On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map

Abstract

A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos, 2009Chaos, , 19:2123Chaos, -2129 and (Nonlinear. Dyn. 2013, 73:1155-1163, the authors presented the two kinds of fractal sets of a class of alternated complex map and left some visually observations to be proved about the boundedness and symmetry properties of these fractal sets. In this paper, we improve the previous results by giving the strictly mathematical proofs of the two properties. Some simulations that verify the theoretical proofs are also included.

On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

“…To further reveal the complex internal structure of the M-J set, some supplementary studies, including quantitative methods and visualization methods, were proposed and added to ETA [28][29][30][31][32][33][34][35]. For instance, by analyzing the trajectory of the critical points, Danca et al [28,29] pointed out that the Julia set showed different connectivity by alternating the iteration of switched systems [30].…”

“…For instance, by analyzing the trajectory of the critical points, Danca et al [28,29] pointed out that the Julia set showed different connectivity by alternating the iteration of switched systems [30]. Afterwards, a kind of fractals paradox phenomenon "disconnected+disconnected=connected" and the corresponding graphical investigations were proposed in [31,32]. The Julia deviation distance method and Julia deviation plot method were given in the study of noise-perturbed Julia sets [34,35].…”

“…As mentioned above, although some progress has been made in the research of modifying the ETA to explore M-J sets' properties, such as the bounded trajectories [20,21,28,29,36], symmetry property [19,22,24,31,32], and region location [9,17,18,24,[33][34][35], few studies have been done on the connectivity measurement of M-J sets. For some high-order polynomial maps [29,31,37], the system generally has multiple critical points, which means that the parameter spaces of this type of map is more complicated than the classical Mandelbrot set. Therefore, the motivation of this work is to quantitatively and visually analyze the distribution of Julia's connected regions.…”

The Symmetry in the Noise-Perturbed Mandelbrot Set

“…Early studies on fractals analyzed the symmetry property of the planar M-set generated from map (1) [2] and some generalized maps [19]. Recently, scholars have focused their attention on the symmetry property of the spatial M-set [11,20]. In [20], the authors proved the symmetry of the 3D slice of the M-set generated by an alternative map.…”

“…Recently, scholars have focused their attention on the symmetry property of the spatial M-set [11,20]. In [20], the authors proved the symmetry of the 3D slice of the M-set generated by an alternative map.…”