New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates
 | International Journal of Computer Trends and Technology (IJCTT) |  |
| © 2014 by IJCTT Journal | ||
| Volume-9 Number-5 | ||
| Year of Publication : 2014 | ||
| Authors : Suman Joshi , Dr. Yashwant Singh Chauhan , Dr. Ashish Negi | ||
| DOI : 10.14445/22312803/IJCTT-V9P141 |
Suman Joshi , Dr. Yashwant Singh Chauhan , Dr. Ashish Negi." New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates". International Journal of Computer Trends and Technology (IJCTT) V9(5):209-216, March 2014. ISSN:2231-2803. www.ijcttjournal.org. Published by Seventh Sense Research Group.
Abstract -
The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting field of research nowadays. We introduce in this paper the dynamics of polynomials z n - z + c = 0 for n 2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. In order to solve this function by Jungck –type iterative schemes, we write it in the form of Sz = Tz, where the function T, S are defined as Tz = z n + c and Sz = z. Only mathematical explanations are derived by applying Jungck Ishikawa Iteration for polynomials in the literature but in this paper we have generated Relative Mandelbrot sets and Relative Julia sets.
References
[1] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H.Freeman, New York, 1983.
[2] S. Ishikawa, “Fixed points by a new iteration method”, Proc. Amer. Math. Soc.44 (1974), 147-150.
[3] M. O. Bosede, “Strong convergence results for the Jungck-Ishikawa and Jungck-Mann iteration processes”, Bulletin of Mathematical analysis and applications, vol. 2, no. 3, pp. 65-73, 2010.
[4] Rajeshri Rana, Yashwant S Chauhan and Ashish Negi.,”Non Linear Dynamics of Ishikawa Iteration”,International Journal of Computer Applications 7(13):43–49, October 2010.
[5] M. O. Olatinwo, “Some stability and strong convergence results for the Jungck-Ishikawa iteration process”, Creative Mathematics and Informatics, vol. 17, pp. 33-42, 2008.
[6] R. Chugh and V. Kumar, “Strong Convergence and Stability results for Jungck-SP iterative scheme” , Ínternational Journal of Computer Applications, vol. 36,no. 12, 2011.
[7] G. Julia, “Sur 1’ iteration des functions rationnelles”, JMath Pure Appli. 8 (1918), 737-747
[8] G. Jungck, “commuting mappings and fixed points”,The American Mathematical Monthly, vol. 83, no. 4, pp.261-263, 1976.
[9] Eike Lau and Dierk Schleicher, “Symmetries of fractals revisited”, Math. Intelligencer (18) (1) (1996), 45-51.MR1381579 Zbl 0847.30018.
[10] J. Milnor, “Dynamics in one complex variable; Introductory lectures”, Vieweg (1999).
[11] Shizuo Nakane, and Dierk Schleicher, “Non-local connectivity of the tricorn and multicorns”, Dynamical systems and chaos (1) (Hachioji, 1994), 200-203, World Sci. Publ., River Edge, NJ, 1995..
[12] Shizuo Nakane, and Dierk Schleicher, “On multicorns andunicorns: I. Antiholomorphic dynamics. Hyperbolic components and real cubic polynomials”, Internat. J. Bifur. Chaos Appl. Sci. Engrg, (13) (10) (2003), 2825-2844..
[13] Ashish Negi, “Generation of Fractals and Applications”, Thesis, Gurukul Kangri Vishwvidyalaya, (2005).
[14] M. O. Osilike, “Stability results for Ishikawa fixed point iteration procedure”, Indian Journal of Pure and Appl. Math., 26(1995), 937-945.
[15] A.G. D. Philip“Wrapped midgets in the Mandelbrot set”, Computer and Graphics 18 (1994), 239-248.
Keywords
Complex dynamics, Relative Superior Mandelbrot set, Relative Julia set, Jungck Ishikawa Iteration