A novel approach to generate Mandelbrot sets, Julia sets and biomorphs via viscosity approximation method

Abstract

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a new approach to visualize Mandelbrot and Julia sets for complex polynomials of the form W(z)=zn+mz+r; n≥2 where m,r∈ℂ, and biomorphs for any complex function through a viscosity approximation method which is among the most widely used iterative methods for finding fixed points of non-linear operators. We derive novel escape criterion for generating Julia and Mandelbrot sets via proposed viscosity approximation method. Moreover, we visualize the sets using the escape time algorithm and the proposed iteration. Then, we discuss the shape change of the obtained sets depending on the parameters of the iteration using graphical and numerical experiments. The presented examples reveal that this change can be very complex, and we are able to obtain a great variety of shapes.