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February  2010, 27(1): 285-300. doi: 10.3934/dcds.2010.27.285

Heterodimensional tangencies on cycles leading to strange attractors

Shin Kiriki 1, , Yusuke Nishizawa 2, and  Teruhiko Soma 2,

1. 

Department of Mathematics, Kyoto University of Education, Fushimi, Kyoto 612-8522, Japan
2. 

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan, Japan

Received  December 2008 Revised  October 2009 Published  February 2010

In this paper, we study a two-parameter family $\{\varphi_{\mu,\nu}\}$ of three-dimensional diffeomorphisms which have a bifurcation induced by simultaneous generation of a heterodimensional cycle and a heterodimensional tangency associated to two saddle points. We show that such a codimension-$2$ bifurcation generates a quadratic homoclinic tangency associated to one of the saddle continuations which unfolds generically with respect to some one-parameter subfamily of $\{\varphi_{\mu,\nu}\}$. Moreover, from this result together with some well-known facts, we detect some nonhyperbolic phenomena (i.e., the existence of nonhyperbolic strange attractors and the $C^{2}$ robust tangencies) arbitrarily close to the codimension-$2$ bifurcation.
Keywords: strange attractors, robust tangencies., heterodimensional cycles, homoclinic tangencies.
Mathematics Subject Classification: 37D30, 37D45, 37G2.
Citation: Shin Kiriki, Yusuke Nishizawa, Teruhiko Soma. Heterodimensional tangencies on cycles leading to strange attractors. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 285-300. doi: 10.3934/dcds.2010.27.285
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