The Universal $\alpha$-Family of Maps

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 show that many well-known regular maps, like integer n- dimensional (area/volume preserving for $n>1$) quadratic maps (including for $n=1$ the Logistic Map which is not measure preserving) and n-dimensional (volume preserving for $n>2$) standard maps (including the non-measure preserving Circle Map and the area preserving Standard Map), can be considered as particular forms of the Universal $\alpha$-Family of Maps. In the case of the fractional $\alpha$ corresponding maps, which are maps with memory, demonstrate various types of attractors including cascade of bifurcation types trajectories. Maps with memory can be applied for modeling biological systems and circuit elements with memory.