<![CDATA[This paper provides a comprehensive analysis of Julia and Mandelbrot sets through the lens of iterative mappings in the complex plane. We investigate the dynamic behavior of quadratic polynomials, focusing on the fundamental connection between the connectedness of Julia sets and parameters within the Mandelbrot set. Employing the escape-time algorithm, we present detailed computational visualizations and explore key characteristics like critical points, self-similarity, and the topological features of both sets. This paper analyzes filled Julia and Mandelbrot sets based on iterative mappings, including the behavior of fixed points and 2-periodic points. Each definition is rigorously explained from a mathematical perspective, and visual representations are presented via bifurcation curves and cobweb diagrams. 2010 Mathematical Subject Classification: Primary: 54H20, Secondary: MSC 2010: 37B25, 37C25, 37C27]]>
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