Abstract. 
We introduce the notion of α-Families 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. The α-Families of Maps represent a 
very general form of the multidimensional maps with power law memory 
and may be applicable to studying such systems with memory as viscoelastic
 materials, electromagnetic fields in dielectric media, Hamiltonian 
systems, adaptation in biological systems, human memory, etc. 
Fractional maps are invaluable tools in studying the general properties 
of the nonlinear fractional dynamical systems. On the examples 
of the fractional Logistic and Standard α-Families of Maps we demonstrate 
that the nonlinear fractional dynamical systems may have solutions in 
the form of periodic sinks, attracting slow diverging trajectories, 
attracting accelerator mode trajectories, chaotic attractors, and 
cascade of bifurcations type trajectories with some new properties.