Abstract
The invention of the latest tools and technology toward computer aided graphics and drawing completely change the thinking view of researchers in analyzing and studying the behavior of a dynamical system. Inspired by work already performed and by adopting the experimental mathematical methods of combining the theory of analytic function with computer aided drawing technology, we generated generalized superior Mandelbrot sets (SM-sets). Also, we analyzed the effect of dynamic noise on SM-sets.
Similar content being viewed by others
References
Andreadis, I., Karakasidis, T.E.: On probabilistic Mandelbrot maps. Chaos Solitons Fractals 42(3), 1577–1583 (2009)
Andreadis, I., Karakasidis, T.E.: On a topological closeness of perturbed Mandelbrot sets. Appl. Math. Comput. 10(215), 3674–3683 (2010)
Andreadis, I., Karakasidis, T.E.: On a topological closeness of perturbed Julia sets. Appl. Math. Comput. 217(6), 2883–2890 (2010)
Argyris, J., Andreadis, I., Pavlos, G., Athanasiou, M.: The influence of noise on the correlation dimension of chaotic attractors. Chaos Solitons Fractals 9(3), 343–361 (1998). Zbl 0933.37045
Argyris, J., Andreadis, I., Karakasidis, T.E.: On perturbations of the Mandelbrot map. Chaos Solitons Fractals 11(7), 1131–1136 (2000). Zbl 0962.37022
Argyris, J., Karakasidis, T.E., Andreadis, I.: On the Julia set of the perturbed Mandelbrot map. Chaos Solitons Fractals 11(13), 2067–2073 (2000). Zbl 0954.37024
Argyris, J., Karakasidis, T.E., Andreadis, I.: On the Julia set of a noise-perturbed Mandelbrot map. Chaos Solitons Fractals 13(2), 245–252 (2002). Zbl 0987.37038
Barnsley, M.F., Devaney, R.L., Mandelbrot, B.B., Peitgen, H.O., Saupe, D., Voss, R.: The Science of Fractal Images. Springer, New York (1988). Zbl 0683.58003
Beck, C.: Physical meaning for Mandelbrot and Julia sets. Physica D 125(3–4), 171–182 (1999). Zbl 0988.37060
Geum, Y.H., Kim, Y.I.: Accurate computation of component centers in the degree -n bifurcation set. Comput. Math. Appl. 48(1–2), 163–175 (2004). Zbl 1154.37348
Glynn, E.F.: The evolution of the Gingerbread Man. Comput. Graph. 15(4), 579–582 (1991)
Gujar, U.G., Bhavsar, V.C.: Fractals from z←z α+c in the complex c-plane. Comput. Graph. 15(3), 441–449 (1991)
Kapitaniak, T., Chua, O.L., Zhong, Q.G.: Experimental synchronization of chaos using continuous control. Int. J. Bifurc. Chaos 4(2), 483–488 (1994). Zbl 0825.93298
Lasota, A., Makey, C.M.: Chaos, Fractals and Noise: Stochastic Aspects of Dynamics, pp. 69–106. Springer, New York (1994). Zbl 0784.58005
Negi, A., Rani, M.: A new approach to dynamic noise on superior Mandelbrot set. Chaos Solitons Fractals 36(4), 1089–1096 (2008)
Pastor, G., Romera, M., Álvarez, G., Arroyo, D., Montoya, F.: Equivalence between subshrubs and Chaotic bands in the Mandelbrot set. Discrete Dyn. Nat. Soc. 5, Article ID 70471, 1–25 (2006). Zbl 1130.37377
Rani, M.: Ph.D. Thesis, Iterative procedure in fractals and chaos, Gurukala Kangri Vishwavidyalaya, Hardwar, India (2002)
Rani, M., Agarwal, R.: Effect of stochastic noise on superior Julia Sets. J. Math. Imaging Vis. 36(1), 63–68 (2010). doi:10.1007/s10851-009-0171-0
Rani, M., Kumar, V.: Superior Julia set. J. Korea Soc. Math. Educ. Ser. D, Res. Math. Educ. 8(4), 261–277 (2004)
Rani, M., Kumar, V.: Superior Mandelbrot set. J. Korea Soc. Math. Educ. Ser. D, Res. Math. Educ., 8(4), 279–291 (2004)
Romera, M., Pastor, G., Álvarez, G., Montoya, F.: External arguments of Douady cauliflowers in the Mandelbrot set. Comput. Graph. 28(3), 437–449 (2004)
Sasmor, J.C.: Fractals for functions with rational exponent. Comput. Graph. 28(4), 601–615 (2004)
Wang, X.Y.: The Fractal Mechanism of Generalized M–J Set, pp. 1–58. Dalian University of Technology Press, Dalian (2002)
Wang, X.Y., Chang, P.J.: Research on fractal structure of generalized M–J sets utilized Lyapunov exponents and periodic scanning techniques. Appl. Math. Comput. 175(2), 1007–1025 (2006). Zbl 1102.37031
Wang, X.Y., Gu, L.N.: Research fractal structures of generalized M–J sets using three algorithms. Fractals 16(1), 79–88 (2008)
Wang, X.Y., Meng, Q.Y.: Research on physical meaning for the generalized Mandelbrot–Julia sets based on Langevin problem. Acta Phys. Sin. 388–395 (2004)
Wang, X., Ruihong, J.: Rendering of the inside structure of the generalized M set period bulbs based on the pre-period. Fractals 16(4), 351–359 (2008)
Wang, X.-Y., Sun, Y.-Y.: The general quaternionic M–J sets on the mapping z←zα+c (αεN). Comput. Math. Appl. 53(11), 1718–1732 (2007)
Wang, X., Zhang, X.: The divisor periodic point of escape-time N of the Mandelbrot set. Appl. Math. Comput. 187(2), 1552–1556 (2007)
Wang, X.Y., Liu, X.D., Zhu, W.Y., Gu, S.: Analysis of c-plane fractal images from z←z α+c for α<0. Fractals 8(3), 307–314 (2000)
Wang, X.Y., Chang, P.J., Gu, N.N.: Additive perturbed Mandelbrot-Julia sets. Appl. Math. Comput. 189(1), 754–765 (2007)
Wang, X., Wang, Z., Lang, Y., Zhang, Z.: Noise perturbed generalized Mandelbrot sets. J. Math. Anal. Appl. 347(1), 179–187 (2008)
Wang, X., Zhang, X., Sun, Y., Li, F.: Dynamics of the generalized M set on escape-line diagram. Appl. Math. Comput. 206(1), 474–484 (2008)
Wang, X., Ruihong, J., Sun, Y.: The generalized Julia set perturbed by composing noise of additive and multiplicative. Discrete Dyn. Nat. Soc. 2009, Article ID 781976 (2009)
Wang, X., Ruihong, J., Zhang, Z.: The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative. Appl. Math. Comput. 210(1), 107–118 (2009)
Wang, X., Ruihong, J., Sun, Y.: Accurate computation of periodic regions’ centers in the general M-set with integer index number. Discrete Dyn. Nat. Soc. 2010, Article ID653816 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Agarwal, R., Agarwal, V. Dynamic noise perturbed generalized superior Mandelbrot sets. Nonlinear Dyn 67, 1883–1891 (2012). https://doi.org/10.1007/s11071-011-0115-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0115-2