Fixed Point Results for Fractal Generation of Complex Polynomials Involving Sine Function via Non-Standard Iterations

Hengxiao Qi,Waqas Nazeer,Yuming Chu,Muhammad Tanveer

doi:10.1109/access.2020.3018090

Abstract

Due to the uniqueness and self-similarity, fractals became most attractive and charming research field. Nowadays researchers use different techniques to generate beautiful fractals for a complex polynomial $z^{n}+c$ . This article demonstrates some fixed point results for a sine function (i.e. $\sin (z^{n}) +c$ ) via non-standard iterations (i.e. Mann, Ishikawa and Noor iterations etc.). Since each two steps iteration (i.e. Ishikawa and S iterations) or each three steps iteration (i.e. Noor, CR and SP iterations) have same escape radii for any complex polynomial, so we use these results for S, CR and SP iterations also to apply for the generation of Julia and Mandelbrot sets with $\sin (z^{n}) +c$ . At some fixed input parameters, we observe the engrossing behavior of Julia and Mandelbrot sets for different $n$ .

Highlights

T O draw the graphs via escape time algorithms in the form of unique and self-similar images by using some electronic tools became an attractive field named as fractals
In Julia set we study the behaviour of the iterates for each z, and in the Mandelbrot set we study connectedness of Julia set for each c defining those sets
We found some images on Internet and papers in which authors just generated Julia and Mandelbrot sets with sin(zn) + c, but they could not prove the escape criteria for this complex function

Summary

Introduction

T O draw the graphs via escape time algorithms in the form of unique and self-similar images by using some electronic tools became an attractive field named as fractals. In 1985, he sketched the Julia set and studied its features He observed that for different values of c the Julia sets have diversity in their nature. His work was extended by Lakhtakia et al [3] in 1987 They generalized the Mandelbrot set for f : z −→ zp + c where p ≥ 2. Crowe et al [4] defined the anti Julia and anti Mandelbrot sets in 1989 and discussed their connected locus. They generated complex graphs for z2 +c and later on these graphs called "tricorn" [5].

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