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conditions. The logistic map was brought to the attention of population biologists in a 1971 article by the mathematical biologist Robert May in the prestigious journal Nature. It had been previously analyzed in detail by several mathematicians, including Stanislaw Ulam, John von Neumann, Nicholas Metropolis, Paul Stein, and Myron Stein. But it really achieved fame in the 1980s when the physicist Mitchell Feigenbaum used it to demonstrate universal properties common to a very large class of chaotic systems. Because of its apparent simplicity and rich history, it is a perfect vehicle to introduce some of the major concepts of dynamical systems theory and chaos.

The logistic map gets very interesting as we vary the value of R. Let’s start with R = 2. We need to also start out with some value between 0 and 1 for x0, say 0.5. If you plug those numbers into the logistic map, the answer for x1 is 0.5. Likewise, x2 = 0.5, and so on. Thus, if R = 2 and the population starts out at half the maximum size, it will stay there forever.

Now let’s try x0 = 0.2. You can use your calculator to compute this one. (I’m using one that reads off at most seven decimal places.) The results are more interesting:

FIGURE 2.6. Behavior of the logistic map for R = 2 and x0 = 0.2.

The same eventual result (xt = 0.5 forever) occurs but here it takes five iterations to get there.

It helps to see these results visually. A plot of the value of xt at each time t for 20 time steps is shown in figure 2.6. I’ve connected the points by lines to better show how as time increases, x quickly converges to 0.5.

What happens if x0 is large, say, 0.99? Figure 2.7 shows a plot of the results.

Again the same ultimate result occurs, but with a longer and more dramatic path to get there.

You may have guessed it already: if R = 2 then xt eventually always gets to 0.5 and stays there. The value 0.5 is called a fixed point: how long it takes to get there depends on where you start, but once you are there, you are fixed.

If you like, you can do a similar set of calculations for R = 2.5, and you will find that the system also always goes to a fixed point, but this time the fixed point is 0.6.

For even more fun, let R = 3.1. The behavior of the logistic map now gets more complicated. Let x0 = 0.2. The plot is shown in figure 2.8.

In this case x never settles down to a fixed point; instead it eventually settles into an oscillation between two values, which happen to be 0.5580141 and 0.7645665. If the former is plugged into the formula the latter is produced, and vice versa, so this oscillation will continue forever. This oscillation will be reached eventually no matter what value is given for x0. This kind of regular final behavior (either fixed point or oscillation) is called an “attractor,” since, loosely speaking, any initial condition will eventually be “attracted to it.”

FIGURE 2.7. Behavior of the logistic map for R = 2 and x0 = 0.99.

FIGURE 2.8. Behavior of the logistic map for R = 3.1 and x0 = 0.2.

For values of R up to around 3.4 the logistic map will have similar behavior: after a certain number of iterations, the system will oscillate between two different values. (The final pair of values will be different for each value of R.) Because it oscillates between two values, the system is said to have period equal to 2.

FIGURE 2.9. Behavior of the logistic map for R = 3.49 and x0 = 0.2.

But at a value between R = 3.4 and R = 3.5 an abrupt change occurs. Given any value of x0, the system will eventually reach an oscillation among four distinct values instead of two. For example, if we set R = 3.49, x0 = 0.2, we see the results in figure 2.9.

Indeed, the values of x fairly quickly reach an oscillation among four different values (which happen to be approximately 0.872, 0.389, 0.829, and 0.494, if you’re interested). That is, at some R between 3.4 and 3.5, the period of the final oscillation has abruptly doubled from 2 to 4.

Somewhere between R = 3.54 and R = 3.55 the period abruptly doubles again, jumping to 8. Somewhere between 3.564