The graph representation problem for the investigation of synchronies in networks
Autor(a) principal: | |
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Data de Publicação: | 2024 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
Texto Completo: | https://www.teses.usp.br/teses/disponiveis/55/55135/tde-14052024-140535/ |
Resumo: | A coupled cells network is a graph endowed with an input-equivalence relation on the set of vertices that enables a characterization of the admissible vector fields that rules the network dynamics according to the coupling types of that graph. In this context, this thesis has two targets. The first one goes in the direction of answering an inverse problem: for n ≥ 2, any mapping on Rn can be realized as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) n. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to an appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realize couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators. As a second target, from the impact of the results about synchronization in Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix as a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian networks on a ring with some extra couplings. |
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The graph representation problem for the investigation of synchronies in networksO problema da construção de grafos para o estudo de sincronias em redesAdmissible vector fieldCampo vetorial admissívelGrafoGraphLaplacian matrixMatrix LaplacianaNetworkRedeSimetriaSincroniaSingularidadeSingularitySymmetrySynchronyA coupled cells network is a graph endowed with an input-equivalence relation on the set of vertices that enables a characterization of the admissible vector fields that rules the network dynamics according to the coupling types of that graph. In this context, this thesis has two targets. The first one goes in the direction of answering an inverse problem: for n ≥ 2, any mapping on Rn can be realized as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) n. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to an appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realize couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators. As a second target, from the impact of the results about synchronization in Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix as a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian networks on a ring with some extra couplings.Uma rede de células é um grafo dotado de uma relação de equivalência que preserva o conjunto de entrada dos vértices que permite uma caracterização dos campos vetoriais admissíveis que regem a dinâmica da rede de acordo com os tipos de acoplamento desse grafo. Neste contexto, esta tese tem dois objetivos. O primeiro vai no sentido de responder ao problema inverso: para n ≥ 2, qualquer mapa em Rn pode ser realizado como um campo vetorial admissível para algum grafo com o número de vértices dependendo de (mas não necessariamente igual a) n. Dado um mapa, apresentamos um procedimento para construir todos os grafos admissíveis não equivalentes, para uma relação de equivalência apropriada. Também fornecemos um limite superior para o número de tais grafos. Como consequência, subespaços invariantes sob o campo vetorial podem ser investigados como o lugar geométrico dos estados de sincronia de um grafo admissível, no sentido de que um grafo adequado pode ser escolhido para realizar acoplamentos com mais (ou menos) sincronia do que outro grafo admissível para o mesmo campo vetorial. A abordagem fornece, em particular, uma investigação sistemática da ocorrência de estados de quimera em uma rede de osciladores idênticos de van der Pol. Como segundo objetivo, a partir do impacto dos resultados de sincronização das redes de Kuramoto, introduzimos a classe generalizada de redes Laplacianas, governadas por mapas cujo Jacobiano em qualquer ponto é uma matriz simétrica no qual cada linha tem soma nula de suas entradas. Ao reconhecer esta matriz como um Laplaciano com pesos do grafo associado, deduzimos estimativas ótimas de seus autovalores positivos, nulos e negativos diretamente da topologia do grafo. Além disso, fornecemos uma caracterização dos mapas que definem as redes Laplacianas. Por último, discutimos a estabilidade do equilíbrio dentro de subespaços de sincronia para dois tipos de redes Laplacianas em um anel com alguns acoplamentos extras.Biblioteca Digitais de Teses e Dissertações da USPManoel, Miriam GarciaAmorim, Tiago de Albuquerque2024-03-04info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/55/55135/tde-14052024-140535/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2024-05-14T17:20:02Zoai:teses.usp.br:tde-14052024-140535Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212024-05-14T17:20:02Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
dc.title.none.fl_str_mv |
The graph representation problem for the investigation of synchronies in networks O problema da construção de grafos para o estudo de sincronias em redes |
title |
The graph representation problem for the investigation of synchronies in networks |
spellingShingle |
The graph representation problem for the investigation of synchronies in networks Amorim, Tiago de Albuquerque Admissible vector field Campo vetorial admissível Grafo Graph Laplacian matrix Matrix Laplaciana Network Rede Simetria Sincronia Singularidade Singularity Symmetry Synchrony |
title_short |
The graph representation problem for the investigation of synchronies in networks |
title_full |
The graph representation problem for the investigation of synchronies in networks |
title_fullStr |
The graph representation problem for the investigation of synchronies in networks |
title_full_unstemmed |
The graph representation problem for the investigation of synchronies in networks |
title_sort |
The graph representation problem for the investigation of synchronies in networks |
author |
Amorim, Tiago de Albuquerque |
author_facet |
Amorim, Tiago de Albuquerque |
author_role |
author |
dc.contributor.none.fl_str_mv |
Manoel, Miriam Garcia |
dc.contributor.author.fl_str_mv |
Amorim, Tiago de Albuquerque |
dc.subject.por.fl_str_mv |
Admissible vector field Campo vetorial admissível Grafo Graph Laplacian matrix Matrix Laplaciana Network Rede Simetria Sincronia Singularidade Singularity Symmetry Synchrony |
topic |
Admissible vector field Campo vetorial admissível Grafo Graph Laplacian matrix Matrix Laplaciana Network Rede Simetria Sincronia Singularidade Singularity Symmetry Synchrony |
description |
A coupled cells network is a graph endowed with an input-equivalence relation on the set of vertices that enables a characterization of the admissible vector fields that rules the network dynamics according to the coupling types of that graph. In this context, this thesis has two targets. The first one goes in the direction of answering an inverse problem: for n ≥ 2, any mapping on Rn can be realized as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) n. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to an appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realize couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators. As a second target, from the impact of the results about synchronization in Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix as a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian networks on a ring with some extra couplings. |
publishDate |
2024 |
dc.date.none.fl_str_mv |
2024-03-04 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://www.teses.usp.br/teses/disponiveis/55/55135/tde-14052024-140535/ |
url |
https://www.teses.usp.br/teses/disponiveis/55/55135/tde-14052024-140535/ |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
|
dc.rights.driver.fl_str_mv |
Liberar o conteúdo para acesso público. info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Liberar o conteúdo para acesso público. |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.coverage.none.fl_str_mv |
|
dc.publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
instname_str |
Universidade de São Paulo (USP) |
instacron_str |
USP |
institution |
USP |
reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
collection |
Biblioteca Digital de Teses e Dissertações da USP |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
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1809090710340632576 |