Paper detail

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.