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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 $α$-Family of Maps depending on a single parameter $α> 0$ which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We consider two particular $α$-families corresponding to the Standard and Logistic Maps. For fractional $α<2$ 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.