Variants of Julia and Mandelbrot sets as fractals via Jungck-Ishikawa fixed point iteration system with $ s $-convexity

Swati Antal; Anita Tomar; Darshana J. Prajapati; Mohammad Sajid; Swati Antal; Anita Tomar; Darshana J. Prajapati; Mohammad Sajid

doi:10.3934/math.2022611

AIMS Mathematics

2022, Volume 7, Issue 6: 10939-10957. doi: 10.3934/math.2022611

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Variants of Julia and Mandelbrot sets as fractals via Jungck-Ishikawa fixed point iteration system with $ s $-convexity

1.
H.N.B. Garhwal University, B.G.R. Campus, Pauri Garhwal 246001, India
2.
Sri Dev Suman Uttarakhand University, Pt.L.M.S. Campus, Rishikesh, Uttarakhand, India
3.
M.B. Patel Institute of Technology, New Vallabh Vidya Nagar, Gujarat 388120, India
4.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi Arabia

Received: 10 December 2021 Revised: 19 March 2022 Accepted: 25 March 2022 Published: 02 April 2022
MSC : 28A10, 31E05, 37F10, 37F46, 47H10

In this paper, we generate some non-classical variants of Julia and Mandelbrot sets, utilizing the Jungck-Ishikawa fixed point iteration system equipped with $ s $-convexity. We establish a novel escape criterion for complex polynomials of a higher degree of the form $ z^n + az^2 -bz + c $, where $ a, \; b $ and $ c $ are complex numbers and furnish some graphical illustrations of the generated complex fractals. In the sequel, we discuss the errors committed by the majority of researchers in developing the escape criterion utilizing distinct fixed point iterations equipped with $ s $-convexity. We conclude the paper by examining variation in images and the impact of parameters on the deviation of dynamics, color and appearance of fractals. It is fascinating to notice that some of our fractals represent the traditional Kachhi Thread Works found in the Kutch district of Gujarat (India) which is useful in the Textile Industry.
- escape criterion,
- fixed point,
- Jungck-Ishikawa orbit,
- $ s $-convexity
Citation: Swati Antal, Anita Tomar, Darshana J. Prajapati, Mohammad Sajid. Variants of Julia and Mandelbrot sets as fractals via Jungck-Ishikawa fixed point iteration system with $ s $-convexity[J]. AIMS Mathematics, 2022, 7(6): 10939-10957. doi: 10.3934/math.2022611

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Abstract

In this paper, we generate some non-classical variants of Julia and Mandelbrot sets, utilizing the Jungck-Ishikawa fixed point iteration system equipped with $ s $-convexity. We establish a novel escape criterion for complex polynomials of a higher degree of the form $ z^n + az^2 -bz + c $, where $ a, \; b $ and $ c $ are complex numbers and furnish some graphical illustrations of the generated complex fractals. In the sequel, we discuss the errors committed by the majority of researchers in developing the escape criterion utilizing distinct fixed point iterations equipped with $ s $-convexity. We conclude the paper by examining variation in images and the impact of parameters on the deviation of dynamics, color and appearance of fractals. It is fascinating to notice that some of our fractals represent the traditional Kachhi Thread Works found in the Kutch district of Gujarat (India) which is useful in the Textile Industry.

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