Fractional Maps and Fractional Attractors. Part I: $\alpha$-Families of Maps

TL;DR

This paper introduces a unified method to derive families of maps from differential equations with periodic kicks, depending on a parameter alpha, and explores how increasing alpha affects system complexity and chaos.

Contribution

It presents a novel framework for generating alpha-dependent map families from differential equations, linking system memory and dimension to dynamic complexity.

Findings

Higher alpha leads to more complex and chaotic behavior

The framework applies to physical and biological systems

Increased alpha correlates with increased system memory

Abstract

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