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Effect of Stochastic Noise on Superior Julia Sets

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Abstract

Julia sets are considered one of the most attractive fractals and have wide range of applications in science and engineering. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck (Physica D 125(3–4):171–182, 1999) to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter. Argyris et al. (Chaos Solitons Fractals 11(13):2067–2073, 2000) have studied the effect of noise on Julia sets and concluded that Julia sets are stable for noises of low strength, and a small increment in the strength of noise may cause considerable deterioration in the configuration of the Julia sets. It is well-known that the method of function iterates plays a crucial role in discrete dynamics utilizing the techniques of fractal theory. However, recently Rani and Kumar (J. Korea Soc. Math. Edu. Ser. D: Res. Math. Edu. 8(4):261–277, 2004) introduced superior iterations as a generalization of function iterations in the study of Julia sets and studied superior Julia sets. This technique is further utilized to study effectively new Mandelbrot sets and related properties (see, for instance, Negi and Rani, Chaos Solitons Fractals 36(2):237–245, 2008; 36(4):1089–1096, 2008, Rani and Kumar, J. Korea Soc. Math. Edu. Ser. D: Res. Math. Edu. 8(4):279–291, 2004). The intent of this paper is to study certain effects of noise on superior Julia sets. We find that the superior Julia sets are drastically more stable for higher strength of noises than the classical Julia sets. Finally, we make a humble attempt to discuss some applications of superior orbit in discrete dynamics and of superior Julia sets in particle dynamics.

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Authors and Affiliations

  1. Faculty of Computer Systems & Software Engineering, University Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Kuantan, Pahang Darul Makmur, Malaysia

    Mamta Rani

  2. IEC College of Engg. & Tech., Greater Noida, U.P., India

    Rashi Agarwal

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  1. Mamta Rani
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  2. Rashi Agarwal
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Correspondence to Mamta Rani.

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Rani, M., Agarwal, R. Effect of Stochastic Noise on Superior Julia Sets. J Math Imaging Vis 36, 63–68 (2010). https://doi.org/10.1007/s10851-009-0171-0

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  • DOI: https://doi.org/10.1007/s10851-009-0171-0

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