Dynamics of Julia and Mandelbrot sets: viscosity approximation method with <i>h</i> -convexity

Abstract This study introduces a two-step iterative strategy [40], incorporating the viscosity approximation technique and h -convexity to refine the algorithm’s precision. Central to analysis is the complex function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> <mml:msup> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>sin</mml:mi> <mml:msup> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">N</mml:mi> <mml:mo>\</mml:mo> <mml:mo stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">}</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mo>\</mml:mo> <mml:mo stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">}</mml:mo> </mml:math> with ∣ s ∣ &lt; ∣ w ∣ r −2 and t ∈ [1, ∞ ). A convergence condition is established in the form of an escape criterion, enabling the adaptation of the escape-time algorithm to the specific iteration scheme under consideration. This approach facilitates the generation of intricate fractal structures and analyzes the impact of parameter variations on their morphology. The refined algorithmic framework enriches the computational aesthetics of fractals, particularly in the visualization of Julia and Mandelbrot sets. Furthermore, the relationship between the iterations parameters and two numerical measures, average number of iterations (ANI) and execution time (ET) in MATLAB, measured in seconds, offers valuable insights into the influence of these parameters on the graphical representation of fractal formation.