Effect of Stochastic Noise on Superior Julia Sets

Rani, Mamta; Agarwal, Rakesh

doi:10.1007/s10851-009-0171-0

Cited by 44 publications

(19 citation statements)

References 14 publications

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“…This will give the ability of changing a biomorph's shape in a predictable way. Negi et al [20] and Rani et al [26] have made a research on the noise in superior Mandelbrot set and superior Julia sets, so we will try to extend their work on the modified biomorphs introduced in this paper.…”

Section: Discussionmentioning

confidence: 99%

Biomorphs via modified iterations

Gdawiec¹,

Kotarski²,

Lisowska³

2016

J. Nonlinear Sci. Appl.

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The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two directions are proposed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm. Biomorphs generated by the modified algorithm are essentially different in comparison to those obtained by the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant.

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Section: Discussionmentioning

confidence: 99%

Biomorphs via modified iterations

Gdawiec¹,

Kotarski²,

Lisowska³

2016

J. Nonlinear Sci. Appl.

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“…Also, they have studied the fractal structure and discontinuity law of the generalized Julia sets generated from the extended complex mapping z n + c, where n ∈ R [25]. Further, Julia and Mandelbrot sets have been studied under the effect of noises [19,4,14,15,27]. In 2004, Rani and Kumar [17,18] introduced the superior iterate and created superior Julia and Mandelbrot sets for quadratic [17,18] and cubic [17,18] polynomials.…”

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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“…Recently, the attention of scientists has focused on the use of different iteration schemes from fixed point theory in the generation of patterns. The iteration schemes have been mainly used in the generation of fractal patterns defined in the complex plane, such as the well‐known Mandelbrot and Julia sets [AA12, ARC14, RA10], and in polynomiography [GKL15] (a method that is based on the root‐finding methods of complex polynomials).…”

Section: Introductionmentioning

confidence: 99%

Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns

Gdawiec

2015

Computer Graphics Forum

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In this paper, we generalize the idea of star‐shaped set inversion fractals using iterations known from fixed point theory. We also extend the iterations from real parameters to so‐called q‐system numbers and proposed the use of switching processes. All the proposed generalizations allowed us to obtain new and diverse fractal patterns that can be used, e.g. as textile and ceramics patterns. Moreover, we show that in the chaos game for iterated function systems—which is similar to the inversion fractals generation algorithm—the proposed generalizations do not give interesting results.

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Effect of Stochastic Noise on Superior Julia Sets

Cited by 44 publications

References 14 publications

Biomorphs via modified iterations

Biomorphs via modified iterations

New Julia and Mandelbrot Sets for a New Faster Iterative Process

Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns

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