An innocent formula where order tips into chaos.
A pocket-sized model for next year's population that, with one knob turned, becomes the most studied piece of chaos in mathematics. The Explorer lets you turn that knob in real time.
Imagine a meadow with a maximum number of rabbits it can feed. Each year next year's rabbits = (growth rate) × (this year's rabbits) × (room still left in the meadow). Write that in symbols and you get xₙ₊₁ = r·xₙ·(1−xₙ). One line, one knob r. Out of that single line the whole modern theory of chaos was built.
Start with x₀ = 0.2 and crank r. At r = 2.5 the meadow settles to a single steady population — boring. At r = 3.1 it starts flickering between two values, high year, low year, forever. At r = 4 the sequence is wild — it leaps around between 0 and 1 and never repeats. Same formula, three completely different worlds.
The cascade you are about to see — single value, then 2, then 4, then 8, then chaos — also governs heart arrhythmias, predator-prey populations, lasers slipping into instability, and convection cells in fluids. The constant that controls how fast the doublings pile up, Feigenbaum's δ ≈ 4.6692…, is the same in every one of them. It is one of the deepest universalities in science.
Treat x as the fraction of carrying capacity used up, between 0 (empty meadow) and 1 (completely full). Pick a growth rate r between 0 and 4. Then next year's fraction is xₙ₊₁ = r · xₙ · (1 − xₙ). Worked example: r = 2.5, xₙ = 0.4 ⇒ xₙ₊₁ = 2.5 · 0.4 · 0.6 = 0.60. Run the line again: 2.5 · 0.60 · 0.40 = 0.60. Already at the fixed point. The 1845 logistic equation of Pierre-François Verhulst, sampled once per generation, is exactly this map.
x₀ = 0.40 x₁ = 2.5 · 0.40 · (1 − 0.40) = 0.60 x₂ = 2.5 · 0.60 · (1 − 0.60) = 0.60 x₃ = x₂ = 0.60
Below r = 3 the iteration always settles to one steady population — a fixed point. Push r just past 3 and the fixed point stops being stable: instead the orbit jumps every year between a higher value and a lower value. The dot has become a 2-cycle. This is a period-doubling bifurcation, the simplest way a steady state can give way to oscillation, and it is the first crack in the calm.
Keep nudging r upward. At r ≈ 3.449 the 2-cycle splits into a 4-cycle. At r ≈ 3.544 the 4-cycle becomes an 8-cycle. Then 16, 32, 64 — and the windows where each new doubled cycle is stable get shorter every time. Plotted, this is the famous bifurcation diagram: the trunk that forks, then forks again, then forks faster than the eye can resolve.
| r | what happens |
|---|---|
| 0 ≤ r < 1 | Extinction. Every starting population dies out. |
| 1 ≤ r < 3 | Stable fixed point. One steady population. |
| r = 3 | First bifurcation. Splits into a 2-cycle. |
| r ≈ 3.449 | 2-cycle → 4-cycle. |
| r ≈ 3.544 | 4-cycle → 8-cycle. |
| r ≈ 3.5644 | 8-cycle → 16-cycle. |
| r ≈ 3.56995 | Onset of chaos (Feigenbaum point). |
| r ≈ 3.8284 | Period-3 window opens (1 + √8). |
| r = 4 | Full chaos; orbit covers [0, 1]. |
Measure how much the windows shrink between doublings: the ratio (rₙ − rₙ₋₁) / (rₙ₊₁ − rₙ) approaches the same limit. Mitchell Feigenbaum computed it on a pocket calculator at Los Alamos in 1975 and got δ ≈ 4.66920160910… The astonishing fact, proved a few years later by Lanford and others, is that this constant is universal: it shows up in the Hénon map, in the Duffing oscillator, in experimental fluid convection, in dripping faucets. Period-doubling is a route, and δ is the speed limit.
Ratio of successive doubling-interval widths. Universal across period-doubling systems.
After infinitely many doublings, all crammed into a finite r-window, the system finally tips into chaos at the Feigenbaum point r∞ ≈ 3.56994567… Past it, the orbit no longer settles to any cycle — it visits an infinite set of values, and two starting points that differ by 10⁻⁶ end up uncorrelated within thirty iterations. At r = 4 the map is conjugate to the tent map on [0,1]: pure deterministic randomness, generated by one line of arithmetic.
Slide r up to about 1 + √8 ≈ 3.8284 and the chaos suddenly stops: a stable 3-cycle appears. From there the doublings start again — period 6, 12, 24 — and the system re-enters chaos. In 1975 Tien-Yien Li and James Yorke proved a startling theorem in the American Mathematical Monthly: for any continuous map of an interval, period three implies chaos. A single 3-cycle anywhere forces orbits of every other period and uncountably many that never repeat. The phrase coined for the title — "chaos" — has been the field's name ever since.
The Explorer renders the full bifurcation diagram from r = 2.5 to 4, draws the live time series, and lets you sweep through every regime in the story above. Pick the famous values from the side panel; or just drag.
How and where this technique lives in the world today.
Topics in the same vein.
