Time Recurrence Analysis of a Near Singular Billiard
Autor(a) principal: | |
---|---|
Data de Publicação: | 2019 |
Outros Autores: | , , |
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. |
id |
UNSP_1f235370e127d437f4cc0882eb997191 |
---|---|
oai_identifier_str |
oai:repositorio.unesp.br:11449/186839 |
network_acronym_str |
UNSP |
network_name_str |
Repositório Institucional da UNESP |
repository_id_str |
2946 |
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 |
|
_version_ |
1808128350726651904 |