### Abstract

Let {f_{a,b}} be the (original) Hénon family. In this paper, we show that, for any b near 0, there exists a closed interval J_{b} which contains a dense subset J' such that, for any a ∈ J' , f _{a,b} has a quadratic homoclinic tangency associated with a saddle fixed point of f_{a,b} which unfolds generically with respect to the oneparameter family {f_{a,b}}_{a∈Jb} . By applying this result, we prove that J_{b} contains a residual subset A^{(2)} _{b} such that, for any a ∈ A^{(2)} _{b} , f _{a,b} admits the Newhouse phenomenon. Moreover, the interval J _{b} contains a dense subset Ã_{b} such that, for any a ∈ Ã_{b}, f_{a,b} has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously.

Original language | English |
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Pages (from-to) | 2253-2269 |

Number of pages | 17 |

Journal | Nonlinearity |

Volume | 23 |

Issue number | 9 |

DOIs | |

State | Published - 1 Sep 2010 |

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## Cite this

*Nonlinearity*,

*23*(9), 2253-2269. https://doi.org/10.1088/0951-7715/23/9/010