Application of CR Iteration Scheme in the Generation of Mandelbrot Sets of zp + log ct Function

• Muhammad Tanveer ^[1] ; Krzysztof Gdawiec ^[2]
  1. [1] University of Agriculture Faisalabad

     University of Agriculture Faisalabad

     Pakistán

     
  2. [2] University of Silesia

     University of Silesia

     Katowice, Polonia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01160-3
• Enlaces
  • Texto completo
• Resumen

  • Mandelbrot set is one of the most fascinating objects in mathematics, so in the literature, one can find many studies related to this set. Moreover, there are many different extensions and generalizations of the classical Mandelbrot set. The two most popular extensions are the use of various kinds of functions and iterations. In this paper, firstly, we replace the constant c in the classical z p +c function used in the definition of Mandelbrot set with log ct , where t ∈ R and t ≥ 1. Secondly, we replace the Picard iteration with the CR iteration scheme. For this combination of function and iteration, we prove the escape criterion for the escape-time algorithm used to generate some images of the proposed sets. Moreover, we study the dependency between the iteration’s parameters and two numerical measures (average number of iterations and generation time). We show that this dependency is complex and non-linear.

• Referencias bibliográficas
  • 1. Mandelbrot, B.B.: The Fractal Geometry ofNature.W.H. Freeman andCompany, San Francisco (1982)
  • 2. Kumari, S., Gdawiec, K., Nandal, A., Postolache, M., Chugh, R.: A novel approach to generate Mandelbrot sets, Julia sets and biomorphs...
  • 3. Gdawiec, K., Kotarski,W.: Polynomiography for the polynomial infinity norm via Kalantari’s formula and nonstandard iterations. Appl. Math....
  • 4. Barnsley, M.F.: Fractals Everywhere, 3rd edn. Dover Publications, New York (2012)
  • 5. Draves, S., Reckase, E.: The fractal flame algorithm. http://flam3.com/flame_draves.pdf, 2003. (accessed on 01.08.2023)
  • 6. Peherstorfer, F., Stroh, C.: Connectedness of Julia sets of rational functions. Comput. Methods Funct. Theory 1(1), 61–79 (2001)
  • 7. Domínguez, P., Fagella, N.: Residual Julia sets of rational and transcendental functions. In: Rippon, P.J., Stallard, G.M. (eds.) Transcendental...
  • 8. Koss, L.: Elliptic functions with disconnected Julia sets. Int. J. Bifurc. Chaos 26(6), 1650095 (2016)
  • 9. Crowe, W.D., Hasson, R., Rippon, P.J., Strain-Clark, P.E.D.: On the structure of the Mandelbar set. Nonlinearity 2(4), 541–553 (1989)
  • 10. Rani, M., Kumar, V.: Superior Mandelbrot set. J. Korea Soc. Math. Edu. Series D: Res. Math. Edu. 8(4), 279–291 (2004)
  • 11. Prajapati, D.J., Rawat, S., Tomar, A., Sajid,M., Dimri, R.C.: A brief study on Julia sets in the dynamics of entire transcendental function...
  • 12. Zou, C., Shahid, A.A., Tassaddiq, A., Khan, A., Ahmad, M.: Mandelbrot sets and Julia sets in Picard- Mann orbit. IEEE Access 8, 64411–64421...
  • 13. Hamada, N., Kharbat, F.:Mandelbrot and Julia sets of complex polynomials involving sine and cosine functions via Picard–Mann orbit. Complex...
  • 14. Abbas, M., Iqbal, H., De la Sen, M.:Generation of Julia andMandelbrot sets via fixed points. Symmetry 12(1), 86 (2020)
  • 15. Tassaddiq, A., Tanveer, M., Azhar, M., Nazeer, W., Qureshi, S.: A four step feedback iteration and its applications in fractals. Fractal...
  • 16. Zhou, H., Tanveer, M., Li, J.: Comparative study of some fixed-point methods in the generation of Julia and Mandelbrot sets. Journal of...
  • 17. Özgür, N., Antal, S., Tomar, A.: Julia and Mandelbrot sets of transcendental function via Fibonacci– Mann iteration. Journal of Function...
  • 18. Li, X.,Tanveer,M., Abbas, M.,Ahmad,M., Kwun,Y.C.,Liu, J.: Fixed point results for fractal generation in extended Jungck-SP orbit. IEEE...
  • 19. Zhang, H., Tanveer, M., Li, Y.-X., Peng, Q., Shah, N.A.: Fixed point results of an implicit iterative scheme for fractal generations....
  • 20. Tassaddiq, A.: General escape criteria for the generation of fractals in extended Jungck-Noor orbit. Math. Comput. Simul. 196, 1–14 (2022)
  • 21. Guran, L., Shabbir, K., Ahmad, K., Bota, M.-F.: Stability, data dependence, and convergence results with computational engendering of...
  • 22. Tomar, A., Prajapati, D.J., Antal, S., Rawat, S.: Variants of Mandelbrot and Julia fractals for higherorder complex polynomials. Mathematical...
  • 23. Panwar, S.S., Singh, K., Mishra, P.K.: Analysis of fangled Mandelbrot and Julia sets controlled by logarithmic function. Int. J. Adv....
  • 24. Tanveer, M., Nazeer,W., Gdawiec, K.: On the Mandelbrot set of z p +log ct via the Mann and Picard- Mann iterations. Math. Comput....
  • 25. Devaney, R.L.: A First Course in Chaotic Dynamical Systems: Theory and Experiment, 2nd edn. CRC Press, Boca Raton (2020)
  • 26. Xiangdong, L.,Zhiliang, Z.,Guangxing,W.,Weiyong, Z.:Composed accelerated escape time algorithm to construct the general Mandelbrot set....
  • 27. Picard, E.: Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J. de Mathématiques...
  • 28. Mann, W.R.: Mean value methods in iteration. Proceed. Am. Math. Soc. 4(3), 506–510 (1953)
  • 29. Khan, S.H.: A Picard–Mann hybrid iterative process. Fixed Point Theory and Applications, 2013:Article ID 69, (2013)
  • 30. Gürsoy, F., Karakaya, V.: A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, 2014. arXiv:...
  • 31. Chugh, R., Kumar, V., Kumar, S.: Strong convergence of a new three step iterative scheme in Banach spaces. Am. J. Comput. Math. 2(04),...
  • 32. Shahid, A.A., Nazeer,W., Gdawiec, K.: The Picard-Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. Monatshefte...