Urban Systems as Examples of Bounded Chaos: Exploring the Relationship between Fractal Dimension, Rank-Size, and Rural-to-Urban Migration

Fractal structures appear to be closely related to systems whose growth to a large extent is the result of a series of unpredictable events. Examples of such structures include river networks, certain political boundaries, the growth of a city, and the development of urban systems. In this paper, we demonstrate how systems of cities have a fractal dimension and that this fractal dimension is related to a very commonly used index of urban systems, the rank-size parameter. Urban systems can be thought of as examples of ordered or bounded chaos in that the growth of a particular urban system results from countless, unpredictable decisions which makes it unique but the range of urban systems that we observe in reality is rather limited. We demonstrate a possible explanation for the bounded nature of urban system development which is based on rural-to-urban migration. It is shown that the controlling influence on the development of an urban system is the choice process by which migrants select alternatives. Both the fractal dimension and rank-size parameter of an urban system can be related to the parameters of a spatial choice model. Several examples are given of simulated urban systems, each of which is unique but yet has the same fractal dimension and rank-size parameter when derived from the same spatial choice model.