Dynamical fractal: Theory and case study

• Published in: Chaos, solitons and fractals solitons and fractals, 2023-11, Vol.176, p.114190, Article 114190
• Main Author: Yin, Junze
• Format: Article
• Language: English
• Subjects: Difference equation; Fractal; Fractal dimension; Logistic differential equation; Urbanization
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2023.114190

Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but none for this specific problem, nor extrapolating the future trend. In the first part of this paper, we construct a theoretical framework for analyzing dynamic (changing) fractals and extrapolating their future trends based on their fractal dimension—a measure of the complexity of the fractal. We believe this approach holds enormous potential for applications in analyzing changing fractals in the real world, such as urban growth, cells, cancers, etc., all of which are invaluable to research. This theoretical framework may shed light on a factor overlooked in past research: the trend of how fractals change. In the second part of this paper, we apply this theoretical framework to study the urbanization of Boston. We compare several maps and measurements of fractal dimensions, produce code11https://github.com/yinj66/fractal-dimension. that reads the maps and divides the city into subsections, and ultimately graph the fractal dimension over time using both differential and difference equations. Finally, we postulate the logistic equation as a model to fit the evolution of the fractal dimension, as well as the total population derived from census data, which serves as a component for comparing dynamical fractals. •Formally define dynamical fractal and analyze its mathematical properties.•Develop models of extrapolating short & long-term trends of dynamical fractals.•Use the concept of curvature to compare the similarity of two dynamical fractals.•Apply all the theoretical concepts to our case study: The urbanization of Boston.•Develop new tools approximating fractal dimension and compare them with existing ways.