Julia sets and Mandelbrot sets in Noor orbit

Ashish, .; Rani, Mamta; Chugh, Renu

doi:10.1016/j.amc.2013.11.077

Cited by 46 publications

(24 citation statements)

References 40 publications

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“…Similar investigations to those presented in this paper can be carried out for complex fractals (Julia and Mandelbrot sets) and biomorphs. Some aspects of such investigations have been reported in the literature [9,[32][33][34]. Another interesting direction rely on replacing complex numbers by more general: dual and double numbers used in [35] for defining the Q-Systems Fractals.…”

Section: Discussionmentioning

confidence: 99%

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Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Gdawiec

Kotarski

Lisowska

2015

Abstract and Applied Analysis

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A survey of some modifications based on the classic Newton's and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard's iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari's visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

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Section: Discussionmentioning

confidence: 99%

“…Earlier, the other types of iterations have been used in [7] for superfractals and in [8] for fractals generated by IFS. Julia sets and Mandelbrot sets [9] and the antifractals [10] have been also investigated using Noor iteration instead of the standard Picard iteration.…”

Section: Introductionmentioning

confidence: 99%

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Gdawiec

Kotarski

Lisowska

2015

Abstract and Applied Analysis

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“…Buraczewski et al studied Mandelbrot set cascades in multidimensions [5]. Ashish et al researched features of Mandelbrot set in Noor orbit [3]. Tiozzo computed topological entropy and dimensions of sections of Mandelbrot set [28].…”

Section: Introductionmentioning

confidence: 99%

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

Liu¹,

Zheng²,

Fu³

et al. 2017

J. Nonlinear Sci. Appl.

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Generalized Mandelbrot set (k-M set) is the basis of fractal analysis. This paper presents a novel method to generate k-M set, which generates k-M set precisely by constructing its asymptote family. Correctness of the proposed method is proved as well as computational complexity is researched. Further, application of the generation method is studied, which is used to analyze distribution of boundary points and periodic points of k-M set. Finally, experiments have been implemented to evaluate the theoretical results.

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“…Recently, Ashish et al [8] used Noor iterative process (an example of fourstep feedback process) for creating new Julia and Mandelbrot sets for quadratic, cubic and higher degree polynomials. Further they have shown anti Julia, Multicorns [6] and cubic Julia sets [7] using Noor iterative process.…”

Section: Introductionmentioning

confidence: 99%

New Julia and Mandelbrot Sets for a New Faster Iterative Process

Kumari¹,

Ashish²,

Chugh³

2016

Int. J. of Pure and Appl. Math.

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Fixed point iterative procedures are the backbones of fractal geometry. In existing literature Julia sets, Mandelbrot sets and their variants have been studied using one -step, two -step, three -step and four -step iterative process. Recently, M. Abbas and T. Nazir[12] introduced a new iterative process (a four-step iterative process) which is faster than all of Picard, Mann and Agarwal processes. In this paper, we obtain further generalizations of Julia and Mandelbrot sets using this faster iterative process for quadratic, cubic and higher degree polynomials. Further, we analyze that few Julia and Mandelbrot sets took the shape of Lord Ganesha (name of Hindu God), Dragon and Urn.

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Julia sets and Mandelbrot sets in Noor orbit

Cited by 46 publications

References 40 publications

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

New Julia and Mandelbrot Sets for a New Faster Iterative Process

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