Investigation of polynomial differential systems: classification, bifurcations and applications

• Grant number: 24/13454-4
• Support Opportunities: Regular Research Grants
• Start date: June 01, 2025
• End date: May 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Alex Carlucci Rezende
• Grantee: Alex Carlucci Rezende
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

We present this research project to request an Individual Research Grant from FAPESP. The object of study of this research project is the polynomial differential systems defined in the plane R^2 and in space R^3. As particular and well-known examples, we highlight Hilbert's 16th Problem, which deals with the search for the number of limit cycles in polynomial differential systems in the plane, and the Lorenz Attractor, defined by a quadratic differential system in space and which presents chaos. Using various tools and approaches, our goal is to contribute to the investigation of the behavior of continuous and piecewise polynomial differential systems defined in the plane and in space. This project employs diverse mathematical tools and encompasses several subjects within the area of Dynamical Systems, such as the classification of structurally unstable quadratic systems of codimension two modulo limit cycles, the classification of quadratic systems possessing invariant algebraic conics, the global behavior of polynomial differential systems in R^3, and the investigation of periodic orbits in smooth and piecewise systems using averaging theory and Lyapunov constants, among others. (AU)