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.

Highlights

  • Chaos is one of the important subjects of science in the twentieth century

  • The pattern of phase space cannot be revealed by the one-dimension mapping diagram based on ecological systems, but it can be displayed by the two-dimension mapping diagram based on the rural–urban population migration and transition

  • By the study of urbanization dynamics, we can obtain three aspects of new knowledge about bifurcation and chaos. It is interaction rather than the intrinsic randomicity of dynamic systems that leads to bifurcation and chaos

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Summary

Introduction

Chaos is one of the important subjects of science in the twentieth century. the problems of origin and essence of chaos were not really solved in last century, and they are passed on to the new century. Using the formula of urbanization level, we can derive the logistic equation from the rural–urban population interaction model, which can be discretized to a two-dimensional map. As will be shown below, the two-dimensional rural–urban map can create the bifurcation and chaos that are identical in patterns to those produced by the one-dimensional logistic map. This suggests that the origin of bifurcation and chaos is two-population coupling and interaction rather than intrinsic randomicity of determinate models [8].

The two-population interaction model
Phase portraits of two-dimension map
Data and method
Parameter estimation and model selection
Numerical experiment
Generalization and supposition
Scaling in bifurcation diagrams
Dynamics of fractal dimension evolution of urban growth
Replacement dynamics
Conclusions
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