Complex Dynamics of Jungck Ishikawa Iteration and its Applications

Abstract

The main aim of this study is to give the description of fractals and complex dynamics of Mandelbrot and Julia sets using Jungck Ishikawa Iterations. Finding out after how many iterations different functions are attaining fixed point and analyzing the dynamics of different polynomials for n is greater than 2 and applied Jungck Ishikawa Iteration to generate new Julia and Mandelbrot sets. We have discussed the complex dynamics of the sine function and iterate it to generate Relative Superior Mandelbrot set and Julia set. We introduce the hyperbolic cosine function and applied iterative scheme to generate beautiful images with symmetry about x axis and y axis. Escape criteria theorems are also proved for different relational maps using JI iterative scheme .We have also introduces an algorithm and proved theorems for Jungck Ishikawa iteration for non self mappings pair and analyzed its convergence property and stability and dependency of data. newlineNumerical examples are also given in support of the validity of our results .By taking complex values we have also discuss the complex dynamics of various functions like increasing functions decreasing functions oscillating functions and biquadratic functions and compared their convergence speeds and generated beautiful images and analyzed their symmetry. newlineWe have also discussed applications of Jungck Ishikawa Iteration in various fields and the future work that can be done after this study. To study complex dynamics and analyzing the fixed point of quadratic cubic trigonometric exponential logarithmic rational function using Jungck Ishikawa Iteration we used ULTRA FRACTAL and MATLAB software. newline newline