From Math Images
- A representation of the regions of chaos and stability over the space of two population growth rates.
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When mathematicians see a system that changes size, we like to try to model it with formulas. This is easy for systems that always change in one way -- say, by always increasing -- but such systems rarely show up in the real world. Much more common are systems such as those created by animal populations. A population cannot grow infinitely, but rather is constrained by the amounts of food, space etc., available to it. To model the pattern of growth and reduction that occurs as populations approach and retreat from their maximum sizes, mathematicians have developed the logistic formula, which models population growth fairly accurately by including a factor that diminishes as population size grows, just as food and space would diminish.
The logistic formula is driven by the initial population size and by the potential rate of change of that population. Mathematically, the more powerful of these values is the rate of change; it will determine whether the size of the population settles to a specific value, oscillates between two or more values, or becomes chaotic. To help determine which of these outcomes would occur, a mathematician named Aleksandr Lyapunov developed a method for comparing changes in growth and time in order to calculate what has been dubbed the Lyapunov exponent. This is a handy little indicator, and here's why:
- If it is zero, the population change is neutral -- at some point in time, it reaches a fixed point and remains there.
- If it is less than zero, the population will become stable. The lower the number, the faster and more thoroughly the population will stabilize.
- If it is positive, the population will become chaotic.
What does all this have to do with the fantastical shapes of the Markus-Lyapunov fractal? Well, a scientist named Mario Markus wanted a way to visualize the potential represented by the Lyapunov exponent as a population moved between two different rates of growth. So he created a graphical space with one rate of growth measured along the x-axis and the other along the y. Thus for any point, (x,y), there is one specific Lyapunov exponent that predicts how a population with those rates of change will behave. Markus then assigned a color to every possible Lyapunov exponent -- one color for positive numbers and another for negative numbers and zero. This second color he placed on a gradient, so that lower negative numbers are lighter and those closer to zero are darker, with zero itself being black. Some Markus-Lyapunov fractals also display superstable points in a third color or black. By this code, Markus could color every point on his graph space based on its Lyapunov exponent.
Consider the main image on this page. The blue "background" shows all the points where the combination of the rate of change on the x and y axes will result in chaotic population growth. The "floating" yellow shapes show where the population will move toward stability. The lighter the yellow, the more stable the population.
The movements from light to dark and the dramatic curves of the boundaries between stability and chaos here create an astonishing 3D effect. But the image is striking not only for its beauty but also for its self-similarity. Self-similarity is that trait that makes fractals what they are -- zooming in on the image reveals smaller and smaller parts that resemble the whole. Consider the image to the right, enlarged from a section of the main image above. Here we see several shapes that repeat in smaller and smaller iterations. Perhaps ironically, this type of pattern is a common property of chaos.
A More Mathematical Explanation
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