds. A global version of this result should assure the persistence of such invariant manifolds in the so called Poincaré Ball, that is, in the finite and in the infinite part of the ambient space. The second problem that this project address is the study of canard solutions in the context of slow-fast systems obtained after regularizations of transition type of discontinuous foliations. Many advances has been done in the context of monotonic regularizations, that is, regularizations that uses monotonic transition functions. However, regularizations of transition type are more general and they can lead to other non normally hyperbolic singularities of slow fast systems that do not appear in the monotonic case. Therefore, conditions on the existence of canards solutions will depend on the transition function adopted.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

PEREZ, OTAVIO HENRIQUE; RONDON, GABRIEL; DA SILVA, PAULO RICARDO. Slow-Fast Normal Forms Arising from Piecewise Smooth Vector Fields. JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, v. N/A, p. 18-pg., 2023-07-17. (21/10198-9, 19/10269-3, 22/12123-9, 20/06708-9, 16/22310-0)