Analysis of Functional Integral Equations, Generalized Ordinary Differential Equations, Impulsive Semidynamic Systems and Applications

• Grant number: 25/01812-6
• Support Opportunities: Regular Research Grants
• Start date: July 01, 2025
• End date: June 30, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Suzete Maria Silva Afonso
• Grantee: Suzete Maria Silva Afonso
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

This project aims at theoretical and applied advances in the study of functional integral equations, generalized ordinary differential equations, and impulsive semidynamical systems-key research lines for the mathematical modeling of complex dynamic phenomena. The research is structured around three main axes: (A) stability analysis of solutions to functional integral equations, (B) topological dynamics in linear generalized ordinary differential equations, and (C) ergodic theory applied to impulsive semidynamical systems.In the first axis, we will investigate the stability of solutions using fixed-point theorems and integration in the sense of Perron, expanding on classical Lyapunov approaches. For generalized ordinary differential equations, we will explore the existence of local flows and dynamic properties, with applications in mathematical modeling and control systems. Regarding the study of impulsive semidynamical systems, we will analyze topological pressure, the uniqueness of equilibrium states, and the relationship between sets of periodic points, strengthening the connection between ergodic theory and discontinuous dynamical systems.The project will involve strategic collaborations, bringing together researchers from renowned institutions such as the University of Seville, USP, UFRJ, and Masaryk University, fostering a high-level scientific exchange. The expected outcomes include the development of new stability criteria, the characterization of the dynamic structure of generalized ordinary differential equations, and the formulation of sufficient conditions for impulsive semiflows to exhibit properties such as expansivity and specification. Additionally, we aim to establish connections between the sets of periodic points in continuous and impulsive semiflows, expanding the understanding of their dynamics.Beyond its theoretical impact, the research to be conducted anticipates applications in biology, economics, and physics, contributing to the understanding of complex systems in fields such as epidemiology, neural networks, and the modeling of phenomena with memory effects. The results will be disseminated through publications in high-standard peer-reviewed journals and presentations at national and international conferences. The support of FAPESP will be crucial to consolidating these advances and expanding the global projection of Brazilian research in Mathematics. (AU)