Fractional order logistic map; Numerical approach

TL;DR

This paper investigates the fractional order logistic map using Caputo's fractional differences, revealing that many iterations are needed to reach steady states and that each initial condition can produce a unique bifurcation diagram, highlighting challenges in numerical analysis.

Contribution

It introduces a numerical approach to the fractional order logistic map and uncovers the dependence of bifurcation diagrams on initial conditions, a phenomenon not previously documented.

Findings

Thousand-plus iterations are necessary to avoid transients.

Different initial conditions lead to distinct bifurcation diagrams.

The phenomenon may affect other FO difference systems.

Abstract

In this paper the fractional order logistic map in the sense of Caputo's fractional differences is numerically approached. It is shown that the necessary iterations number to avoid transients must be of order of thousand, not of order of hundreds as commonly used in several works. Also, it is revealed an interesting phenomenon according to which for every initial condition it correspond a different bifurcation diagram. This phenomenon seems to appear also in other FO difference systems, fact which could represent an obstacle for the numerical analysis. A short Matlab code is used to obtain the results.