Mandelbrot set and Julia sets of fractional order

TL;DR

This paper introduces fractional-order Mandelbrot and Julia sets based on $q$-th Caputo-like fractional differences, revealing unique properties and behaviors that differ from classical sets, supported by analytical and extensive numerical studies.

Contribution

It is the first to define and analyze fractional-order Mandelbrot and Julia sets using $q$-th Caputo-like differences, exploring their properties and differences from classical sets.

Findings

Fractional-order sets exhibit properties different from classical sets.

Numerical experiments support conjectures about set similarities at different $q$ values.

An adapted escape-time algorithm for fractional sets is developed and presented.

Abstract

In this paper the fractional-order Mandelbrot and Julia sets in the sense of $q$-th Caputo-like discrete fractional differences, for $q\in(0,1)$, are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for $q\uparrow1$, contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for $q\downarrow0$. Also, we conjecture that for $q\downarrow0$, the fractional-order Mandelbrot set is similar to the integer-order Mandelbrot set, while for $q\downarrow0$ and $c=0$, one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.