The single most famous picture in chaos theory, drawn live in your browser. Slowly turn up the growth rate of the logistic map and watch one steady value split into two, then four, then a blur of chaos — the period-doubling cascade. It's the calm, step-by-step route into the chaos you see in our double pendulum and Lorenz attractor. Drag a box to zoom into the fractal; pick any point to see the orbit behind it.
Chaos arrives by doubling, and it does so at a universal rate. The logistic map settles to one value until r ≈ 3, flips between two until r ≈ 3.449, then four, eight, sixteen — each split crowding closer than the last. They pile up at r ≈ 3.5699, where chaos begins. The ratio of the shrinking gaps approaches the Feigenbaum constant δ ≈ 4.6692 — the same number that governs the doubling route to chaos in dripping taps, heart rhythms and electronic circuits. Zoom into any fork and you find the whole diagram again.
How It Works
One equation, swept across a parameter, plotting where it settles.
1
Iterate the logistic map
Start with a population x between 0 and 1 and apply x → r·x·(1−x) over and over. The growth rate r pushes it up; the (1−x) term pulls it back as the population fills its environment. After a few hundred steps the transient dies away and only the long-term pattern remains.
2
Sweep r across the canvas
Each vertical slice of the plot is one value of r, increasing from left to right. Above it we plot the handful of values the population keeps returning to. One dot means a steady state; two means it alternates; a smear means it never repeats — chaos.
3
Read the forks, zoom the fractal
Pick any r with the cursor and the inset draws its orbit as a cobweb, with the detected period beside it. Drag a box over a fork and the simulator recomputes just that window — revealing a shrunken copy of the entire diagram, because the cascade is self-similar at every scale.
What is a bifurcation diagram?
A bifurcation diagram plots the long-term behaviour of a system against a parameter you slowly change. For the logistic map, the horizontal axis is the growth rate r and, above each r, the diagram shows the values the population eventually settles onto. Where a single line splits into two, then four, then a dense band, you can read off exactly where steady behaviour gives way to oscillation and then to chaos.
What is the logistic map?
The logistic map is the simple rule x → r·x·(1−x), a toy model of a population that grows in proportion to its size but is held back as it fills its environment. Despite having just one parameter and one multiplication, iterating it produces fixed points, cycles and full chaos as r increases — which is why it became the textbook example of how simple equations can behave unpredictably.
What is period doubling?
Period doubling is the route the logistic map takes into chaos. Below r ≈ 3 the population settles to one value. Past 3 it flips between two values (period 2), then four, then eight, with each split happening over a shorter and shorter range of r. The splits pile up at r ≈ 3.5699, beyond which the motion is chaotic — punctuated by narrow windows where order briefly returns, such as the period-3 window near r ≈ 3.83.
What is the Feigenbaum constant?
As the period-doubling forks crowd together, the ratio of the gaps between successive splits approaches a fixed number, δ ≈ 4.6692, discovered by Mitchell Feigenbaum. Remarkably the same constant appears in the period-doubling route to chaos of many completely different systems, from dripping taps to electronic circuits, which is why it is called a universal constant of chaos.