The Lorenz model in discrete time

A two-dimensional noninvertible map T:(x′,y′)=(x(1+aτ−y),(1−τ)y+τx2) proposed by Lorenz in 1989, depending on two parameters a and τ, is reconsidered. We show the two different bifurcation scenarios occurring for a>0 and aτ=1 and τ=2, describing the related bifurcations as a function of the parameter a>0. For τ=2, a straight line filled with 2-cycles and the occurrence of a resonant case with rotation number π/2 in the Neimark–Sacker bifurcation of two fixed points determine particular bifurcations, associated with the existence of four invariant sets of the map. For τ=1, the critical curve degenerates into one point, the origin, which is also fixed for map T and the focal point for the inverse map, leading to chaotic attracting sets with infinitely many lobes issuing from the origin. The transition to chaos from attracting closed curves is also analysed, evidencing the occurrence of homoclinic tangles.