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Existence of generic cubic homoclinic tangencies for Hénon maps

Published online by Cambridge University Press:  08 May 2012

SHIN KIRIKI  and
TERUHIKO SOMA
SHIN KIRIKI
Affiliation:
Department of Mathematics, Kyoto University of Education, 1 Fukakusa-Fujinomori, Fushimi-ku, Kyoto, 612-8522, Japan (email: skiriki@kyokyo-u.ac.jp)
TERUHIKO SOMA
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan (email: tsoma@tmu.ac.jp)
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Abstract

In this paper, we show that the Hénon map $\varphi _{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko, Shilnikov and Turaev [On models with non-rough Poincaré homoclinic curves. Physica D 62(1–4) (1993), 1–14; Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6(1) (1996), 15–31; On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle. Proc. Steklov Inst. Math.216 (1997), 70–118; Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. 67 (1999), 69–128, translation in J. Math. Sci. 105 (2001), 1738–1778; Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20 (2007), 241–275]. Combining this fact with theorems in Kiriki and Soma [Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies. Nonlinearity 21(5) (2008), 1105–1140], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1029 - 1051
Copyright
Copyright © 2012 Cambridge University Press 

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