Spatial interaction creates period-doubling bifurcation and chaos of urbanization

This paper provides a new way of looking at complicated dynamics of simple mathematical models. The complicated behavior of simple equations is one of the headstreams of chaos theory. However, a recent study based on dynamical equations of urbanization shows that there are still some undiscovered secrets behind the simple mathematical models such as logistic equation. The rural-urban interaction model can also display varied kinds of complicated dynamics, including period-doubling bifurcation and chaos. The two-dimension map of urbanization presents the same dynamics as that from the one-dimension logistic map. In theory, the logistic equation can be derived from the two-population interaction model. This seems to suggest that the complicated behavior of simple models results from interaction rather than pure intrinsic randomicity. In light of this idea, the classical predator-prey interaction model can be revised to explain the complex dynamics of logistic equation in physical and social sciences. (C) 2009 Elsevier Ltd. All rights reserved.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000268987400003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701Mathematics, Interdisciplinary ApplicationsPhysics, MultidisciplinaryPhysics, MathematicalSCI(E)EI10ARTICLE31316-13254