### 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) |
---|---|

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 2011 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability.** / Wang, Qiu-Dong; Ott, William.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 64, no. 11, pp. 1439-1496. https://doi.org/10.1002/cpa.20379

}

TY - JOUR

T1 - Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability

AU - Wang, Qiu-Dong

AU - Ott, William

PY - 2011/11

Y1 - 2011/11

N2 - 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 $${{d^2 q} \over {dt^2 }} + (\lambda - \gamma q^2){{dq} \over {dt}} - q + q^3 = \mu \sin (\omega t).$$

AB - 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 $${{d^2 q} \over {dt^2 }} + (\lambda - \gamma q^2){{dq} \over {dt}} - q + q^3 = \mu \sin (\omega t).$$

UR - http://www.scopus.com/inward/record.url?scp=80052023566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052023566&partnerID=8YFLogxK

U2 - 10.1002/cpa.20379

DO - 10.1002/cpa.20379

M3 - Article

VL - 64

SP - 1439

EP - 1496

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 11

ER -