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2010, Volume 27, Issue 1: 285-300. Doi: 10.3934/dcds.2010.27.285
This issue Previous Article Asymptotic behavior of a discrete turing model Next Article Measure of full dimension for some nonconformal repellers

Heterodimensional tangencies on cycles leading to strange attractors

  • 1.

    Department of Mathematics, Kyoto University of Education, Fushimi, Kyoto 612-8522

  • 2.

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

Received:  December 2008
Revised:  October 2009
Published:  March 2010
Abstract / Introduction Related Papers Cited by
    Abstract
  • 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.
    Mathematics Subject Classification: 37D30, 37D45, 37G25.

    Citation:
    shu

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