Escape criterion for generating fractals using Picard–Thakur hybrid iteration

Abstract

Fractals are important to explore the objects of our natural environment, such as mountains, trees, ferns, plants, lightning, rainbows, and snowflakes. Therefore, taking motivation from recent research about a new iteration scheme that converges more quickly than other popular hybrid iterative methods, we apply it to study the escape criterion. Using this criterion, we establish three algorithms. These algorithms are used to generate the important classes of fractals. Moreover, we present the analysis of how the fractal image changes with variations in different input parameters using proposed algorithms. More specifically, the complex function ℷ(z)=zm+b where m≥2,b∈ℂ is considered for the proof and application of the results. Also, we calculate the image generation time by considering different values of the involved parameters. Fascinating J-sets (i.e., Julia sets), M-sets (i.e., Mandelbrot sets), and multicorns (i.e., Mandelbrot sets of conjugate complex polynomial functions) at different values of m can be visualized. The successful application of a novel iteration scheme highlights the significance of this research over the existing one.

Keywords

• Escape criterion
• Fractals
• Picard–Thakur hybrid iteration