An example of bifurcation to homoclinic orbits

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Abstract

Consider the equation ẍ − x + x2 = −λ1x + λ2ƒ(t) where ƒ(t + 1) = ƒ(t) and λ = (λ1, λ2) is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations.

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This research was supported in part by the Nètional Science Foundation, under MCS76-06739.

This research was supported in part by the National Science Foundation under MCS-79-05774, in part by the United States Army under AROD DAAG 27-76-G-0294 and in part by the United States Air Force under AF-AFOSR 76-3092C.