2011/06/06 by Yanguang Chen · 4 citations
Earth and Planetary Sciences · Economics, Econometrics and Finance · Environmental Science · Mathematics · Physics and Astronomy · #Artificial intelligence #Bifurcation #Chaotic #Complex Systems and Time Series Analysis #Computer science #Correlation dimension #Dimension (graph theory) #Fractal #Fractal dimension #Geometry #Land Use and Ecosystem Services #Logistic function #Logistic map #Mathematical analysis #Mathematics #Nonlinear system #Physics #Pure mathematics #Remote Sensing and Land Use #Statistical physics #Statistics #physics.soc-ph
paper · pdf · doi:10.1016/j.chaos.2011.10.007
published as Chaos, Solitons & Fractals, 2012, 45 (2): 115-124 · 17 pages, 5 figures, 2 tables
arxiv created 2011/06/06 · openalex publication_date 2011/12/09 · arxiv updated 2018/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city system will fall into disorder. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos seem to be related with some kind of replacement process.