Bifurcation of limit cycles in smooth piecewise systems and an application in Medicine

• Grant number: 21/14987-8
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: March 01, 2022
• End date: February 05, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Regilene Delazari dos Santos Oliveira
• Grantee: Leonardo Pereira Costa da Cruz
• 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): 22/14484-9 - Applications in biology of piecewise differential equations, BE.EP.PD

Abstract

This project concern to the Piecewise-Smooth dynamical system (PWS). The PWS studies have gained prominence in recent years because they proved to be more efficient than smooth systems in modeling physical phenomena in different areas of knowledge. More details on this topic and its applications can be found in [1,2]. This project aims to contribute to the understanding of the PWS. For that, the current project is divided into three sub-projects: the first one is an extension of the center problem, and it aims to investigate a perturbative problem. More specifically, the subproject intends to investigate the bifurcation of limit cycles for a PWS family of centers, looking for the Hilbert number. The second subproject proposes an application of the piecewise of smooth ordinary differential equations to treat prostate cancer and other alternative problems. Finally, the third subproject is related to the normal forms and bifurcation of limit cycles in a symmetrical piecewise smooth system in four zones in this case for linear and quadratic polynomial systems. Therefore, through the use, among others, of classic tools such as averaging theory, Lyapunov constants, and recent advances, we intend to develop this project. (AU)

Scientific publications

(5)

(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)

DA CRUZ, LEONARDO P. C.; LLIBRE, JAUME. Global centres in a class of quintic polynomial differential systems. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v. N/A, p. 16-pg., 2024-04-11. (21/14987-8, 22/14484-9)

DA CRUZ, LEONARDO PEREIRA COSTA; TORREGROSA, JOAN; BERDUGO, MIGUEL; SARDANYES, JOSEP. Resource-consumer dynamics in drylands: Modeling the role of plant-plant facilitation-competition shifts with a piecewise system. PHYSICA D-NONLINEAR PHENOMENA, v. 473, p. 11-pg., 2025-03-01. (22/14484-9, 21/14987-8)

BRAUN, FRANCISCO; DA CRUZ, LEONARDO PEREIRA COSTA; TORREGROSA, JOAN. On the number of limit cycles in piecewise planar quadratic differential systems. NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, v. 79, p. 15-pg., 2024-05-04. (22/14484-9, 19/07316-0, 21/14987-8, 20/14498-4, 23/00376-2)

CARVALHO, YAGOR ROMANO; DA CRUZ, LEONARDO P. C.; GOUVEIA, LUIZ F. S.. New lower bound for the Hilbert number in low degree Kolmogorov systems. CHAOS SOLITONS & FRACTALS, v. 175, p. 9-pg., 2023-08-19. (22/14484-9, 20/04717-0, 21/14987-8, 22/03800-7, 21/14695-7, 22/03801-3)

DA CRUZ, LEONARDO P. C.; REZENDE, ALEX C.; TORREGROSA, JOAN. Coexistence of Analytic and Piecewise Analytic Limit Cycles in Planar Piecewise Quadratic Differential Systems. Qualitative Theory of Dynamical Systems, v. 24, n. 2, p. 19-pg., 2025-04-01. (19/21181-0, 21/14987-8, 22/14484-9)