Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures

Solutions of most nonlinear differential equations have to be obtained numerically. The time series obtained by numerical integration will be a solution of the differential equation only if it is independent of the integration step h. A numerical solution is considered to have converged, when the di...

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Autores: Bastos de Figueiredo, Julio Cesar, Diambra, Luis Anibal, Coraci Pereira, Malta
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2011
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/101960
Acceso en línea:http://hdl.handle.net/11336/101960
Access Level:acceso abierto
Palabra clave:Chaotic Attractor
Statistical Measure
Numerical Integration
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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spelling Convergence Criterium of Numerical Chaotic Solutions Based on Statistical MeasuresBastos de Figueiredo, Julio CesarDiambra, Luis AnibalCoraci Pereira, MaltaChaotic AttractorStatistical MeasureNumerical Integrationhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Solutions of most nonlinear differential equations have to be obtained numerically. The time series obtained by numerical integration will be a solution of the differential equation only if it is independent of the integration step h. A numerical solution is considered to have converged, when the difference between the time series for steps h and h/2 becomes smaller as h decreases. Unfortunately, this convergence criterium is unsuitable in the case of a chaotic solution, due to the extreme sensitivity to initial conditions that is characteristic of this kind of solution. We present here a criterium of convergence that involves the comparison of the attractors associated to the time series for integration time steps h and h/2. We show that the probability that the chaotic attractors associated to these time series are the same increases monotonically as the integration step h is decreased. The comparison of attractors is made using 1) the method of correlation integral, and 2) the method of statistical distance of probability distributions.Fil: Bastos de Figueiredo, Julio Cesar. Escola Superior de Propaganda e Marketing; BrasilFil: Diambra, Luis Anibal. Universidad Nacional de La Plata. Centro Regional de Estudios Genómicos; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Coraci Pereira, Malta. Universidade de Sao Paulo; BrasilScientific Research Publishing2011-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/101960Bastos de Figueiredo, Julio Cesar; Diambra, Luis Anibal; Coraci Pereira, Malta; Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures; Scientific Research Publishing; Applied Mathematics; 2; 4; 4-2011; 436-4432152-7385CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.scirp.org/journal/paperinformation.aspx?paperid=4505info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T14:00:43Zoai:ri.conicet.gov.ar:11336/101960instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 14:00:43.599CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
title Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
spellingShingle Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
Bastos de Figueiredo, Julio Cesar
Chaotic Attractor
Statistical Measure
Numerical Integration
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
title_short Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
title_full Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
title_fullStr Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
title_full_unstemmed Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
title_sort Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures
dc.creator.none.fl_str_mv Bastos de Figueiredo, Julio Cesar
Diambra, Luis Anibal
Coraci Pereira, Malta
author Bastos de Figueiredo, Julio Cesar
author_facet Bastos de Figueiredo, Julio Cesar
Diambra, Luis Anibal
Coraci Pereira, Malta
author_role author
author2 Diambra, Luis Anibal
Coraci Pereira, Malta
author2_role author
author
dc.subject.none.fl_str_mv Chaotic Attractor
Statistical Measure
Numerical Integration
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic Chaotic Attractor
Statistical Measure
Numerical Integration
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description Solutions of most nonlinear differential equations have to be obtained numerically. The time series obtained by numerical integration will be a solution of the differential equation only if it is independent of the integration step h. A numerical solution is considered to have converged, when the difference between the time series for steps h and h/2 becomes smaller as h decreases. Unfortunately, this convergence criterium is unsuitable in the case of a chaotic solution, due to the extreme sensitivity to initial conditions that is characteristic of this kind of solution. We present here a criterium of convergence that involves the comparison of the attractors associated to the time series for integration time steps h and h/2. We show that the probability that the chaotic attractors associated to these time series are the same increases monotonically as the integration step h is decreased. The comparison of attractors is made using 1) the method of correlation integral, and 2) the method of statistical distance of probability distributions.
publishDate 2011
dc.date.none.fl_str_mv 2011-04
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/101960
Bastos de Figueiredo, Julio Cesar; Diambra, Luis Anibal; Coraci Pereira, Malta; Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures; Scientific Research Publishing; Applied Mathematics; 2; 4; 4-2011; 436-443
2152-7385
CONICET Digital
CONICET
url http://hdl.handle.net/11336/101960
identifier_str_mv Bastos de Figueiredo, Julio Cesar; Diambra, Luis Anibal; Coraci Pereira, Malta; Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures; Scientific Research Publishing; Applied Mathematics; 2; 4; 4-2011; 436-443
2152-7385
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.scirp.org/journal/paperinformation.aspx?paperid=4505
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Scientific Research Publishing
publisher.none.fl_str_mv Scientific Research Publishing
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 15,812429

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