Reinterpreting the Origin of Bifurcation and Chaos by Urbanization Dynamics

Abstract

Chaos associated with bifurcation makes a new science, but the origin and essence of chaos are not yet clear. Based on the well-known logistic map, chaos used to be regarded as intrinsic randomicity of determinate dynamics systems. However, urbanization dynamics indicates new explanation about it. Using mathematical derivation, numerical computation, and empirical analysis, we can explore chaotic dynamics of urbanization. The key is the formula of urbanization level. The urbanization curve can be described with the logistic function, which can be transformed into 1-dimensional map and thus produce bifurcation and chaos. On the other hand, the logistic model of urbanization curve can be derived from the rural-urban population interaction model, and the rural-urban interaction model can be discretized to a 2-dimensional map. An interesting finding is that the 2-dimensional rural-urban coupling map can create the same bifurcation and chaos patterns as those from the 1-dimensional logistic map. This suggests that the urban bifurcation and chaos come from spatial interaction between rural and urban population rather than pure intrinsic randomicity of determinate models. This discovery provides a new way of looking at origin and essence of bifurcation and chaos. By analogy with urbanization models, the classical predator-prey interaction model can be developed to interpret the complex dynamics of the logistic map in physical and social sciences.