Abstract
This paper investigates the destruction of the symmetrical structure of the noise-perturbed Mandelbrot set (M-set). By applying the “symmetry criterion” method, we quantitatively compare the damages to the symmetry of the noise-perturbed Mandelbrot set resulting from additive and multiplicative noises. Because of the uneven distribution between the core positions and the edge positions of the noise-perturbed Mandelbrot set, the comparison results reveal a paradox between the visual sense and quantified result. Thus, we propose a new “visual symmetry criterion” method that is more suitable for the measurement of visual asymmetry.
Highlights
In the early 20th Century, French mathematician Gaston Julia [1] focused on the following simple map: (xn+1 = xn2 − y2n + c1 f : (1)yn+1 = 2xn yn + c2, where xn, yn, c1, c2 ∈ R
To analyze the connectivity property of the Julia set with different parameters (c1, c2 ), Mandelbrot [2] revealed another set of classical fractals, the Mandelbrot set (M-set), that was formed with all the values (c1, c2 ) that make its corresponding Julia set connected
The present paper demonstrates that the symmetry of the noise-perturbed M-set can be measured in two ways: (1) on the basis of the virtual density distribution and (2) on the basis of the visual sense
Summary
As one of the most basic sets of fractals and one of the hottest topics of nonlinear science theory, the Mandelbrot set has drawn increasing attention for its theoretical investigations, such as the topological structure analysis [3,4,5], properties [6,7], control [8,9], and high-dimensional developments of such sets [9,10,11]. Studies on fractals analyzed the symmetry property of the planar
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