Time Recurrence Analysis of a Near Singular Billiard

Detalhes bibliográficos
Autor(a) principal: Baroni, Rodrigo Simile [UNESP]
Data de Publicação: 2019
Outros Autores: Carvalho, Ricardo Egydio de [UNESP], Castaldi, Bruno [UNESP], Furlanetto, Bruno [UNESP]
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Institucional da UNESP
Texto Completo: http://dx.doi.org/10.3390/mca24020050
http://hdl.handle.net/11449/186839
Resumo: Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent lambda was calculated using the FTLE method, which for conservative systems, lambda > 0 indicates chaotic behavior and lambda = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater's theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of lambda, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.
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spelling Time Recurrence Analysis of a Near Singular Billiardrecurrence timeSlater's theoremLyapunov exponentpoint scattererannular billiardBilliards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent lambda was calculated using the FTLE method, which for conservative systems, lambda > 0 indicates chaotic behavior and lambda = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater's theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of lambda, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Sao Paulo State Univ UNESP, Inst Geosci & Exact Sci IGCE, Av 24A 1515, Rio Claro, SP, BrazilSao Paulo State Univ UNESP, Inst Geosci & Exact Sci IGCE, Av 24A 1515, Rio Claro, SP, BrazilCNPq: 306034/2015-8MdpiUniversidade Estadual Paulista (Unesp)Baroni, Rodrigo Simile [UNESP]Carvalho, Ricardo Egydio de [UNESP]Castaldi, Bruno [UNESP]Furlanetto, Bruno [UNESP]2019-10-06T07:30:30Z2019-10-06T07:30:30Z2019-06-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article16http://dx.doi.org/10.3390/mca24020050Mathematical And Computational Applications. Basel: Mdpi, v. 24, n. 2, 16 p., 2019.1300-686Xhttp://hdl.handle.net/11449/18683910.3390/mca24020050WOS:00048330740001774977815566223280000-0002-2684-5058Web of Sciencereponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengMathematical And Computational Applicationsinfo:eu-repo/semantics/openAccess2021-10-23T00:00:02Zoai:repositorio.unesp.br:11449/186839Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T14:21:13.474756Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv Time Recurrence Analysis of a Near Singular Billiard
title Time Recurrence Analysis of a Near Singular Billiard
spellingShingle Time Recurrence Analysis of a Near Singular Billiard
Baroni, Rodrigo Simile [UNESP]
recurrence time
Slater's theorem
Lyapunov exponent
point scatterer
annular billiard
title_short Time Recurrence Analysis of a Near Singular Billiard
title_full Time Recurrence Analysis of a Near Singular Billiard
title_fullStr Time Recurrence Analysis of a Near Singular Billiard
title_full_unstemmed Time Recurrence Analysis of a Near Singular Billiard
title_sort Time Recurrence Analysis of a Near Singular Billiard
author Baroni, Rodrigo Simile [UNESP]
author_facet Baroni, Rodrigo Simile [UNESP]
Carvalho, Ricardo Egydio de [UNESP]
Castaldi, Bruno [UNESP]
Furlanetto, Bruno [UNESP]
author_role author
author2 Carvalho, Ricardo Egydio de [UNESP]
Castaldi, Bruno [UNESP]
Furlanetto, Bruno [UNESP]
author2_role author
author
author
dc.contributor.none.fl_str_mv Universidade Estadual Paulista (Unesp)
dc.contributor.author.fl_str_mv Baroni, Rodrigo Simile [UNESP]
Carvalho, Ricardo Egydio de [UNESP]
Castaldi, Bruno [UNESP]
Furlanetto, Bruno [UNESP]
dc.subject.por.fl_str_mv recurrence time
Slater's theorem
Lyapunov exponent
point scatterer
annular billiard
topic recurrence time
Slater's theorem
Lyapunov exponent
point scatterer
annular billiard
description Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent lambda was calculated using the FTLE method, which for conservative systems, lambda > 0 indicates chaotic behavior and lambda = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater's theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of lambda, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.
publishDate 2019
dc.date.none.fl_str_mv 2019-10-06T07:30:30Z
2019-10-06T07:30:30Z
2019-06-01
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://dx.doi.org/10.3390/mca24020050
Mathematical And Computational Applications. Basel: Mdpi, v. 24, n. 2, 16 p., 2019.
1300-686X
http://hdl.handle.net/11449/186839
10.3390/mca24020050
WOS:000483307400017
7497781556622328
0000-0002-2684-5058
url http://dx.doi.org/10.3390/mca24020050
http://hdl.handle.net/11449/186839
identifier_str_mv Mathematical And Computational Applications. Basel: Mdpi, v. 24, n. 2, 16 p., 2019.
1300-686X
10.3390/mca24020050
WOS:000483307400017
7497781556622328
0000-0002-2684-5058
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Mathematical And Computational Applications
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 16
dc.publisher.none.fl_str_mv Mdpi
publisher.none.fl_str_mv Mdpi
dc.source.none.fl_str_mv Web of Science
reponame:Repositório Institucional da UNESP
instname:Universidade Estadual Paulista (UNESP)
instacron:UNESP
instname_str Universidade Estadual Paulista (UNESP)
instacron_str UNESP
institution UNESP
reponame_str Repositório Institucional da UNESP
collection Repositório Institucional da UNESP
repository.name.fl_str_mv Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)
repository.mail.fl_str_mv
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