On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system
Autor(a) principal: | |
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Data de Publicação: | 2017 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1007/s11071-016-3277-0 http://hdl.handle.net/11449/173965 |
Resumo: | We consider the well-known Sprott A system, which depends on a single real parameter a and, for a= 1 , was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for a= 0 , the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For a≠ 0 , the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for a> 0 small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the α- and ω-limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for a< 1. Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that for a> 0 , the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered. |
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Repositório Institucional da UNESP |
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On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A systemHidden chaotic attractorHomoclinic and heteroclinic orbitsInvariant algebraic surfacesNested invariant toriSprott A systemWe consider the well-known Sprott A system, which depends on a single real parameter a and, for a= 1 , was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for a= 0 , the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For a≠ 0 , the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for a> 0 small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the α- and ω-limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for a< 1. Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that for a> 0 , the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered.Departamento de Matemática e Computação Faculdade de Ciências e Tecnologia – FCT UNESP Univ Estadual PaulistaDepartamento de Matemática Instituto de Biociências Letras e Ciências Exatas – IBILCE UNESP Univ Estadual PaulistaDepartamento de Matemática e Computação Faculdade de Ciências e Tecnologia – FCT UNESP Univ Estadual PaulistaDepartamento de Matemática Instituto de Biociências Letras e Ciências Exatas – IBILCE UNESP Univ Estadual PaulistaUniversidade Estadual Paulista (Unesp)Messias, Marcelo [UNESP]Reinol, Alisson C. [UNESP]2018-12-11T17:08:33Z2018-12-11T17:08:33Z2017-04-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article807-821application/pdfhttp://dx.doi.org/10.1007/s11071-016-3277-0Nonlinear Dynamics, v. 88, n. 2, p. 807-821, 2017.1573-269X0924-090Xhttp://hdl.handle.net/11449/17396510.1007/s11071-016-3277-02-s2.0-850068733382-s2.0-85006873338.pdf3757225669056317Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengNonlinear Dynamicsinfo:eu-repo/semantics/openAccess2023-10-13T06:04:15Zoai:repositorio.unesp.br:11449/173965Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462023-10-13T06:04:15Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
title |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
spellingShingle |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system Messias, Marcelo [UNESP] Hidden chaotic attractor Homoclinic and heteroclinic orbits Invariant algebraic surfaces Nested invariant tori Sprott A system |
title_short |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
title_full |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
title_fullStr |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
title_full_unstemmed |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
title_sort |
On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system |
author |
Messias, Marcelo [UNESP] |
author_facet |
Messias, Marcelo [UNESP] Reinol, Alisson C. [UNESP] |
author_role |
author |
author2 |
Reinol, Alisson C. [UNESP] |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) |
dc.contributor.author.fl_str_mv |
Messias, Marcelo [UNESP] Reinol, Alisson C. [UNESP] |
dc.subject.por.fl_str_mv |
Hidden chaotic attractor Homoclinic and heteroclinic orbits Invariant algebraic surfaces Nested invariant tori Sprott A system |
topic |
Hidden chaotic attractor Homoclinic and heteroclinic orbits Invariant algebraic surfaces Nested invariant tori Sprott A system |
description |
We consider the well-known Sprott A system, which depends on a single real parameter a and, for a= 1 , was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for a= 0 , the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For a≠ 0 , the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for a> 0 small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the α- and ω-limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for a< 1. Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that for a> 0 , the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered. |
publishDate |
2017 |
dc.date.none.fl_str_mv |
2017-04-01 2018-12-11T17:08:33Z 2018-12-11T17:08:33Z |
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/10.1007/s11071-016-3277-0 Nonlinear Dynamics, v. 88, n. 2, p. 807-821, 2017. 1573-269X 0924-090X http://hdl.handle.net/11449/173965 10.1007/s11071-016-3277-0 2-s2.0-85006873338 2-s2.0-85006873338.pdf 3757225669056317 |
url |
http://dx.doi.org/10.1007/s11071-016-3277-0 http://hdl.handle.net/11449/173965 |
identifier_str_mv |
Nonlinear Dynamics, v. 88, n. 2, p. 807-821, 2017. 1573-269X 0924-090X 10.1007/s11071-016-3277-0 2-s2.0-85006873338 2-s2.0-85006873338.pdf 3757225669056317 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Nonlinear Dynamics |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
807-821 application/pdf |
dc.source.none.fl_str_mv |
Scopus 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_ |
1799964540747644928 |