Fractional Maps and Fractional Attractors. Part I: α-Families of Maps

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 α > 0. We investigate general properties of such families and how they vary with the increase in α 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 α-families we use examples from physics (Standard α-family of maps) and population biology (Logistic α-family of maps).We show that with the increase in α systems demonstrate more complex and chaotic behavior.