  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.
