### Abstract

We prove that when subjected to periodic forcing of the form $$p-{\mu ,\rho ,\omega } (t) = \mu (\rho h(x,y) + \sin (\omega t)),$$ certain two-dimensional vector fields with dissipative homoclinic loops generate strange attractors with Sinai-Ruelle-Bowen measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov and applies the recent theory of rank 1 maps developed by Wang and Young. We prove a general theorem and then apply this theorem to an explicit model: a forced Duffing equation of the form $${{d^2 q} \over {dt^2 }} + (\lambda - \gamma q^2){{dq} \over {dt}} - q + q^3 = \mu \sin (\omega t).$$

Original language | English (US) |
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Pages (from-to) | 1439-1496 |

Number of pages | 58 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 64 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2011 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics