Non Linear Dynamics of Ishikawa Iteration

ABSTRACT We introduce in this paper the dynamics for Ishikawa iteration procedure. The geometry of Relative Superior Mandelbrot sets are explored for Ishikawa iterates. allowed to be complex numbers. In other words, i Keywords 2 Complex dynamics, Relative Superior Mandelbrot Set, Ishikawa Iteration.. 1. INTRODUCTION Complex graphics of nonlinear dynamical systems have been a subject of intense research nowadays. These graphics are generally obtained by “coloring” the escape speed of the seed points within the certain regions of the complex plane that give rise to the unbounded orbits. The complexity of the mathematical objects such as Julia sets and Mandelbrot sets, in spite of their deceitful simplicity of equations that generate them is truly overwhelming. Perhaps, the Mandelbrot set is the most popular object in the fractal theory. It is believed to be the most beautiful object not only in the real but also in the complex plane. This object was given by Benoit B. Mandelbrot in 1979 and has been the subject of intense research right from its advent. Mandelbrot set and its various extensions and variants have been extensively studied using Picard’s iterations. Recently M. Rani and V. Kumar[21] introduced the superior Mandelbrot sets using Mann iteration procedure. We introduce in this paper a new class of Mandelbrot sets named as Relative Superior Mandelbrot sets using Ishikawa iterations. Our study shows that Relative Superior Mandelbrot sets are exclusively elite and effectively different from other Mandelbrot sets existing in the present literature.

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