Generalized fuzzy Mandelbrot and Mandelbar sets

Abstract

In this article, we introduce the generalized fuzzy Mandelbrot and Mandelbar sets. In this regard, we assign a membership value to each complex number under the iterations, even if the orbit of any complex number is not limited. We give a good deal of examples for selected points in detail computationally and visually. We form a number of classifications for the membership values of these particular points in these generalized sets with each degree of integers p>2. We reveal a difference between the generalized fuzzy Mandelbrot set and the generalized fuzzy Mandelbar set by proving the symmetry properties of these fuzzy sets according to the membership function. Also, we prove that the membership value of each complex number in a fuzzy generalized Mandelbrot (Mandelbar) set with degree p remains invariant whenever it is rotated through 2πp−1 (2πp+1) radians about the origin.