Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order

Abstract

This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined.

Keywords

• Caputo forward difference operator
• Mandelbrot set of fractional order
• Julia sets of fractional order
• fractional equipotential lines
• fractional external rays