The Logistic Map
One line of arithmetic applied over and over. At a low growth rate it settles on a single value, then splits into two, then four, then dissolves into chaos.
Take a number between 0 and 1, multiply it by r, multiply that by one minus the number, and do it again to whatever comes out. That is the whole system. Below r = 3 every start walks to the same single value and stays. Past 3 it lands on two values and alternates, then four, then eight, and the doublings crowd together and stop being countable at r = 3.5699…, after which the values never repeat at all. There is no machinery here, not even a simulated one. Every other panel in this set has something doing work: a field, a shaker, a solver. This one has a rule and someone applying it. Stop applying it and the number sits exactly where it was left, forever.
Everything the Rule Does, at Every Rate
The Rule
That is it. It was written down as a population model, where x is a population as a fraction of the most the habitat can hold, r is how fast it breeds, and the (1 − x) is crowding. What it produces has almost nothing to do with populations and everything to do with the shape of the rule.
Because there is no physics in it, there is nothing to compare against experiment. So this page compares against arithmetic that was worked out independently, and the panel had to reproduce it without being told:
| quantity | this panel | known value |
|---|---|---|
| period 2 appears | 2.000000000000 | 2 exactly * |
| period 4 appears | 3.236067977500 | 1 + √5 exactly * |
| λ at r = 2.5 | −0.693147 | −ln 2 exactly |
| λ at r = 4 | +0.69314718 | ln 2 exactly |
| period-3 window opens | 3.8284271305 | 1 + √8 = 3.8284271247 |
| Feigenbaum δ | 4.669201 | 4.6692016091 |
| the doubling stops being countable | 3.5699456719 | 3.5699456718 |
* These two are the superstable rates, where the cycle passes exactly through x = ½. They are the well-conditioned way to measure the cascade, and both have closed forms, which is what makes them a check rather than a result.
The last two rows are the interesting ones. Feigenbaum’s δ is the ratio by which each doubling crowds closer than the last, and it is the same number for a huge class of rules that have nothing else in common. Measuring it here from ten superstable rates gives 4.669201 against a known 4.6692016. Extrapolating that same ratio forward gives the rate at which the doublings accumulate, 3.5699456719 against a known 3.5699456718: agreement in the tenth digit, from a constant that was never put in.
Why This Is an Attractlet
The kernel’s Attractlet Model class (ATTRACTLET, § 7.4)
covers formal and symbolic models outright, and this is about as formal as a model gets. But the
reason to put it in this set is that it strips the question bare.
Look at what the other panels needed. Ferrofluid spikes need a field. A Chladni figure needs a shaker. The magnetic pendulum needs friction. Even the Lorenz attractor needs a solver advancing it. This one needs a person. Press Stop applying the rule: the number does not decay, drift, relax or wander. It sits at exactly the value it had, and it would sit there for the rest of time. The pull toward the cycle was never in the number. It was in the hand that kept applying the arithmetic.
The Shape of the Basin
Below the accumulation point the basin is as complete as a basin gets. Four hundred starting values scattered across the interval, run to settle: at r = 2.8, 3.2, 3.5 and 3.55, all four hundred land on exactly the same set of values, agreeing to the last bit of double precision. Where you start is not merely unimportant; it leaves no trace at all.
Above it the basin does something worth being careful about. The same four hundred starts still go to the same attractor, the same band of the interval the values are confined to. But zero of four hundred land on the same values. Press Release twins at r = 4 and watch two starts a billionth apart take about thirty steps to be as far apart as the interval allows, which is exactly what λ = ln 2 means: every step buys one more bit of separation and throws away one bit of what you knew. Returning to a set and returning to a state are different things, and this is the cleanest place in the whole set to see it.
And the shading in the top panel is not decoration. It is how often the rule returns to each value, so at r = 4 the picture is brightest at the two ends. That has a closed form too: the density is 1 / π√(x(1−x)), the arcsine law. Four million iterates into forty bins match it to within 0.7% in the worst bin.
What This Model Is Not
The period is a measurement. It is the smallest repeat within 10⁻⁹ after a transient. Near a bifurcation, convergence is algebraically slow and no reachable transient is long enough, so the detector reads the period late; measured against the exact values, it lands about 0.001 low at r = 3 and about 0.0004 low at 1 + √6. That bias is why the cascade above is measured from the superstable rates instead, where the condition is a clean sign change with no tolerance in it.
Double precision runs out. Past the tenth doubling the rates differ in the eleventh decimal and the measured δ stops improving. Nothing here is claimed past that.
The Lyapunov exponent is a finite average. 60,000 steps for the live readout, 2,400 for each column of the curve. It is noisy inside the narrow periodic windows, which is why those windows look thinner in the curve than they are.
“Chaos” is not proven by a panel. A positive measured λ and a failure to find a repeat are evidence, not proof. For this map it happens that the proof exists independently, which is not a thing the panel is entitled to take credit for.
May, R. M. (1976). Simple mathematical models with very complicated dynamics.
Nature, 261, 459–467.
Feigenbaum, M. J. (1978). Quantitative universality for a class of nonlinear transformations.
Journal of Statistical Physics, 19(1), 25–52.
Feigenbaum constants, Wikipedia.
Source of δ = 4.6692 and of the published table of period-doubling parameters this panel was
checked against.
Li, T.-Y. & Yorke, J. A. (1975). Period three implies chaos.
American Mathematical Monthly, 82(10), 985–992.