Complex and Inverse Complex Dynamics of Fractals using Ishikawa Iteration

Complex graphics of dynamical system have been a subject of intense research nowadays. The fractal geometry is the base of these beautiful graphical images. Many researchers and authors have worked to study the complex nature of the two most popular sets in fractal geometry, the Julia set and the Mandelbrot set, and proposed their work in various forms using existing tools and techniques. Still researches are being conducted to study and reveal the new concepts unexplored in the complexities of these two most popular sets of fractal geometry. Recently, Ashish Negi, Rajeshri Rana and Yashwant S. Chauhan are among those researchers who have contributed a lot in the area of Fractal Geometry applications. In this paper we review the recently done work on complex and inverse complex functions for producing beautiful fractal graphics. The reviewed work mainly emphasizes on the study of the nature of complex and inverse complex functional dynamics using Ishikawa iterates and existence of relative superior Mandel-bar set.

[1]  P. J. Rippon,et al.  On the structure of the Mandelbar set , 1989 .

[2]  Ken Shirriff,et al.  An investigation of fractals generated by z --> 1/zn + c , 1993, Comput. Graph..

[3]  J. Milnor.  Dynamics in one complex variable , 2000 .

[4]  G. Julia.  Mémoire sur l'itération des fonctions rationnelles , 1918 .

[5]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[6]  Ashish Negi,et al.  Inverse Complex Function Dynamics of Ishikawa Iterates , 2010 .

[7]  Ashish Negi,et al.  Mandel-Bar Sets of Inverse Complex Function , 2010 .

[8]  Ashish Negi,et al.  Complex Dynamics of Ishikawa Iterates for Non Integer Values , 2010 .

[9]  Uday G. Gujar,et al.  Fractal images from z <-- z alpha + c in the complex z-plane , 1992, Comput. Graph..

[10]  Uday G. Gujar,et al.  Analysis of z-plane fractal images from z <-- z alpha + c for alpha < 0 , 1993, Comput. Graph..

[11]  Dierk Schleicher,et al.  Symmetries of fractals revisited , 1996 .

[12]  Dierk Schleicher,et al.  On Multicorns and unicorns I: Antiholomorphic Dynamics, Hyperbolic Components and Real cubic Polynomials , 2003, Int. J. Bifurc. Chaos.

[13]  Xingyuan Wang,et al.  ANALYSIS OF C-PLANE FRACTAL IMAGES FROM z ← zα + c FOR (α < 0) , 2000 .

[14]  S. Ishikawa.  Fixed points by a new iteration method , 1974 .

[15]  Robert L. Devaney,et al.  A First Course In Chaotic Dynamical Systems: Theory And Experiment , 1993 .

[16]  Luis Garrido.  Dynamical System and Chaos , 1983 .

[17]  J. Milnor,et al.  Dynamics in One Complex Variable: Introductory Lectures , 2000 .

[18]  Uday G. Gujar,et al.  Fractals from z <-- z alpha + c in the complex c-plane , 1991, Comput. Graph..