The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets

Shahid, Abdul Aziz; Nazeer, Waqas; Gdawiec, Krzysztof

doi:10.1007/s00605-021-01591-z

Cited by 20 publications

(5 citation statements)

References 36 publications

(35 reference statements)

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“…Analysing dependency between multicorn sets and the iteration input parameters is very complex. See, for example, [22,24,25], wherein the authors present the dependency via numerical measures.…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

Multicorn Sets of z¯k+cm via S-Iteration with h-Convexity

Tassaddiq,

Tanveer,

Israr

et al. 2023

Fractal Fract

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Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets and their dynamic nature. In this paper, we study the dynamics of multicorn sets of z¯k+cm, where k≥2, c≠0∈C, and m∈R, by using S-iteration with h-convexity instead of standard S-iteration. We develop escape criterion z¯k+cm for S-iteration with h-convexity. We analyse the dynamical behaviour of the proposed conjugate complex function and discuss the variation of iteration parameters along with function parameter m. Moreover, we discuss the effects of input parameters of the proposed iteration and conjugate complex functions of the behaviour of multicorn sets with numerical simulations.

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Section: Numerical Simulations and Discussionmentioning

confidence: 99%

Multicorn Sets of z¯k+cm via S-Iteration with h-Convexity

Tassaddiq,

Tanveer,

Israr

et al. 2023

Fractal Fract

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“…Over time, mathematicians have developed several generalizations of these sets, including the use of different iteration methods derived from fixed-point theory. These techniques have been utilized to generalize Julia and Mandelbrot sets [30][31][32][33][34].…”

Section: Escape Criteria For Complex Fractal Generationmentioning

confidence: 99%

“…We can break down Q c into two mappings, U and V, such that Q c = V − U and U is an injective function. In addition to this reconstruction, a new escape criterion for the mappings and Equation (32) must also be derived.…”

Section: Escape Criteria For Complex Fractal Generationmentioning

confidence: 99%

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Stability, Data Dependence, and Convergence Results with Computational Engendering of Fractals via Jungck–DK Iterative Scheme

et al. 2023

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We have developed a Jungck version of the DK iterative scheme called the Jungck–DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck–DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck–DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration.

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“…Later on, various researchers ( [8][9][10][11][12][13][14][15][16][17][18][19][20][21]) used different iterative processes and obtained variants of these sets to study their behavior and pattern for different polynomials because it is known that the shape, size, color, and other characteristics vary with the iterative procedures for the same function. Shahid et al [22] introduced a new iterative scheme to study the behavior of orbits and dynamics for a (k + 1)st degree complex polynomial. Sajid and Kapoor [23] generated Julia sets of a family of transcendental meromorphic functions having rational Schwarzian derivatives.…”

Section: Introductionmentioning

confidence: 99%

A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme

Prajapati¹,

Rawat

Tomar

et al. 2022

Fractal Fract

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In this research, we look at the Julia set patterns that are linked to the entire transcendental function f(z)=aezn+bz+c, where a,b,c∈C and n≥2, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function.

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The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets

Cited by 20 publications

References 36 publications

Multicorn Sets of z¯k+cm via S-Iteration with h-Convexity

Multicorn Sets of z¯k+cm via S-Iteration with h-Convexity

Stability, Data Dependence, and Convergence Results with Computational Engendering of Fractals via Jungck–DK Iterative Scheme

A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme

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