Fractals through modified iteration scheme

Abstract

In this paper we study the geometry of relative superior Mandelbrot sets through S-iteration scheme. Our results are quit significant from other Mandelbrot sets existing in the literature. Besides this, we also observe that S-iteration scheme converges faster than Ishikawa iteration scheme. We believe that the results of this paper can be inspired those who are interested in creating automatically aesthetic patterns.

Generation of Julia and Mandelbrot Sets via Fixed Points

“…Our results are different from comparable existing results as we obtain escape criterion and fractals for polynomials of the form T(x) = x n + mx + r where m, r ∈ C without using the Jungck iterative process. It is also worth mentioning that the behaviour of the polynomial and shape of the fractal generated under the iterative process (2) is different and unique as compared to the iterative process studied before in the literature [1,24,29,32]. Acknowledgments: All the authors are grateful to the referees for their critical remarks and valuable suggestions which helped to improve the presentation of the paper.…”

“…The modified Ishikawa process, S-iteration, was employed by Kang et al [24,25] to study relative superior Mandelbrot sets, tricorn and multicorns. The S-orbit, for any x 0 ∈ C, is a sequence {x i } given by…”

“…It is noteworthy that for each iterative process the behaviour and dynamics of the Julia and Mandelbrot sets differ. For some thought-provoking and fascinating comparisons, the reader may refer to [1,24,[27][28][29] and references therein.…”

Generation of Antifractals via Hybrid Picard-Mann Iteration

“…These sets are most complex sets till the date and can not be seen without computer [9]. There are many ways to generate Fractals and fixed point theory is one of them in which iterative algorithms are used to find fixed point of complex polynomials [10], [11]. Different researcher used different iterative algorithms to generate Fractals, for example, see [10]- [14].…”

“…There are many ways to generate Fractals and fixed point theory is one of them in which iterative algorithms are used to find fixed point of complex polynomials [10], [11]. Different researcher used different iterative algorithms to generate Fractals, for example, see [10]- [14]. Generation of Fractals is an aesthetic endeavor, a sooting diversion or only a numerical model and fractal art is totally different from other computer activities [15].…”