Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach

Detalhes bibliográficos
Autor(a) principal: Duarte, Jorge
Data de Publicação: 2012
Outros Autores: Januário, Cristina, Martins, Nuno, Sardanyés, Josep
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10400.21/5137
Resumo: The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.
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spelling Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approachScaling LawSaddle-Node BifurcationsOne-Dimensional MapsComplex VariableCritical slowing-downIntermittencyCooperationTransitionsHypercyclesExtinctionsModelsGhostsThe study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.SpringerRCIPLDuarte, JorgeJanuário, CristinaMartins, NunoSardanyés, Josep2015-09-10T09:54:23Z2012-012012-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfapplication/pdfhttp://hdl.handle.net/10400.21/5137engDUARTE, J.; [et al] – Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach. Nonlinear Dynamics. ISSN: 0924-090X. Vol. 67, nr. 1 (2012), pp. 541-5470924-090Xmetadata only accessinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-08-03T09:48:07Zoai:repositorio.ipl.pt:10400.21/5137Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T20:14:27.553950Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
title Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
spellingShingle Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
Duarte, Jorge
Scaling Law
Saddle-Node Bifurcations
One-Dimensional Maps
Complex Variable
Critical slowing-down
Intermittency
Cooperation
Transitions
Hypercycles
Extinctions
Models
Ghosts
title_short Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
title_full Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
title_fullStr Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
title_full_unstemmed Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
title_sort Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
author Duarte, Jorge
author_facet Duarte, Jorge
Januário, Cristina
Martins, Nuno
Sardanyés, Josep
author_role author
author2 Januário, Cristina
Martins, Nuno
Sardanyés, Josep
author2_role author
author
author
dc.contributor.none.fl_str_mv RCIPL
dc.contributor.author.fl_str_mv Duarte, Jorge
Januário, Cristina
Martins, Nuno
Sardanyés, Josep
dc.subject.por.fl_str_mv Scaling Law
Saddle-Node Bifurcations
One-Dimensional Maps
Complex Variable
Critical slowing-down
Intermittency
Cooperation
Transitions
Hypercycles
Extinctions
Models
Ghosts
topic Scaling Law
Saddle-Node Bifurcations
One-Dimensional Maps
Complex Variable
Critical slowing-down
Intermittency
Cooperation
Transitions
Hypercycles
Extinctions
Models
Ghosts
description The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.
publishDate 2012
dc.date.none.fl_str_mv 2012-01
2012-01-01T00:00:00Z
2015-09-10T09:54:23Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10400.21/5137
url http://hdl.handle.net/10400.21/5137
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv DUARTE, J.; [et al] – Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach. Nonlinear Dynamics. ISSN: 0924-090X. Vol. 67, nr. 1 (2012), pp. 541-547
0924-090X
dc.rights.driver.fl_str_mv metadata only access
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eu_rights_str_mv openAccess
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dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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