Non-linearity of the dynamics of transcendental function using Fibonacci–Mann iteration and some critical remarks

With a generalized setting of transcendental functions associated with a non-linear variable, we have observed the combinatorial structures of the fractals of Julia sets and Mandelbrot sets due to the function z κ c o s ( z κ ) + p z 2 + q , in the complex plane with p , q ∈ ℂ , κ > 2 . We emphasized the dependency of the combinatorial structure of the fractals on the parameter values of the equation and the iterations. We set up an escaping criteria for the group of points in the dynamical plane to generate the fractals of Julia and use it to generate the Mandelbrot set in the parameter plane. We put some constructive remarks on the calculations due to Fibonacci–Mann iteration of a family of a transcendental map discussed in related literature to enhance the applicability of the article. Furthermore, we explored average number of iterations (ANI) on the parameters we used for the iteration.