Generation of Julia and Mandelbrot Sets via Fixed Points

Mujahid Abbas,Manuel De La Sen,Hira Iqbal

doi:10.3390/sym12010086

Abstract

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T ( x ) = x n + m x + r where m , r ∈ C and n ≥ 2 . Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals.

Highlights

Fixed point theory provides a suitable framework to investigate various nonlinear phenomena arising in the applied sciences including complex graphics, geometry, biology and physics [1,2,3,4]
Fractals can be treated as self similar mathematical structures which have similarity and symmetry such that considerably small parts of the shape are geometrically akin to the whole shape
Julia [5] who is considered as one of the pioneers of fractal geometry, studied iterated complex polynomials and introduced Julia set as a classical example of fractals

Summary

Introduction

Fixed point theory provides a suitable framework to investigate various nonlinear phenomena arising in the applied sciences including complex graphics, geometry, biology and physics [1,2,3,4].Complex graphical shapes such as fractals, were discovered as fixed points of certain set maps [1].Informally, fractals can be treated as self similar mathematical structures which have similarity and symmetry such that considerably small parts of the shape are geometrically akin to the whole shape.Fractals are known as expanding symmetries or unfolding symmetries. Fixed point theory provides a suitable framework to investigate various nonlinear phenomena arising in the applied sciences including complex graphics, geometry, biology and physics [1,2,3,4]. Complex graphical shapes such as fractals, were discovered as fixed points of certain set maps [1]. Fractals can be treated as self similar mathematical structures which have similarity and symmetry such that considerably small parts of the shape are geometrically akin to the whole shape. The behaviour of the iterates T i ( x ) for large i determine the Julia set (see [1,6,7,8])

Objectives

Results

Conclusion