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Fractal dimension evolution and spatial replacement dynamics of urban growth

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  • Chen, Yanguang

Abstract

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 interpret 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 just 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 thus made in this article 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. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos mirror a process of complex replacement.

Suggested Citation

  • Chen, Yanguang, 2012. "Fractal dimension evolution and spatial replacement dynamics of urban growth," Chaos, Solitons & Fractals, Elsevier, vol. 45(2), pages 115-124.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:2:p:115-124
    DOI: 10.1016/j.chaos.2011.10.007
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    Cited by:

    1. Chen, Yanguang, 2014. "An allometric scaling relation based on logistic growth of cities," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 65-77.
    2. Man, Wang & Nie, Qin & Li, Zongmei & Li, Hui & Wu, Xuewen, 2019. "Using fractals and multifractals to characterize the spatiotemporal pattern of impervious surfaces in a coastal city: Xiamen, China," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 520(C), pages 44-53.
    3. Chen, Yanguang, 2013. "Fractal analytical approach of urban form based on spatial correlation function," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 47-60.
    4. Doménech-Carbó, Antonio & Doménech-Casasús, Clara, 2021. "The evolution of COVID-19: A discontinuous approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 568(C).
    5. Chen, Yanguang & Feng, Jian, 2012. "Fractal-based exponential distribution of urban density and self-affine fractal forms of cities," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1404-1416.
    6. Zhonghao Zhang & Rui Xiao & Weixuan Yu & Yue Liu & Meng Lin & Meng Wang, 2017. "Characterizing Factors Associated with Built-Up Land Expansion in Urban and Non-Urban Areas from a Morphological Perspective," Sustainability, MDPI, vol. 9(8), pages 1-15, August.
    7. Chen, Yanguang & Huang, Linshan, 2019. "Modeling growth curve of fractal dimension of urban form of Beijing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1038-1056.
    8. Chen, Yanguang, 2014. "Urban chaos and replacement dynamics in nature and society," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 373-384.
    9. Chen, Yanguang, 2013. "A set of formulae on fractal dimension relations and its application to urban form," Chaos, Solitons & Fractals, Elsevier, vol. 54(C), pages 150-158.
    10. Lü Ye & Yanguang Chen & Yuqing Long, 2023. "Exploring the Relationship between Urbanization and Ikization," Sustainability, MDPI, vol. 15(12), pages 1-17, June.
    11. Chen, Yanguang & Feng, Jian, 2017. "Spatial analysis of cities using Renyi entropy and fractal parameters," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 279-287.
    12. Lei Gong & Jianzhu Yang & Chong Wu & Hui Zhou, 2023. "Fractal Characteristics of the Spatial Texture in Traditional Miao Villages in Qiandongnan, Guizhou, China," Sustainability, MDPI, vol. 15(17), pages 1-23, September.
    13. Doménech-Carbó, Antonio, 2019. "Rise and fall of historic tram networks: Logistic approximation and discontinuous events," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 522(C), pages 315-323.
    14. Chen, Yanguang, 2020. "Equivalent relation between normalized spatial entropy and fractal dimension," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    15. Liu, Jie & Zhang, Lang & Zhang, Qingping & Li, Chao & Zhang, Guilian & Wang, Yuncai, 2022. "Spatiotemporal evolution differences of urban green space: A comparative case study of Shanghai and Xuchang in China," Land Use Policy, Elsevier, vol. 112(C).
    16. Stamov, Gani Tr. & Simeonov, Stanislav & Stamova, Ivanka M., 2018. "Uncertain impulsive Lotka–Volterra competitive systems: Robust stability of almost periodic solutions," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 178-184.

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