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