From steady to chaotic
A very simple rule that goes from calm, to flip-flopping, to unpredictable.
What you are looking at
Dots spread across a graph. For small values on the left there is one line. Moving right, the line splits in two, then four, then eight, until it breaks into a cloud.
The idea
Here is a rule for a population of animals. Take this year's number (as a fraction of the most the land can hold). Multiply it by how much room is left, then multiply by a growth number, r. That gives next year's number. Few animals means growth. Too many means a crash.
If r is small, the population settles down to one steady number. Turn r up, and the population starts to flip-flop between two values, year after year. Turn it up more, and it cycles between four. Then eight. The jumps come faster and faster.
Past a certain point the population never repeats and looks random, even though the rule has no randomness in it. The animation draws all of this as one picture: the steady line, then the splits, then the cloud.
Where else you see it
- Animal populations that boom and bust.
- A dripping faucet going from steady drips to an uneven pattern as you open it.
- The same pattern of splitting shows up in many other systems, not just populations.
Go deeper: the proper names and the math
The logistic map is xn+1 = r xn(1 − xn). For 1 < r < 3 the fixed point x* = 1 − 1/r is stable. At r = 3 it becomes unstable and a 2-cycle appears; further doublings occur at r ≈ 3.449, 3.544, 3.564… and accumulate at r∞ ≈ 3.5699, where chaos begins. Inside the chaos there are windows of order, such as a period-3 window near r ≈ 3.83.
The gaps between doublings shrink by a factor that approaches the Feigenbaum constant δ ≈ 4.6692. Feigenbaum (1975) found that the same number appears for a whole class of one-hump maps, and it has been measured in real experiments, which is why it is called universal.