operationally defined “energy states.” Smooth changes along the path of the nonlinear parameter manifold generated discontinuous changes in energy levels indicating states of the observable. Crossing a wrinkle in an “independent variable” (some call it “order parameter” to indicate its emergence rather than availability for predictable manipulation) such as the nonlinear parameter surface of the countervailing influences of survival fear and financial cost, may lead to a bifurcation
in behavior from peace (“low energy”) to war (“high energy”) (Zeeman, 1977).
In a similar geometric spirit but dealing with nonequilibrium systems. in motion, the conditions such that one could “smoothly” embed a trajectory like a continuously recorded EEG record, a complicatedly coiled snake into a three or higher dimensional box without loss of its essential dynamical or statistically measureable properties, was settled by Whitney in what is now referred to as the “embedding theorem” (Whitney, 1936). Starting with a tangled knot of overlapping vectorial orbits with apparent “non-invertable points” (given a point, one cannot chose among or between the more than one point that it apparently came from), it can always be unwrapped into a non-crossing trajectory satisfying uniqueness when reconstructed in a box of a little more than twice the parameter-determined
dimension of the original space of observables.
A common technique for the spatial reconstruction of the output of a dynamical system is called a “time delay embedding.” This approach, first
suggested by Ruelle (1987, pg. 28) replaced the value, x, versus the time
derivative, = phase portrait plot described for a continuously perturbed bob on a
spring above. A sequence of observables over time, in, for example, three dimensional “phase space” (Packard et al, 1980; Takens, 1981; Sauer et al, 1991), is depicted by a curve representing the system’s trajectory at times 4, f, fs, by sliding one-by-one down the series and plotting each pj, Po, ps, location with respect to each other along the x, y and z axes respectively. The choice of time interval between the points, the delay, can be delicate and usually some standard fraction of the decay time of the sequence’s autocorrelation length, “the decay time
of mutual information” is chosen. There are many technical considerations,
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