Dissipative Homoclinic Loops of 2-dimensional Maps and Strange Attractors with 1 Direction of Instability

We prove that when subjected to periodic forcing of the form pμ,ρ,ω(t) = μ(ρh(x, y) + sin(ωt)), certain 2-dimensional vector fields with dissipative homoclinic loops admit strange attractors with SRB measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov [1] and applies the recent theory of rank 1 maps developed by Wang and Young [48, 52, 53]. We prove a general theorem and we then apply this theorem to an explicit model: a forced Duffing equation of the form dq dt2 + (λ− γq2) dt − q + q = μ sin(ωt).