
Lyapunov Fractals

formulas: 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

