On the bifurcation in a "modulated" logistic map

It is known that in the bifurcation diagram of a logistic map with a modulated parameter (denoted as A) a cross-over of the inner bifurcation branches in the four-cycle region appears, thus showing a three-cycle (at A = A₃). In this paper we present some numerical evidences showing that in the four-cycle region, before the cross-over of the inner bifurcation branches, there may be a cross-over within both the upper branch and the lower branch (at A= A ̃₂), thus giving the appearance of a two-cycle in this region. We also find that the value of A ̃₂ depends upon the initial condition.