  In this paper we present a uniform way to derive families of maps from the
corresponding differential equations describing systems which experience
periodic kicks. The families depend on a single parameter - the order of a
differential equation $\alpha > 0$. We investigate general properties of such
families and how they vary with the increase in $\alpha$ which represents
increase in the space dimension and the memory of a system (increase in the
weights of the earlier states). To demonstrate general properties of the
$\alpha$-families we use examples from physics (Standard $\alpha$-family of
maps) and population biology (Logistic $\alpha$-family of maps). We show that
with the increase in $\alpha$ systems demonstrate more complex and chaotic
behavior.
