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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
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
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
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https://www.mathnet.ru/eng/rcd623 https://www.mathnet.ru/eng/rcd/v12/i3/p233
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