Abstract
In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.
Highlights
Since the concept of fractals was proposed in the mid-1970s [1,2], it has gradually become an active research hotspot of nonlinear science
As one of the important branches of fractal, the research on Julia sets can be traced back to the early 20th century, when French mathematician Gaston Julia [12] considered the characteristics of a simple complex map zn+1 = z2n + c, z, c ∈ C in the case n → ∞
The main novelties of this work can be summarized as follows: (1). This is the first attempt to solve the problem of measuring the connectivity of Julia sets generated from polynomial maps
Summary
Since the concept of fractals was proposed in the mid-1970s [1,2], it has gradually become an active research hotspot of nonlinear science. Some follow-up studies are proposed for the Julia set construction algorithm of generalized polynomial mapping, zn+1 = zkn + c, k ∈ N+ [18,19]. 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]. The main novelties of this work can be summarized as follows: This is the first attempt to solve the problem of measuring the connectivity of Julia sets generated from polynomial maps.
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