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# Dissipative homoclinic loops of two‐dimensional maps and strange attractors with one direction of instability

Communications on Pure and Applied Mathematics (impact factor: 2.58). 10/2011; 64(11):1439 - 1496. DOI:10.1002/cpa.20379 pp.1439 - 1496

ABSTRACT We prove that when subjected to periodic forcing of the form 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 © 2011 Wiley Periodicals, Inc.

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##### Article:How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency
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### Keywords

forced Duffing equation

form certain two-dimensional vector fields

Inc

periodic

positive Lebesgue measure

rank 1 maps

recent theory

Sinai-Ruelle-Bowen measures

strange attractors

Wang