This bifurcation diagram draws the single most famous picture in chaos theory, 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.
Reading the simulation
What the forks, the vertical slices and the cobweb inset are telling you.
1
Each column is the long-term set at one r
Read the diagram column by column. Every vertical slice sits above one growth rate r, increasing from left to right, and the dots stacked in that column are the values the population keeps returning to after the transient dies away. One dot is a single steady state (a fixed point); a short stack of dots is a repeating cycle; a solid vertical smear is chaos, where the orbit never settles. Move the yellow r-cursor and the "Long-term behavior at r" readout names exactly which case you are on.
2
Single lines fork into 2, 4, 8 …
Follow one line from the left and watch it split. It holds a single value until r ≈ 3, then forks into two (the population now alternates high–low), then into four, then eight — the period-doubling cascade. Each split crowds closer to the last, and they pile up at the onset of chaos, r ≈ 3.5699. The rate at which the gaps shrink approaches the Feigenbaum constant, δ ≈ 4.6692, shown live in the readout.
3
The cobweb inset shows the orbit itself
The inset plots the map's parabola with the y = x line and a yellow staircase — axes xₙ across and xₙ₊₁ up. Watch it settle: the staircase spirals into a point where the diagram shows one dot, traces a closed loop of 2 or 4 corners where it cycles, and wanders without ever closing where the column is chaotic. It is the same orbit as the column above the cursor, drawn one step at a time.
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. The tool discards a few hundred warm-up steps so the transient dies away, then records only the long-term values — that is what each column plots.
2
Sweep r across the canvas
The engine steps r across every pixel column from left to right, iterating the map at each and plotting the values it keeps returning to. The whole sweep is only a few hundred thousand multiplications, so it renders in a single instant pass rather than an animation — which is why a preset or a zoom repaints the diagram immediately.
3
Pick presets, read the period, zoom the fractal
Use the preset views to jump to the full range, the period-doubling zone, the period-3 window near r ≈ 3.83, or a Feigenbaum zoom into the accumulation point. Click to set the r-cursor and the tool iterates that r to detect its period (1, 2, 4, … or chaos). Drag a box over any fork and it recomputes just that window — revealing a shrunken copy of the whole diagram, because the cascade is self-similar at every scale.
What is a bifurcation diagram?
A bifurcation diagram plots the long-term behavior 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 behavior 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. The growth rate r is its only parameter, and every dot on this diagram comes from iterating that one line of arithmetic a few hundred times and recording where it settles. 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?
The Feigenbaum constant, δ ≈ 4.6692, measures how fast the period-doubling forks crowd together: the range of r between one split and the next shrinks by roughly that factor every time. Because the gaps shrink geometrically they add up to a finite total, which is why the whole cascade piles up at r ≈ 3.5699 instead of continuing forever. It was identified 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.
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Classroom activity
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Describe the logistic map and its long-term behavior.
Identify period-doubling as a route to chaos.
Locate where chaos begins.
Variables to change
Growth parameter r
Initial population
Procedure
Set a low r and observe the population settle to a single steady value.
Increase r and watch it split into 2, then 4, then more values.
Increase r past about 3.57 and observe chaotic behavior.
Observations
Record the number of stable values as r increases and where the pattern turns chaotic.
Questions
What does each branch split (bifurcation) represent?
What is the logistic map equation?
Roughly where does chaos begin?
Is the chaotic region completely disordered?
What does this simple equation demonstrate?
Explanation
The logistic map x_{n+1} = r·x_n·(1−x_n) settles to one value at low r, then period-doubles (2, 4, 8, …) as r rises, reaching chaos near r ≈ 3.57. This simple rule shows how deterministic systems can become chaotic through period doubling.
Answer key (for instructors)
1. A period doubling — the population now cycles between twice as many values.
2. x_{n+1} = r·x_n·(1 − x_n).
3. Around r ≈ 3.57, after an infinite cascade of period doublings.
4. No — it contains narrow "windows" of periodic behavior (e.g. period-3) amid the chaos.
5. That very simple deterministic rules can produce complex, chaotic behavior.
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