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 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.