Julia and Mandelbrot Sets of Transcendental Function via Fibonacci-Mann Iteration

Abstract

In this paper, utilizing the Fibonacci-Mann iteration process, we explore Julia and Mandelbrot sets by establishing the escape criteria of a transcendental function, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="normal">sin</a:mi> <a:mfenced open="(" close=")"> <a:mrow> <a:msup> <a:mrow> <a:mi>z</a:mi> </a:mrow> <a:mrow> <a:mi>n</a:mi> </a:mrow> </a:msup> </a:mrow> </a:mfenced> <a:mo>+</a:mo> <a:mi>a</a:mi> <a:mi>z</a:mi> <a:mo>+</a:mo> <a:mi>c</a:mi> </a:math> , <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"> <f:mi>n</f:mi> <f:mo>≥</f:mo> <f:mn>2</f:mn> </f:math> ; here, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"> <h:mi>z</h:mi> </h:math> is a complex variable, and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:mi>a</j:mi> </j:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M5"> <l:mi>c</l:mi> </l:math> are complex numbers. Also, we explore the effect of involved parameters on the deviance of color, appearance, and dynamics of generated fractals. It is well known that fractal geometry portrays the complexity of numerous complicated shapes in our surroundings. In fact, fractals can illustrate shapes and surfaces which cannot be described by the traditional Euclidean geometry.

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