2.40 Lyapunov Fractals
(type=lyapunov)
The Bifurcation fractal illustrates what happens in a simple population
model as the growth rate increases. The Lyapunov fractal expands that
model into two dimensions by letting the growth rate vary in a periodic
fashion between two values. Each pair of growth rates is run through a
logistic population model and a value called the Lyapunov Exponent is
calculated for each pair and is plotted. The Lyapunov Exponent is
calculated by adding up log | r - 2*r*x| over many cycles of the
population model and dividing by the number of cycles. Negative Lyapunov
exponents indicate a stable, periodic behavior and are plotted in color.
Positive Lyapunov exponents indicate chaos (or a diverging model) and
are colored black.
Order parameter. Each possible periodic sequence yields a two
dimensional space to explore. The Order parameter selects a sequence.
The default value 0 represents the sequence ab which alternates between
the two values of the growth parameter. On the screen, the a values run
vertically and the b values run horizontally. Here is how to calculate
the space parameter for any desired sequence. Take your sequence of a's
and b's and arrange it so that it starts with at least 2 a's and ends
with a b. It may be necessary to rotate the sequence or swap a's and
b's. Strike the first a and the last b off the list and replace each
remaining a with a 1 and each remaining b with a zero. Interpret this
as a binary number and convert it into decimal.
An Example. I like sonnets. A sonnet is a poem with fourteen lines
that has the following rhyming sequence: abba abba abab cc. Ignoring
the rhyming couplet at the end, let's calculate the Order parameter for
this pattern.
abbaabbaabab doesn't start with at least 2 a's
aabbaabababb rotate it
1001101010 drop the first and last, replace with 0's and 1's
512+64+32+8+2 = 618
An Order parameter of 618 gives the Lyapunov equivalent of a sonnet.
"How do I make thee? Let me count the ways..."
Population Seed. When two parts of a Lyapunov overlap, which spike
overlaps which is strongly dependent on the initial value of the
population model. Any changes from using a different starting value
between 0 and 1 may be subtle. The values 0 and 1 are interpreted in a
special manner. A Seed of 1 will choose a random number between 0 and 1
at the start of each pixel. A Seed of 0 will suppress resetting the seed
value between pixels unless the population model diverges in which case
a random seed will be used on the next pixel.
Filter Cycles. Like the Bifurcation model, the Lyapunov allow you to
set the number of cycles that will be run to allow the model to approach
equilibrium before the lyapunov exponent calculation is begun. The
default value of 0 uses one half of the iterations before beginning the
calculation of the exponent.
Reference. A.K. Dewdney, Mathematical Recreations, Scientific American,
Sept. 1991