| Summary: | Fractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we establish an escape criteria using DK-iteration as well as complex sine function <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mo form="prefix">sin</mo><mrow><mo>(</mo><msup><mi>z</mi><mi>m</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mo>;</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> and complex exponential function <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><msup><mi>e</mi><msup><mi>z</mi><mi>m</mi></msup></msup><mo>+</mo><mi>c</mi><mo>;</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We use this to analyze the dynamical behavior of specific fractals namely Julia set and Mandelbrot set. This is achieved by generalizing the existing algorithms, which led to the visualization of beautiful fractals for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> and 4. Moreover, the image generation time in seconds using different values of input parameters is also computed.
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