This paper presents a new perspective of looking at the relation betweenfractals and chaos by means of cities. Especially, a principle of space fillingand spatial replacement is proposed to explain the fractal dimension of urbanform. The fractal dimension evolution of urban growth can be empiricallymodeled with Boltzmann's equation. For the normalized data, Boltzmann'sequation is equivalent to the logistic function. The logistic equation can betransformed into the well-known 1-dimensional logistic map, which is based on a2-dimensional map suggesting spatial replacement dynamics of city development.The 2-dimensional recurrence relations can be employed to generate thenonlinear dynamical behaviors such as bifurcation and chaos. A discovery ismade that, for the fractal dimension growth following the logistic curve, thenormalized dimension value is the ratio of space filling. If the rate ofspatial replacement (urban growth) is too high, the periodic oscillations andchaos will arise, and the city system will fall into disorder. The spatialreplacement dynamics can be extended to general replacement dynamics, andbifurcation and chaos seem to be related with some kind of replacement process.
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