Human ecology
Discrete mathematics

Pseudo-phase-spaces

A pseudo-phase-space is a variation on a phase-space. A pseudo-phase-space allows a 1 dimensional data-series to be converted into a 2D (or higher) representation. It is used in non-linear systems to look for periodicity or stability within a data-series.

For a 1D dimemsional data-series, such as:

s(0), s(1), s(2), s(3), ..., s(n), ...,

A 2D representation of the data can be produced by plotting s(i) against s(i+1).

For example:

0123456789ABCD EF
0 173 184 127 82 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 20 135 129 92 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 1 103 94 81 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 34 112 95 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0 0 0 125 131 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0 46 199 0
6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 209
7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 203
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200
9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 232
A 0 0 0 0 0 0 0 0 0 0 0 0 0 40 182 0
B 0 0 0 0 0 0 0 0 0 0 0 0 108 84 0 0
C 0 0 0 0 0 0 0 0 0 25 110 98 0 0 0 0
D 0 0 0 0 0 0 1 100 105 95 0 0 0 0 0 0
E 0 0 0 32 121 116 122 0 0 0 0 0 0 0 0 0
F 393 192 152 107 0 0 0 0 0 0 0 0 0 0 0 0

A pseudo-phase-space of a logistic map { x(n+1) = 3.995 * x(n) * (1 - x(n)) }
Here, the first 5000 values of x(n), which varies between 0 and 1, have been assigned to sixteen intervals. The rows represent values of s(n), the column values of s(n+1). Each entry in the table shows the frequency of events which turn a given s(n) to a given s(n+1). This convention is adopted throughout these webpages.



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Created 27/9/99
Last modified 1/8/00