Nonlinear Dyn (2012) 67:1883–1891 DOI 10.1007/s11071-011-0115-2 ORIGINAL PAPER Dynamic noise perturbed generalized superior Mandelbrot sets Rashi Agarwal · Vishal Agarwal Received: 12 January 2011 / Accepted: 29 May 2011 / Published online: 13 July 2011 © Springer Science+Business Media B.V. 2011 Abstract The invention of the latest tools and tech- nology 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 com- puter aided drawing technology, we generated gener- alized superior Mandelbrot sets (SM-sets). Also, we analyzed the effect of dynamic noise on SM-sets. Keywords Mandelbrot set · Superior orbit · Superior Mandelbrot set · Additive noise · Multiplicative noise · General noise 1 Introduction In recent 20 years, researchers have found existed orderly structure in generalized Mandelbrot-sets (M- sets) constructed from the complex map z n+1 = z α n + c(α ∈ R) [8, 23] and the direct relationship between α R. Agarwal ( ) Department of Computer Science, Sharda University, Greater Noida, India e-mail: agarwal_rashi@yahoo.com V. Agarwal Aricent, Gurgoan, India e-mail: agarwal_vishal@yahoo.com and visual structural characteristics of generalized M- sets [12]. Some of the authors found asymmetric and symmetric evolution of generalized M-sets when the phase angle θ ∈ [−π,π), described the fractal growth along certain shock lines [11] and put forward the em- bedded topological distribution theorem of the gen- eralized M-sets [30]. Sasmor analyzed both the dy- namic and parameter spaces and discussed the conse- quences of the discontinuity of the function f (z) [22]. Romera and Pastor et al. researched on the equiv- alence relationship between subshrubs and chaotic bands in the Misiurewicz points [16, 21]. Geum and Wang studied the structure, distribution of the peri- odic bud of generalized M-sets, and the topological rule of their periodic trajectories [10, 24]. Beck and Wang both discussed the physical meaning of gener- alized M-sets [9, 26]. Andreadis and Karakasidis re- searched probabilistic Mandelbrot sets and topological closeness of perturbed Mandelbrot sets and perturbed Julia sets [1–3]. Indeed, a pure deterministic system rarely exists in reality as stochastic noise is ubiquitous. Argyris et al. have discussed the notion of a dynamical noise [4–7] and studied the effect of additive and multiplicative noises on the perturbation of Mandelbrot map [5] and Julia sets [4]. Stochastic aspects of dynamics and syn- chronization of chaos was experimentally studied us- ing continuous control [13, 14]. Further, the effect of noise on the correlation dimension of chaotic attrac- tors was studied [4].