The 3D cusp-cusp singularity for piecewise smooth vector fields and applications.

• Grant number: 25/04699-6
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: June 01, 2025
• End date: December 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Tiago de Carvalho
• Grantee: Gedeana Pantoja da Silva
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant: 22/02819-6 - Intermittent vector fields: theoretical aspects and applications, AP.PNGP.PI

Abstract

In cancer and HIV treatments, where for a period the patient undergoes intensive treatment and for another period the patient is given rest to allow their body to recover from the side effects, Piecewise Smooth Vector Fields (PSVFs) are used to describe the dynamics of the model. In three-dimensional applied models, with 2 zones where the surface of discontinuity is a plane, it was found that each of the fields, in each one of the 2 zones, has a cubic (isolated) contact with the discontinuity. Moreover, such cubic contacts can coincide, leading to the emergence of a cusp-cusp singularity. In the main problem of this research project, we will seek the bifurcation diagram, the basin of attraction, and determine the asymptotic stability of this singularity, so that we can obtain important qualitative properties regarding the dynamics of the applied model. (AU)