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Complete orbits for twist maps on the plane

Abstract Twist maps (θ1,r1)=f(θ,r) are considered in this paper, with no assumption on the periodicity of the map in θ. Under appropriate assumptions, the existence of infinitely many bounded (in r) complete orbits is proven. In particular, our results apply to the class of maps \[ \theta _1=\theta +r,\quad r_1=r+\lambda (\sin \omega _1(\theta +r) +\sin \omega _2(\theta +r)), \] where λ>0 and no arithmetic condition has to be imposed on ω1/ω2.