Universal fractional map and cascade of bifurcations type attractors

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. (C) 2013 AIP Publishing LLC.