On escape criterion of an orbit with s−convexity and illustrations of the behavior shifts in Mandelbrot and Julia set fractals

Our study presents a novel orbit with s −convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type T α , β ( u ) = cos( u m )+ αu + β , for u , α , β ∈ C and m ≥ 2. We also demonstrate t...

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Published inPloS one Vol. 20; no. 1; p. e0312197
Main Authors Alam, Khairul Habib, Rohen, Yumnam, Saleem, Naeem, Aphane, Maggie, Razzaque, Asima
Format Journal Article
LanguageEnglish
Published United States Public Library of Science 07.01.2025
Public Library of Science (PLoS)
Subjects
20th century
Algorithms
Analysis
Computer and Information Sciences
Convex domains
Convexity
Criteria
Data compression
Engineering and Technology
Escape behavior
Fractal geometry
Fractals
Geometry
Illustrations
Mathematicians
Mathematics
Models, Theoretical
Number systems
Physical Sciences
Software
Video compression
India
Online AccessGet full text
ISSN1932-6203
1932-6203
DOI10.1371/journal.pone.0312197

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Abstract Our study presents a novel orbit with s −convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type T α , β ( u ) = cos( u m )+ αu + β , for u , α , β ∈ C and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
AbstractList Our study presents a novel orbit with s−convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type Tα,β(u) = cos(um)+αu + β, for and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
Our study presents a novel orbit with s-convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type T.sub.[alpha],[beta] (u) = cos(u.sup.m )+[alpha]u + [beta], for u, [alpha], [beta] [element of] C and m [greater than or equal to] 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
Our study presents a novel orbit with s-convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type Tα,β(u) = cos(um)+αu + β, for [Formula: see text] and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
Our study presents a novel orbit with s −convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type T α , β ( u ) = cos( u m )+ αu + β , for u , α , β ∈ C and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
Our study presents a novel orbit with s −convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type T α , β ( u ) = cos( u m )+ αu + β , for and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
Our study presents a novel orbit with s-convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type Tα,β(u) = cos(um)+αu + β, for [Formula: see text] and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.Our study presents a novel orbit with s-convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type Tα,β(u) = cos(um)+αu + β, for [Formula: see text] and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.
Audience Academic
Author Aphane, Maggie
Saleem, Naeem
Razzaque, Asima
Alam, Khairul Habib
Rohen, Yumnam
AuthorAffiliation 6 Department of Mathematics, College of Science, King Faisal University, Al-Ahsa, Saudi Arabia
5 Department of Basic Sciences, Preparatory Year, King Faisal University, Al-Ahsa, Saudi Arabia
University of Education, PAKISTAN
4 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria, South Africa
1 Department of Mathematics, National Institute of Technology Manipur, Imphal, Manipur, India
2 Department of Mathematics, Manipur University, Imphal, Manipur, India
3 Department of Mathematics, University of Management and Technology, Lahore, Pakistan
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CitedBy_id crossref_primary_10_3390_fractalfract9010040
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2025 Alam et al. This is an open access article distributed under the terms of the Creative Commons Attribution License: http://creativecommons.org/licenses/by/4.0/ (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
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40063558 - PLoS One. 2025 Mar 10;20(3):e0320234. doi: 10.1371/journal.pone.0320234.
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– reference: 40063558 - PLoS One. 2025 Mar 10;20(3):e0320234. doi: 10.1371/journal.pone.0320234.
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Snippet Our study presents a novel orbit with s −convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape...
Our study presents a novel orbit with s-convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape...
Our study presents a novel orbit with s−convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape...
Our study presents a novel orbit with s −convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape...
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SubjectTerms 20th century
Algorithms
Analysis
Computer and Information Sciences
Convex domains
Convexity
Criteria
Data compression
Engineering and Technology
Escape behavior
Fractal geometry
Fractals
Geometry
Illustrations
Mathematicians
Mathematics
Models, Theoretical
Number systems
Physical Sciences
Software
Video compression
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Title On escape criterion of an orbit with s−convexity and illustrations of the behavior shifts in Mandelbrot and Julia set fractals
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http://dx.doi.org/10.1371/journal.pone.0312197
Volume 20
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