On the Julia sets of a noise-perturbed Mandelbrot map (Q5950948)

The authors present numerical results concerning the application of the smooth decomposition method of \textit{J. Peinke} et al. [Z. Naturforsch. A 43, 14-16 (1987; Zbl 0987.37501)] to the Julia set of a slightly noise perturbed Mandelbrot map subject to several type of noise. In Table 1 they present the correspondings of the Mandelbrot map under the four given types of noise, and the values of the noise parameters in the case of a low-strength noise. In Figs. 1-4, the Mandelbrot maps in the case of a low-strength noise are presented. They say in this paper that the obtained results for all four types of noise are in accordance with that of Peinke et al. in the case of the generalized Mandelbrot set under low-strength noise: in the interior of the Julia sets a \(2^n\)-type bifurcation occurs in the sense of the appearance of empty space from the interior towards the outer region, and that in the case of higher-strength noise a similar bifurcation phenomenon is observed from the outer region of the Julia set, at least in the parameter region that they investigated.