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Regular and Chaotic Dynamics, 2007, Volume 12, Issue 3, Pages 233–266
DOI: https://doi.org/10.1134/S156035470703001X
(Mi rcd623)
 

This article is cited in 28 scientific papers (total in 28 papers)

Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps

S. V. Gonchenkoa, V. S. Gonchenkoa, J. C. Tatjerb

a Institute for Applied Mathematics and Cybernetics, ul. Uljanova 10, Nizhny Novgorod, 603005 Russia
b Departament de Matemática Aplicada i Análisi, Gran Via de les Corts Catalanes 585, Barcelona 08007, Spain
Citations (28)
Abstract: We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers $e^{\pm i \phi}$. On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.
Keywords: homoclinic tangency, rescaling, generalized Henon map, bifurcation.
Received: 03.03.2007
Accepted: 10.05.2007
Bibliographic databases:
Document Type: Article
MSC: 37C05, 37G25, 37G35
Language: English
Citation: S. V. Gonchenko, V. S. Gonchenko, J. C. Tatjer, “Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps”, Regul. Chaotic Dyn., 12:3 (2007), 233–266
Citation in format AMSBIB
\Bibitem{GonGonTat07}
\by S. V. Gonchenko, V. S. Gonchenko, J.~C.~Tatjer
\paper Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple Quadratic Homoclinic Tangencies and Generalized H\'{e}non Maps
\jour Regul. Chaotic Dyn.
\yr 2007
\vol 12
\issue 3
\pages 233--266
\mathnet{http://mi.mathnet.ru/rcd623}
\crossref{https://doi.org/10.1134/S156035470703001X}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2350324}
\zmath{https://zbmath.org/?q=an:1229.37040}
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  • This publication is cited in the following 28 articles:
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