Fractional Maps and Fractional Attractors. Part II: Fractional   Difference $\alpha$-Families of Maps
  In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic $\alpha$-Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory. 
Mark Edelman
nlin.CD physics.bio-ph