Fractional Maps and Fractional Attractors. Part II: Fractional Difference α-Families of Maps

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In this paper we extend the notion of an α-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 α-Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.