Fractal dimension of basin boundaries calculated using the basin entropy

The concept of basin entropy (S-b) was recently introduced as a means to characterize basins of attraction with regard to their complexity. It was also found a connection between S-b and the uncertainty exponent alpha. This connection allows the calculation of the fractal dimension d of the basin boundary between two basins of attraction. However, this method of calculation has not been explored in the literature. In this work we evaluate the performance of the method based upon the basin entropy in the calculation of the fractal dimension of basin boundaries. For that purpose, the method is applied to the calculation of d for several artificial uniform fractals, such as the Koch island and the Sierpinski Carpet, and the values obtained are compared with the exact dimensions obtained by analytical methods. It is concluded that excellent results are generally obtained if the boxes used in the calculation of S-b are chosen adequately, and a simple criterion for this choice is proposed. Numerical arguments are provided to justify the exclusion of small boxes with low resolution in the calculation of d. While the investigation is motivated by the calculation of d for basin boundaries, the method can be applied to any image containing two distinct regions with a boundary, whose dimension has to be determined. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The concept of basin entropy (S-b) is introduced recently to characterize the complexity of basins of attraction, establishing a connection with the uncertainty exponent alpha for calculating the fractal dimension of basin boundaries. The method shows excellent performance in calculating d for various artificial uniform fractals, proposing a simple criterion for choosing boxes and justifying the exclusion of small boxes with low resolution. The approach can be applied to determine the dimension of boundaries in any image with two distinct regions.