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(x _{1}, x_{2, }*, , ,

*x*

_{n})

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

dx_{i}/dy = *f(x _{1}, x_{2, }*, , ,

*x*

_{n})

dx_{2}/dy = *f(x _{1}, x_{2, }*, , ,

*x*

_{n})

. . .

. . .

. . .

dx_{n}/dy = *f(x _{1}, 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(x_{1}’, x_{2}”, , , 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