The foundation for time series analysis methods to detect chaos is the notion that phase spaces and dynamics of a nonlinear dynamical system (NDS) can be reconstructed from a single variable, based on Takens embedding theorem (Takens, 1981). Many years ago (Phillips, 1993) I showed that temporal-domain chaos in the presence of anything other than perfect spatial isotropy (and when does that ever happen in the real world?) leads to spatial-domain chaos. This implies an analogous principle in the spatial domain.

Assume an Earth surface system (ESS) characterized by n variables or components x_i, i = 1, 2, . . , n, which vary as functions of each other:

ESS = f(x₁, x_2, , , , x_n)

If spatial variation is directional along a gradient y (of e.g., elevation, moisture, insolation) then

dx_i/dy = f(x₁, x_2, , , , x_n)

dx₂/dy = f(x₁, x_2, , , , x_n)

.                 .                   .

.                 .                   .

.                 .                   .

dx_n/dy = f(x₁, x_2, , , , x_n)

The system can be converted to one highly nonlinear equation by successive differentiation of any x_i:

dx_i/dy = f(x₁’, x₂”, , , x_n^n-1).

Thus, by logic directly analogous to Takens, the spatial series of any realization of the ESS (any component x_i) represents the dynamics of the entire dynamical system.

At least, I think so. Comments welcome.

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Phillips, J.D.  1993. Spatial-domain chaos in landscapes. Geographical Analysis 25: 101-117.

Takens, F. 1981. Detecting strange attractors in turbulence. Lecture Notes in Mathematics 898, Berlin:Springer-Verlag