Scaling law in Saddle-node bifurcations for one-dimensional maps: a complex variable approach
Autor(a) principal: | |
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Data de Publicação: | 2012 |
Outros Autores: | , , |
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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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 info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
metadata only access |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf application/pdf |
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) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
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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1799133402444595200 |