🌳 The logistic map and the road to chaos

What you’re seeing

Start with a number $x$ between 0 and 1: think of it as the size of an animal population, as a fraction of the most the land can hold. Each year the new size is

$$x_{\text{next}} = r\,x\,(1 - x).$$

The factor $r$ is how fast the animals breed. The $(1 - x)$ part says crowding holds them back.

Left: the cobweb. The green hill is the rule; the dashed line is $y = x$. Start at $x_0$ on the bottom, go up to the hill (that’s next year), across to the line (make it the new $x$), up or down to the hill again, and so on. The orange path is the history of the population.

Right: the bifurcation diagram. For each $r$ across the bottom, we run the rule for hundreds of years, throw away the start, and plot where $x$ keeps landing. One line means the population settles on one size. Two lines mean it alternates between two sizes. A gray smear means it never settles.

Why it works

Settling down. A fixed point is a value that the rule leaves alone: $x^* = r x^*(1 - x^*)$. Besides $0$, the solution is $x^* = 1 - \frac1r$. Whether the population actually settles there depends on the slope of the hill at that point. Using calculus, the slope is $f'(x) = r(1 - 2x)$, so at $x^*$ it is $2 - r$. A fixed point attracts when the slope is between $-1$ and $1$:

$$|2 - r| < 1 \quad\Longleftrightarrow\quad 1 < r < 3.$$

On the cobweb, a gentle slope makes the path spiral in. A slope steeper than $-1$ makes it spiral out.

Period doubling. At $r = 3$ the fixed point stops attracting, and a cycle of two values takes over. At $r = 1 + \sqrt6 \approx 3.449$ that cycle splits into four, near $3.544$ into eight, and so on, faster and faster. The doublings pile up at $r \approx 3.5699$. Beyond that point, for most $r$, the population never repeats: this is chaos.

A universal number. The gaps between doublings shrink by almost the same factor each time, and in the limit the factor is

$$\delta \approx 4.6692.$$

Mitchell Feigenbaum found in 1975 that the same number appears for many other hill-shaped rules, and in real experiments with fluids and circuits. It is now called the Feigenbaum constant.

Windows of order. Inside the chaos are clear stripes. The widest one starts at $r = 1 + \sqrt8 \approx 3.828$, where a cycle of three appears and then doubles again: 6, 12, 24. Choose “the window near 3.83” to see that the picture inside looks like a small copy of the whole diagram.

Why it matters. In 1976 the biologist Robert May used this map to warn that simple rules can give wild results, so erratic data does not have to mean a complicated cause.

Try this

• Set $r = 2.8$ and move the start $x_0$. Does the population always end at the same place? (Check: $1 - 1/2.8 \approx 0.643$.)
• Set $r = 3.2$. The cobweb settles into a square loop: a cycle of two.
• Set $r = 3.83$. The stats line should find a cycle of 3. Now nudge $r$ up to $3.86$.
• Press Play and watch the stats line as $r$ sweeps right.