Networks of coupled dynamical systems

• Grant number: 23/09279-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2023
• End date: July 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Míriam Garcia Manoel
• Grantee: Nícolas da Rocha Brito
• 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

Abstract

This project deals with the introductory study of dynamical systems, with special interest to coupled dynamical systems, also called networks, given from individual systems (cells) that interact and react according to the laws of coupling. The foundations of this project are the qualitative theory of ordinary differential equations and the algebraic theory of graphs. In general terms, a network is a graph whose vertices represent the individual systems and whose edges represent the couplings between these systems. Synchrony, a natural concept in network dynamics and also contemplated in this project, is a partition of graph vertices into subsets (often called clusters) so that all cells in the same cluster are synchronous. An autonomous system that describes such an interaction dynamics is given from a special class of vector fields, called admissible vector fields, which is the central object of this project. From beginning to end, this work should be filled with examples of synchronization phenomena in nature, in physics, and in simple mathematical models. A model to be investigated in greater detail is the Kuramoto's model with a coupling of six oscillators.