Bifurcations and chaos in a three-dimensional generalized Hénon map

Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions...
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Gespeichert in:

Autor*in:	Jingjing Zheng [verfasserIn] Ziwei Wang [verfasserIn] You Li [verfasserIn] Jinliang Wang [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2018

Schlagwörter:	Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle

Übergeordnetes Werk:	In: Advances in Difference Equations - SpringerOpen, 2006, (2018), 1, Seite 14
Übergeordnetes Werk:	year:2018 ; number:1 ; pages:14

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.1186/s13662-018-1622-y

Katalog-ID:	DOAJ077617010

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520			\|a Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.
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allfields	10.1186/s13662-018-1622-y doi (DE-627)DOAJ077617010 (DE-599)DOAJ4ffee127c8334b26a606432200670af1 DE-627 ger DE-627 rakwb eng QA1-939 Jingjing Zheng verfasserin aut Bifurcations and chaos in a three-dimensional generalized Hénon map 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits. Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle Mathematics Ziwei Wang verfasserin aut You Li verfasserin aut Jinliang Wang verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2018), 1, Seite 14 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2018 number:1 pages:14 https://doi.org/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/article/4ffee127c8334b26a606432200670af1 kostenfrei http://link.springer.com/article/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2018 1 14
spelling	10.1186/s13662-018-1622-y doi (DE-627)DOAJ077617010 (DE-599)DOAJ4ffee127c8334b26a606432200670af1 DE-627 ger DE-627 rakwb eng QA1-939 Jingjing Zheng verfasserin aut Bifurcations and chaos in a three-dimensional generalized Hénon map 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits. Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle Mathematics Ziwei Wang verfasserin aut You Li verfasserin aut Jinliang Wang verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2018), 1, Seite 14 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2018 number:1 pages:14 https://doi.org/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/article/4ffee127c8334b26a606432200670af1 kostenfrei http://link.springer.com/article/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2018 1 14
allfields_unstemmed	10.1186/s13662-018-1622-y doi (DE-627)DOAJ077617010 (DE-599)DOAJ4ffee127c8334b26a606432200670af1 DE-627 ger DE-627 rakwb eng QA1-939 Jingjing Zheng verfasserin aut Bifurcations and chaos in a three-dimensional generalized Hénon map 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits. Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle Mathematics Ziwei Wang verfasserin aut You Li verfasserin aut Jinliang Wang verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2018), 1, Seite 14 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2018 number:1 pages:14 https://doi.org/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/article/4ffee127c8334b26a606432200670af1 kostenfrei http://link.springer.com/article/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2018 1 14
allfieldsGer	10.1186/s13662-018-1622-y doi (DE-627)DOAJ077617010 (DE-599)DOAJ4ffee127c8334b26a606432200670af1 DE-627 ger DE-627 rakwb eng QA1-939 Jingjing Zheng verfasserin aut Bifurcations and chaos in a three-dimensional generalized Hénon map 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits. Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle Mathematics Ziwei Wang verfasserin aut You Li verfasserin aut Jinliang Wang verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2018), 1, Seite 14 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2018 number:1 pages:14 https://doi.org/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/article/4ffee127c8334b26a606432200670af1 kostenfrei http://link.springer.com/article/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2018 1 14
allfieldsSound	10.1186/s13662-018-1622-y doi (DE-627)DOAJ077617010 (DE-599)DOAJ4ffee127c8334b26a606432200670af1 DE-627 ger DE-627 rakwb eng QA1-939 Jingjing Zheng verfasserin aut Bifurcations and chaos in a three-dimensional generalized Hénon map 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits. Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle Mathematics Ziwei Wang verfasserin aut You Li verfasserin aut Jinliang Wang verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2018), 1, Seite 14 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2018 number:1 pages:14 https://doi.org/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/article/4ffee127c8334b26a606432200670af1 kostenfrei http://link.springer.com/article/10.1186/s13662-018-1622-y kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2018 1 14
language	English
source	In Advances in Difference Equations (2018), 1, Seite 14 year:2018 number:1 pages:14
sourceStr	In Advances in Difference Equations (2018), 1, Seite 14 year:2018 number:1 pages:14
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle Mathematics
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container_title	Advances in Difference Equations
authorswithroles_txt_mv	Jingjing Zheng @@aut@@ Ziwei Wang @@aut@@ You Li @@aut@@ Jinliang Wang @@aut@@
publishDateDaySort_date	2018-01-01T00:00:00Z
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callnumber-first	Q - Science
author	Jingjing Zheng
spellingShingle	Jingjing Zheng misc QA1-939 misc Hénon map misc Fold bifurcation misc Flip bifurcation misc Naimark–Sacker bifurcation misc Double-cycle misc Mathematics Bifurcations and chaos in a three-dimensional generalized Hénon map
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topic_title	QA1-939 Bifurcations and chaos in a three-dimensional generalized Hénon map Hénon map Fold bifurcation Flip bifurcation Naimark–Sacker bifurcation Double-cycle
topic	misc QA1-939 misc Hénon map misc Fold bifurcation misc Flip bifurcation misc Naimark–Sacker bifurcation misc Double-cycle misc Mathematics
topic_unstemmed	misc QA1-939 misc Hénon map misc Fold bifurcation misc Flip bifurcation misc Naimark–Sacker bifurcation misc Double-cycle misc Mathematics
topic_browse	misc QA1-939 misc Hénon map misc Fold bifurcation misc Flip bifurcation misc Naimark–Sacker bifurcation misc Double-cycle misc Mathematics
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title	Bifurcations and chaos in a three-dimensional generalized Hénon map
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title_sort	bifurcations and chaos in a three-dimensional generalized hénon map
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title_auth	Bifurcations and chaos in a three-dimensional generalized Hénon map
abstract	Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.
abstractGer	Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.
abstract_unstemmed	Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.
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title_short	Bifurcations and chaos in a three-dimensional generalized Hénon map
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Bifurcations and chaos in a three-dimensional generalized Hénon map

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