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

• Published in: PloS one Vol. 20; no. 1; p. e0312197
• Main Authors: Alam, Khairul Habib, Rohen, Yumnam, Saleem, Naeem, Aphane, Maggie, Razzaque, Asima
• Format: Journal Article
• Language: English
• 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
• ISSN: 1932-6203; 1932-6203
• DOI: 10.1371/journal.pone.0312197

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.