On a homoclinic origin of Hénon-like maps

Гонченко, С. В.; Gonchenko, V. S.; Shilnikov, L. P.

doi:10.1134/s1560354710510118

Cited by 6 publications

(23 citation statements)

References 30 publications

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“…представляют особый интерес для нелинейной динамики, поскольку они имеют «гомоклиническое происхождение», то есть являются нормальными формами отображений первого возвращения вблизи гомоклинических касаний некоторых типов (см., например, [12,15,[22][23][24][25]). Содержание работы.…”

Section: нелинейная динамика 2012 T 8 № 1 с 3-28unclassified

Towards scenarios of chaos appearance in three-dimensional maps

Gonchenko¹,

Гонченко²,

Shilnikov³

2012

Nelin. Dinam.

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В настоящей работе изучаются вопросы хаотической динамики трехмерных гладких отображений (диффеоморфизмов). Показывается, что здесь существует два основных сценария развития хаоса от устойчивой неподвижной точки либо к спиральному аттрактору, либо к странному аттрактору лоренцевского или «восьмерочного» типа. Дается качественное описание этих аттракторов, приводятся условия, при которых они могут быть «настоящими» (псевдогиперболическими странными аттракторами). В работу также включены соответствующие результаты численного исследования аттракторов в трехмерных отображениях Эно. Ключевые слова: странный аттрактор, хаотическая динамика, спиральный аттрактор, тор-хаос, гомоклиническая траектория, инвариантная кривая, трехмерное отображение Эно Получено 6 октября 2011 года После доработки 24 декабря 2011 года Работа выполнена в рамках гранта Правительства Российской Федерации для государственной поддержки научных исследований, проводимых под руководством ведущих ученых в российских образовательных учреждениях высшего профессионального образования, договор № 11.G34.31.0039. Работа поддержана грантами РФФИ No. 10-01-00429, No. 11-01-00001 и РФФИ No. 11-01-97017-рповолжье.

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Section: нелинейная динамика 2012 T 8 № 1 с 3-28unclassified

Towards scenarios of chaos appearance in three-dimensional maps

Gonchenko¹,

Гонченко²,

Shilnikov³

2012

Nelin. Dinam.

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“…In its turn, the periodic saddle orbits 𝑃 4 and 𝑃 8 , occurring after the corresponding period-doubling bifurcations, also undergo a cascade of perioddoubling bifurcations. Respectively, the curves of first such period-doubling bifurcations reside below the Neimark-Sacker bifurcation curves NS 4 , NS 8 and coincide with the same curves PD 4 and PD 8 (green dashed lines in Fig. 12b).…”

Section: B=01 Cmentioning

confidence: 74%

“…In particular, we show that hyperchaotic attractors on the base of the period-4 resonant orbit appear in accordance with the scenarios presented in Sec. 2. , and SN 16 are colored in blue, Neimark-Sacker bifurcation curves NS, NS 4 , and NS 8 -in red, and period-doubling bifurcation curves PD 4 , PD 8 , PD 16 , pd 4 , pd 8 , and pd 16 -in green (solid curves are used for representing bifurcations with stable orbits, while dashed -with saddle ones); the points ff 4 , ff 8 , and ff 16 correspond to a fold-flip bifurcation.…”

Section: Three-dimensional Mirá Map: Main Bifurcations and Dynamical ...mentioning

confidence: 99%

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Shykhmamedov¹,

Karatetskaia²,

Казаков³

et al. 2020

Preprint

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We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mirá map.

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“…(2) (for ν = 1), these are so-called Keller or Cremona maps [Essen, 2000]. Such maps are elements of the group of polynomial automorphisms, and while in two dimensions these are known to be conjugate to a composition of affine and elementary maps [Dullin & Meiss, 2000], similar results in higher dimensions are not available [Essen, 2000;Gonchenko et al, 2006Gonchenko et al, , 2010.…”

Section: The Generalized Hénon Mapmentioning

confidence: 99%

The Generalized Time-Delayed Hénon Map: Bifurcations and Dynamics

Bilal

Ramaswamy

2013

Int. J. Bifurcation Chaos

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We analyze the bifurcations of a family of time-delayed Hénon maps of increasing dimension and determine the regions where the motion is attracted to different dynamical states. As a function of parameters that govern nonlinearity and dissipation, boundaries that confine asymptotic periodic motion are determined analytically, and we examine their dependence on the dimension d. For large d these boundaries converge. In low dimensions both the period-doubling and quasiperiodic routes to chaos coexist in the parameter space, but for high dimensions the latter predominates and prior to the onset of chaos, the systems exhibit multistability. When the nonlinearity parameter is varied, the dimension of chaotic attractors in the systems changes smoothly with increasing number of non-negative Lyapunov exponents.

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On a homoclinic origin of Hénon-like maps

Cited by 6 publications

References 30 publications

Towards scenarios of chaos appearance in three-dimensional maps

Towards scenarios of chaos appearance in three-dimensional maps

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

The Generalized Time-Delayed Hénon Map: Bifurcations and Dynamics

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