On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps

We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H^{\pm}_3: \bar x=y, \bar y=−x+M_1+M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues $e^{±i2π/3}$. It follows from [1] that this resonance is d...

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Autores: Gonchenko, Marina, Kazakov, Alexey O., Samylina, Evgeniya A., Shykhmamedov, Aikan
Formato: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2022
País:España
Recursos:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/194357
Acesso em linha:https://hdl.handle.net/2445/194357
Access Level:acceso abierto
Palavra-chave:Teoria de la bifurcació
Sistemes dinàmics diferenciables
Teoria ergòdica
Bifurcation theory
Differentiable dynamical systems
Ergodic theory
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spelling On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon MapsGonchenko, MarinaKazakov, Alexey O.Samylina, Evgeniya A.Shykhmamedov, AikanTeoria de la bifurcacióSistemes dinàmics diferenciablesTeoria ergòdicaSistemes dinàmics diferenciablesBifurcation theoryDifferentiable dynamical systemsErgodic theoryDifferentiable dynamical systemsWe consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H^{\pm}_3: \bar x=y, \bar y=−x+M_1+M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues $e^{±i2π/3}$. It follows from [1] that this resonance is degenerate for $M_1=0, M_2=−1$ when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map $H^+_3$ and elliptic orbits in the case of map $H^−_3$), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map $H^+_3$ and saddles with the Jacobians less than 1 and greater than 1 in the case of map $H^−_3$). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [1] to the case of the p:q resonances with odd q and show that all of them are also degenerate for the maps $H^\pm_3$ with $M_1=0$. .Pleiades Publishing2022info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2445/194357Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésVersió postprint del document publicat a: https://doi.org/10.1134/S1560354722020058Regular and Chaotic Dynamics, 2022, vol. 27, num. 2, p. 198-216https://doi.org/10.1134/S1560354722020058(c) Pleiades Publishing, 2022info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1943572026-05-27T06:46:51Z
dc.title.none.fl_str_mv On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
title On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
spellingShingle On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
Gonchenko, Marina
Teoria de la bifurcació
Sistemes dinàmics diferenciables
Teoria ergòdica
Sistemes dinàmics diferenciables
Bifurcation theory
Differentiable dynamical systems
Ergodic theory
Differentiable dynamical systems
title_short On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
title_full On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
title_fullStr On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
title_full_unstemmed On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
title_sort On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
dc.creator.none.fl_str_mv Gonchenko, Marina
Kazakov, Alexey O.
Samylina, Evgeniya A.
Shykhmamedov, Aikan
author Gonchenko, Marina
author_facet Gonchenko, Marina
Kazakov, Alexey O.
Samylina, Evgeniya A.
Shykhmamedov, Aikan
author_role author
author2 Kazakov, Alexey O.
Samylina, Evgeniya A.
Shykhmamedov, Aikan
author2_role author
author
author
dc.subject.none.fl_str_mv Teoria de la bifurcació
Sistemes dinàmics diferenciables
Teoria ergòdica
Sistemes dinàmics diferenciables
Bifurcation theory
Differentiable dynamical systems
Ergodic theory
Differentiable dynamical systems
topic Teoria de la bifurcació
Sistemes dinàmics diferenciables
Teoria ergòdica
Sistemes dinàmics diferenciables
Bifurcation theory
Differentiable dynamical systems
Ergodic theory
Differentiable dynamical systems
description We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H^{\pm}_3: \bar x=y, \bar y=−x+M_1+M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues $e^{±i2π/3}$. It follows from [1] that this resonance is degenerate for $M_1=0, M_2=−1$ when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map $H^+_3$ and elliptic orbits in the case of map $H^−_3$), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map $H^+_3$ and saddles with the Jacobians less than 1 and greater than 1 in the case of map $H^−_3$). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [1] to the case of the p:q resonances with odd q and show that all of them are also degenerate for the maps $H^\pm_3$ with $M_1=0$. .
publishDate 2022
dc.date.none.fl_str_mv 2022
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/194357
url https://hdl.handle.net/2445/194357
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.1134/S1560354722020058
Regular and Chaotic Dynamics, 2022, vol. 27, num. 2, p. 198-216
https://doi.org/10.1134/S1560354722020058
dc.rights.none.fl_str_mv (c) Pleiades Publishing, 2022
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Pleiades Publishing, 2022
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Pleiades Publishing
publisher.none.fl_str_mv Pleiades Publishing
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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