Development of mandelbrot set for the logistic map with two parameters in the complex plane

In this paper, the study of the dynamical behavior of logistic map has been disused with representing fractals graphics of map, the logistic map depends on two parameters and works in the complex plane, the map defined by f(z,α,β)=αz(1–z)^β. where  and  are complex numbers, and β is a positive integers number, the visualization method used in this work to generate fractals of the map and to inspect the relation between the value of β and the shape of the map, this visualization analysis showed also that, as the value of β increasing, as the number of humps in the function also increasing, and it demonstrate that is true also for the function’s first iteration , f²(x₀)=f(f(x₀)) and the second iteration , f³(x₀)=f(f²(x₀)), beside that , the visualization technique showed that the number of humps in that fractal is less than the ones in the second iteration of the original function ,the study of the critical points and their properties of the logistic map also discussed it, whereas finding the fixed point led to find the critical point of the function f, in addition , it haven proven for the set of all pointsα∈C and β∈N, the iteration function f(f(z) has an attractive fixed points, and belongs to the region specified by the disc |1–β(α–1)|<1. Also, The discussion of the Mandelbrot set of the function defined by the f(f(z)) examined in complex plans using the path principle, such that the path of the critical point z=z₀ is restricted, finally, it has proven that the Mandelbrot set f(z,α,β) contains all the attractive fixed points and all the complex numbers  in which α≤(1/β+1) (1/β+1) and the region containing the attractive fixed points for f²(z,α,β) was identified