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

Qiu-Dong Wang, William Ott

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1439-1496
Number of pages58
JournalCommunications on Pure and Applied Mathematics
Volume64
Issue number11
DOIs
StatePublished - Nov 2011

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Homoclinic Loop
Strange attractor
sin(x+y)
Duffing Equation
Periodic Forcing
Lebesgue Measure
Theorem
Forcing
Vector Field
Form
Model

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.

In: Communications on Pure and Applied Mathematics, Vol. 64, No. 11, 11.2011, p. 1439-1496.

Research output: Contribution to journalArticle

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