Hybrid quadratic control method for fractal analysis: Parameter sensitivity in Mandelbrot and Julia sets

In this paper, we introduce the Hybrid Quadratic Control Method (HQCM), which features a novel quadratic growth control mechanism with quadratic scaling <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>z</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> {t(z)=az²} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> {a} . This method is explicitly designed to enhance iterative stability and regulate escape criteria. The HQCM is particularly effective for visualizing and analyzing fractals, as well as exploring complex dynamical systems, especially functions of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo>+</m:mo> <m:mrow> <m:mi>cos</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>c</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {Q(z)=zᵏ+cos(cᵖ)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {k≥ 2} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {p∈[1,∞)} . Through a combination of graphical visualizations and numerical experiments, we investigate how the shapes and structures of the resulting sets evolve with variations in the iteration parameters. The experiments show that changes in parametric values can produce highly complex and diverse shapes, exhibiting the rich behavior of the fractals. To quantify the impact of iteration parameters on the shape and structure of fractals, two numerical metrics, the Non-Escaping Area Index (NAI) and the Average Escape Time (AET), are utilized. These metrics allow for a comprehensive analysis of the dependence of fractal structures on the underlying iteration process, providing valuable insights into the dynamics of the sets and their sensitivity to changes in parametric values.