Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process

Jolaoso, Lateef Olakunle; Khan, Safeer Hussain

doi:10.3390/math8122172

Cited by 10 publications

(2 citation statements)

References 36 publications

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“…More so, polynomiographs are sensitive to small changes in the control parameters of the iteration functions, polynomials or scales of the graphics, whereas fractals are self-similar, have a typical structure and are independent of scale. In general, polynomiography has found several applications in design, education, art and science [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

2022

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In this paper, we proposed two mixed algorithms of Newton’s and Abbasbandy’s methods using a known iteration scheme from fixed point theory in polynomiography. We numerically investigated some properties of the proposed algorithms using periodic sequence parameters instead of the constant parameters that are mostly used by many authors. Two pseudo-Newton algorithms were introduced based on the mixed iterations for the purpose of generating polynomiographs. The properties of the obtained polynomiographs were studied with respect to their graphics, turning effects and computation time. Moreover, some of these polynomiographs exhibited symmetrical properties when the degree of the polynomial was even.

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Section: Introductionmentioning

confidence: 99%

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

2022

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“…Based on the Jungck-CR iteration process with s-convexity, authors proved new escape criteria for the generation of Mandelbrot and Julia sets and presented some graphical examples obtained by the use of an escape time algorithm and the derived criteria in [32]. In [33], the authors investigated the biomorphs for certain polynomials by using a more general iteration method and examined their graphical behaviour with respect to the variation in parameters. In [34], the authors adjust algorithms according to the developed conditions and draw some attractive Julia and Mandelbrot sets with iterate sequences from proposed fixed-point iterative methods.…”

Section: Introductionmentioning

confidence: 99%

Fractals Parrondo’s Paradox in Alternated Superior Complex System

Zhang

Wang

2021

Fractal Fract

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This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

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Variants of Mandelbrot and Julia fractals for higher‐order complex polynomials

Tomar¹,

Prajapati

Antal

et al. 2022

Math Methods in App Sciences

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We establish some escape criteria via Jungck–Mann fixed point iteration system with ‐convexity for complex‐valued polynomials of higher orders. As a result, we point out errors in the corresponding existing criterion and develop a correct technique to obtain the escape criterion for analogous fixed point iterations equipped with ‐convexity. Further, we derive a novel escape radius for visualizing the alluring fractals. Toward the end, we utilize our conclusions to generate some variants of classical Mandelbrot and Julia sets. We observe that the visualized fractals are similar to beautiful objects found in nature and each point in Mandelbrot set includes a massive image data of a Julia set. Some examples are also provided to demonstrate the variation in images and discuss the impact of parameters on the deviation of color and appearance of fractals.

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Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process

Cited by 10 publications

References 36 publications

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

Fractals Parrondo’s Paradox in Alternated Superior Complex System

Variants of Mandelbrot and Julia fractals for higher‐order complex polynomials

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