Development of mandelbrot set for the logistic map with two parameters in the complex plane

Authors

DOI:

https://doi.org/10.15587/1729-4061.2021.249264

Keywords:

fixed points, logistic map, quadratic map, Mandelbrot set, Zoomer Xaos

Abstract

In this paper, the study of the dynamical behavior of logistic map has been disused with representing fractals graphics of map, the logistic map depends on two parameters and works in the complex plane, the map defined by f(z,α,β)=αz(1–z)β. where  and  are complex numbers, and β is a positive integers number, the visualization method used in this work to generate fractals of the map and to inspect the relation between the value of β and the shape of the map, this visualization analysis showed also that, as the value of β increasing, as the number of humps in the function also increasing, and it demonstrate that is true also for the function’s first iteration , f2(x0)=f(f(x0)) and the second iteration , f3(x0)=f(f2(x0)), beside that , the visualization technique showed that the number of humps in that fractal is less than the ones in the second iteration of the original function ,the study of the critical points and their properties of the logistic map also discussed it, whereas finding the fixed point led to find the critical point of the function f, in addition , it haven proven for the set of all pointsα∈C and β∈N, the iteration function f(f(z) has an attractive fixed points, and belongs to the region specified by the disc |1–β(α–1)|<1. Also, The discussion of the Mandelbrot set of the function defined by the f(f(z)) examined in complex plans using the path principle, such that the path of the critical point z=z0 is restricted, finally, it has proven that the Mandelbrot set f(z,α,β) contains all the attractive fixed points and all the complex numbers  in which α≤(1/β+1) (1/β+1) and the region containing the attractive fixed points for f2(z,α,β) was identified

Author Biographies

Wasan Saad Ahmed, University of Diyala

Master in Computer Science

Department of Computer Science

Saad Qasim Abbas, Bilad Alrafidain University College

Doctor of Mathematics, Professor

Department of Medical Instrumentation of Technology

Muntadher Khamees, University of Diyala

Doctor in Computer Science, Professor

Department of Computer Science

Mustafa Musa Jaber, Dijlah University College; AL-Turath University College

PhD, Lecturer

Department of Medical Instrumentation Techniques Engineering

Department of Computer Technology Engineering

References

  1. Yu, D., Ta, W., Zhou, Y. (2021). Fractal diffusion patterns of periodic points in the Mandelbrot set. Chaos, Solitons & Fractals, 153, 111599. doi: https://doi.org/10.1016/j.chaos.2021.111599
  2. Schilling, H. (1988). Peitgen, H.-O.; Richter, P. H., The Beauty of Fractals. Images of Complex Dynamical Systems. Berlin etc., Springer-Verlag 1986. XII, 199 pp., 184 figs., many in color, DM 78,—. ISBN 3-540-15851-0. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik, 68 (10), 512–512. doi: https://doi.org/10.1002/zamm.19880681015
  3. Brooks, R., Matelski, J. P. (1981). The Dynamics of 2-Generator Subgroups of PSL(2, ℂ). Riemann Surfacese and Related Topics (AM-97), 65–72. doi: https://doi.org/10.1515/9781400881550-007
  4. Devaney, R., Keen, L. (Eds). (1989). Chaos and Fractals: The Mathematics Behind the Computer Graphics. Proceedings of Symposia in Applied Mathematics. doi: https://doi.org/10.1090/psapm/039
  5. Choudhury, S. R. (1994). Dynamics and Bifurcations (Jack K. Hale and Huseyin Kocak). SIAM Review, 36 (2), 297–299. doi: https://doi.org/10.1137/1036075
  6. Liu, S., Pan, Z., Fu, W., Cheng, X. (2017). Fractal generation method based on asymptote family of generalized Mandelbrot set and its application. The Journal of Nonlinear Sciences and Applications, 10 (03), 1148–1161. doi: https://doi.org/10.22436/jnsa.010.03.24
  7. May, R. M., Leonard, W. J. (1975). Nonlinear Aspects of Competition Between Three Species. SIAM Journal on Applied Mathematics, 29 (2), 243–253. doi: https://doi.org/10.1137/0129022
  8. May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261 (5560), 459–467. doi: https://doi.org/10.1038/261459a0
  9. Douady, A., Hubbard, J. H. (2007). Etude´ dynamique des polynomes complexes. Societe Mathematique de France. Available at: https://pi.math.cornell.edu/~hubbard/OrsayFrench.pdf
  10. Hao, B.-L., Zheng, W.-M. (1998). Applied Symbolic Dynamics and Chaos. World Scientific, 460. doi: https://doi.org/10.1142/3830
  11. Introduction (2018). Applied Symbolic Dynamics and Chaos, 1–14. doi: https://doi.org/10.1142/9789813236431_0001
  12. S Chen, S., Feng, S., Fu, W., Zhang, Y. (2021). Logistic Map: Stability and Entrance to Chaos. Journal of Physics: Conference Series, 2014 (1), 012009. doi: https://doi.org/10.1088/1742-6596/2014/1/012009
  13. Kwun, Y. C., Tanveer, M., Nazeer, W., Gdawiec, K., Kang, S. M. (2019). Mandelbrot and Julia Sets via Jungck–CR Iteration With s –Convexity. IEEE Access, 7, 12167–12176. doi: https://doi.org/10.1109/access.2019.2892013
  14. Mandelbrot, B. B., Wheeler, J. A. (1983). The Fractal Geometry of Nature. American Journal of Physics, 51 (3), 286–287. doi: https://doi.org/10.1119/1.13295
  15. Lakhtakia, A., Varadan, V. V., Messier, R., Varadan, V. K. (1987). On the symmetries of the Julia sets for the process z⇒zp+c. Journal of Physics A: Mathematical and General, 20 (11), 3533–3535. doi: https://doi.org/10.1088/0305-4470/20/11/051
  16. Kim, T. (2015). Quaternion Julia Set Shape Optimization. Computer Graphics Forum, 34 (5), 167–176. doi: https://doi.org/10.1111/cgf.12705
  17. Drakopoulos, V., Mimikou, N., Theoharis, T. (2003). An overview of parallel visualisation methods for Mandelbrot and Julia sets. Computers & Graphics, 27 (4), 635–646. doi: https://doi.org/10.1016/s0097-8493(03)00106-7
  18. Sun, Y., Chen, L., Xu, R., Kong, R. (2014). An Image Encryption Algorithm Utilizing Julia Sets and Hilbert Curves. PLoS ONE, 9 (1), e84655. doi: https://doi.org/10.1371/journal.pone.0084655
  19. Abbas, S. Q., Abd Almeer, H. A., Ahmed, W. S., Hammid, A. T. (2020). A novel algorithm for generating an edge-regular graph. Procedia Computer Science, 167, 1038–1045. doi: https://doi.org/10.1016/j.procs.2020.03.403
  20. Izhikevich, E. M. (2006). Dynamical Systems in Neuroscience. MIT Press. doi: https://doi.org/10.7551/mitpress/2526.001.0001
  21. Redona, J. F. (1996). The Mandelbrot set. Theses Digitization Project. Available at: https://scholarworks.lib.csusb.edu/cgi/viewcontent.cgi?article=2166&context=etd-project
  22. Fowler, A. C., McGuinness, M. J. (2019). The size of Mandelbrot bulbs. Chaos, Solitons & Fractals: X, 3, 100019. doi: https://doi.org/10.1016/j.csfx.2019.100019
  23. Milnor, J., Thurston, W. (1988). On iterated maps of the interval. Lecture Notes in Mathematics, 465–563. doi: https://doi.org/10.1007/bfb0082847
  24. Pesin, Y., Climenhaga, V. (2009). Lectures on Fractal Geometry and Dynamical Systems. The Student Mathematical Library. doi: https://doi.org/10.1090/stml/052
  25. Kumari, M., Kumari, S., Chugh, R. (2017). International Journal of Mathematics And its Applications Superior Julia Sets and Superior Mandelbrot Sets in SP Orbit. International Journal of Mathematics And its Applications, 5 (2-A), 67–83. Available at: http://ijmaa.in/v5n2-a/67-83.pdf
  26. Khamees, M., Ahmed, W. S., Abbas, S. Q. (2020). Train the Multi-Layer Perceptrons Based on Crow Search Algorithm. 2020 1st. Information Technology To Enhance e-Learning and Other Application (IT-ELA). doi: https://doi.org/10.1109/it-ela50150.2020.9253073
  27. Ashish, Cao, J., Chugh, R. (2018). Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model. Nonlinear Dynamics, 94 (2), 959–975. doi: https://doi.org/10.1007/s11071-018-4403-y
  28. Kim, Y. I., Feldstein, A. (1997). Bifurcation and k-cycles of a finite-dimensional iterative map, with applications to logistic delay equations. Applied Numerical Mathematics, 24 (2-3), 411–424. doi: https://doi.org/10.1016/s0168-9274(97)00036-6
  29. Fruchter, G., Ben-Haim, S. (1991). Stability analysis of one-dimensional dynamical systems applied to an isolated beating heart. Journal of Theoretical Biology, 148 (2), 175–192. doi: https://doi.org/10.1016/s0022-5193(05)80340-6
  30. Hirsch, M. W., Smale, S., Devaney, R. L. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press. doi: https://doi.org/10.1016/c2009-0-61160-0
  31. Weisstein, E. W. Dottie Number." From MathWorld--A Wolfram Web Resource. Available at: https://mathworld.wolfram.com/DottieNumber.html
  32. Alobaidi, M. H., Idan Kadham, O. (2019). Dynamical Behavior of some families of cubic functions in complex plane. Tikrit Journal of Pure Science, 24 (7), 122. doi: https://doi.org/10.25130/j.v24i7.922
  33. Ahmed, W. S. (2013). Construction a MATLAB Program to Solving the Timetable Scheduling Problem. Journal of Engineering and Applied Sciences, 13 (23), 9976–9984. Available at: http://docsdrive.com/pdfs/medwelljournals/jeasci/2018/9976-9984.pdf

Downloads

Published

2021-12-29

How to Cite

Ahmed, W. S., Abbas, S. Q., Khamees, M., & Jaber, M. M. (2021). Development of mandelbrot set for the logistic map with two parameters in the complex plane. Eastern-European Journal of Enterprise Technologies, 6(3 (114), 47–56. https://doi.org/10.15587/1729-4061.2021.249264

Issue

Vol. 6 No. 3 (114) (2021): Control processes

Section

Control processes

License

Copyright (c) 2021 Wasan Saad Ahmed, Saad Qasim Abbas, Muntadher Khamees, Mustafa Musa Jaber

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.