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Synchrony in coupled systems: a connection between graphs and singularities

Grant number: 19/21230-0
Support type:Scholarships in Brazil - Doctorate
Effective date (Start): August 01, 2020
Effective date (End): February 29, 2024
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Míriam Garcia Manoel
Grantee:Tiago de Albuquerque Amorim
Home 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

The project deals with coupled cell networks, proposing application and advances not only in dynamical systems but also in graph theory and singularities. The relationship between these theories regards the systematic analysis of occurrence of synchrony patterns, which is our main theme of investigation. More precisely, a graph determines the structure of such system, with individual cells represented by its vertices and the coupling between cells represented by the edges. The possible synchrony patterns are determined from the topology of the graph and are configurations that reside in invariant subspaces under the dynamics. Such subspaces are defined through an equivalence relation in the set of vertices of the graph, and it is given by certain symmetries defined from the connections among vertices. Stationary synchronies are of particular interest in the project: they are singularities of the vector field. We intend to develop a systematic treatment of local questions about stability, normal forms and bifurcations, which are theoretical open questions which are, under our view, of significant relevance. As expected results we also include applications of networks that model certain dynamics in the brain. Our questions are in parallel with many recent research carried out in the direction to the understanding of several dynamical aspects in the brain, particularly regarding abnormal behaviour. In theory, certain abnormal synchronies in brain regions characterize the occurrence of Epilepsy. Some references included in the description of this project indicate that this problem can be treated mathematically as a Kuramoto model, which is a model studied recently by the supervisor of the project in the context of coupled networks. We finally mention that this project is related to one of the lines of research coordinated by its supervisor which is part of the FAPESP Thematic project "Singularity theory and applications to differential geometry, differential equations and computer vision". (AU)