  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.
