13.2 Growth in a Population and Chaos Theory
561
even if you have perfect initial data, the initial data stored in a computer will have
an uncertainty at the limits of the computer’s precision, where the number has been
truncated to fit the allocated memory. Again, the uncertainty in the solutions can
grow exponentially.
Systems that exhibit chaos will correspondingly generate entropy in the course
of time through the information they lose.
Now, back to the ‘logistic map’ produced by Eq. (13.5). If r ≤ 4 and 0 ≤ x i ≤ 1,
the new value x i+1 will also range from 0 to r/4 ≤ 1. In effect, the range of x i
has been ‘folded’ onto itself by the logistic map, preparing x i+1 to be ‘acted’ on
again. This folding is shown geometrically in Fig. 13.2 by the horizontal lines drawn
from the ‘y’ axis, representing x i+1 extending to the diagonal line y = x, and then
straight down to the x axis to start a new mapping. Solutions having 0 < x i < 1
occur as i increases for each value of r. If r is selected less than one, iteration of
Fig. 13.2 Logistic mapping for various r. Where the horizontal lines in the figures become dense
and clustered, there is an attractor value within that cluster. (Taken from Wikipedia, Logistic map)
561
even if you have perfect initial data, the initial data stored in a computer will have
an uncertainty at the limits of the computer’s precision, where the number has been
truncated to fit the allocated memory. Again, the uncertainty in the solutions can
grow exponentially.
Systems that exhibit chaos will correspondingly generate entropy in the course
of time through the information they lose.
Now, back to the ‘logistic map’ produced by Eq. (13.5). If r ≤ 4 and 0 ≤ x i ≤ 1,
the new value x i+1 will also range from 0 to r/4 ≤ 1. In effect, the range of x i
has been ‘folded’ onto itself by the logistic map, preparing x i+1 to be ‘acted’ on
again. This folding is shown geometrically in Fig. 13.2 by the horizontal lines drawn
from the ‘y’ axis, representing x i+1 extending to the diagonal line y = x, and then
straight down to the x axis to start a new mapping. Solutions having 0 < x i < 1
occur as i increases for each value of r. If r is selected less than one, iteration of
Fig. 13.2 Logistic mapping for various r. Where the horizontal lines in the figures become dense
and clustered, there is an attractor value within that cluster. (Taken from Wikipedia, Logistic map)