  We modified the way in which the Universal Map is obtained in the regular
dynamics to derive the Universal $\alpha$-Family of Maps depending on a single
parameter $\alpha > 0$ which is the order of the fractional derivative in the
nonlinear fractional differential equation describing a system experiencing
periodic kicks. We consider two particular $\alpha$-families corresponding to
the Standard and Logistic Maps. For fractional $\alpha<2$ in the area of
parameter values of the transition through the period doubling cascade of
bifurcations from regular to chaotic motion in regular dynamics corresponding
fractional systems demonstrate a new type of attractors - cascade of
bifurcations type trajectories.
