Chaos-101

The Logistic Map, or Bifurcation Diagram, is usually a graphical representation of the formula

p1 = r * p * (1 - p)

where p is the initial population of a group (from 0.0 to 1.0), r is the reproduction rate of that population, and p1 is the new population after one interval. (Other formulas can be used however, with the same result.)

The population can range from 0, which would be complete extinction, to 1, which would be the maximum conceivable population for that group. Values for r are usually somewhere in the range of 2 to 4. An r of 2 would mean that the population doubles each interval (each year for example). So, next year’s population would be this year’s population times the reproduction rate. Simple. But there’s that last bit in parentheses. As the population goes higher and higher towards the maximum, overcrowding, competition, food shortage, etc. sets in and death rates rise. If the population is 0.2, then 1-p is 0.8, meaning an 80% survival rate. When the population rises to something like 0.7, only 30% will survive.

So you have these two factors, r making the population rise, and 1-p cutting it down. If r is 1.0 or less, the population is going to lose more than it gains and eventually go to zero. It is said that zero is an attractor of the function. From a little over 1.0, up to 3.0, a state of equilibrium will be found, meaning the population will eventually settle in on a stable number. Again, this number is an attractor.

At 3.0, something strange happens. The graph splits. The population won’t settle into a single number, but oscillates between two different values. You see this in animal species that have high population years followed by low ones, in a regular cycle.

Around 3.5, it doubles again, settling into a 4 period cycle. Then 8, 16, etc.

At about 3.57, the graph enters a state of chaos, meaning that it ceases to settle down into any repeatable cycle. However, as r continues to increase, several windows of order appear briefly.

Applications

This first application shows two views of the bifurcation diagram. Click to start iterating, click to stop. On top is a graph showing the result of each iteration from left to right. You’ll notice that sometimes there’s a little curvature on the far left, then it settles down. The number shown in the top left corner is the current value of r. You’ll see that as this reaches 3.0, the line splits in two, then splits again and again, finally turning into chaos.

The bottom half shows the traditional view of the diagram, where you can see the bifurcation as a line splitting off into two branches.

The next application shows a completed diagram. Click on any point to zoom in by 2x. Hit space to go back to the full view. There are some interesting forms as you zoom in.

Finally we have the “cobweb” map. This shows the formula, y = r * x * (1 - x) plotted directly as a parabola. You can adjust the starting position and the value of r. A vertical line is drawn to the new x, y position on the parabola. The y is now taken as the new x. As a visual shortcut, we draw a horizontal line from the existing point to the 45 degree line representing the formula x=y. This indicates the starting x value of the next iteration. Then another vertical line is drawn to the newly calculated y. The resulting figure bounces off the parabola and the 45 degree line.

You’ll see for low values of r, the result converges on zero, extinction. As r increases, a positive single attractor is found. You’ll see it sometimes overshoots and spirals around, homing in on the attractor. Again, as r hits 3.0, the spiral starts to open up, as it is no longer converging on a single point, but becomes a rectangle, where the attractors are the two corners touching the parabola. Then you’ll see it go into kind of a double loop, with four points touching the parabola. Beyond that, it is hard to follow, but it’s easy to bump up the r and see the ensuing chaos, where it never settles down into any number of points. Again there are windows where it does, but it’s mostly chaos.

Tags:
Topics:
Date: Sunday, January 27th, 2008 1:04 pm