Bifurcation Diagram
Draw the bifurcation diagram of the logistic map x → r x (1 − x): for each growth rate r the tool iterates the map, drops the first iterations and plots the values the orbit keeps visiting, shaded by how often it visits them. Zoom into any range of r and x to see period doubling, the onset of chaos and the period-3 window, with r1 = 3, r2 = 1 + √6 and the window start marked. Switch to a cobweb plot to follow one orbit step by step, with its period and Lyapunov exponent, or plot the Lyapunov exponent against r to see where the map is chaotic. Download a PDF, a 300 DPI PNG, an SVG or the numbers as CSV. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints on one sheet when the drawing fits; the check bar in the footer lets you confirm the scale. The SVG holds the drawing at the printed size. The CSV holds the numbers behind the drawing.
Logistic map x → r x (1 − x)
Look
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every dot, cobweb step and exponent is computed by iterating the logistic map, and r2, the period-3 window and the doubling points are solved numerically. You may use what you make here, including commercially, with no credit needed.
How to draw a bifurcation diagram
- Pick a preset, or choose the diagram, a cobweb plot or the Lyapunov exponent.
- Set the r range (and x range), the columns and the iterations dropped and kept.
- For a cobweb plot, set r, the start value x0 and the number of steps.
- Choose the shading, colours, background and Fitted to one page or Real size.
- Download the PDF and print at 100% (Actual size), or the PNG, SVG or CSV.
What you can use it for
- Lessons on period doubling and chaos
- Zoomed prints of the period-3 window
- Cobweb plots for one value of r
- Comparing stable and chaotic r with the Lyapunov exponent
- Posters and line art of the diagram
- Data for spreadsheets from the CSV
The 16 presets
| Preset | Shows | Setting | Result |
|---|---|---|---|
| Bifurcation diagram, r from 2.8 to 4 | Diagram | r 2.8–4, 700 columns, x 0–1 | 234 columns with more than 64 values |
| Whole range, r from 1 to 4 | Diagram | r 1–4, 700 columns, x 0–1 | 95 columns with more than 64 values |
| Zoom: r from 3.4 to 4 | Diagram | r 3.4–4, 450 columns, x 0–1 | 298 columns with more than 64 values |
| Period doubling cascade: r from 3.54 to 3.575 | Diagram | r 3.54–3.575, 1,000 columns, x 0.3–0.92 | 157 columns with more than 64 values |
| Period-3 window: r from 3.82 to 3.86 | Diagram | r 3.82–3.86, 800 columns, x 0–1 | 370 columns with more than 64 values |
| Inside the window: a small copy (r 3.847 to 3.857) | Diagram | r 3.847–3.857, 450 columns, x 0.44–0.56 | 315 columns with more than 64 values |
| One colour, no density shading | Diagram | r 2.8–4, 700 columns, x 0–1 | 234 columns with more than 64 values |
| Light on a dark background | Diagram | r 3.4–4, 450 columns, x 0–1 | 298 columns with more than 64 values |
| Cobweb: r = 2.8, settles on one point | Cobweb | r = 2.8, 40 steps | period 1, exponent −0.223 |
| Cobweb: r = 3.2, period 2 | Cobweb | r = 3.2, 60 steps | period 2, exponent −0.916 |
| Cobweb: r = 3.5, period 4 | Cobweb | r = 3.5, 80 steps | period 4, exponent −0.873 |
| Cobweb: r = 3.83, period 3 | Cobweb | r = 3.83, 120 steps | period 3, exponent −0.369 |
| Cobweb: r = 4, chaos | Cobweb | r = 4, 200 steps | no cycle, exponent 0.693 |
| Lyapunov exponent, r from 2.5 to 4 | Lyapunov | r 2.5–4, 1,000 columns | 254 of 1,000 r values chaotic |
| Lyapunov exponent, r from 3.4 to 4 | Lyapunov | r 3.4–4, 1,000 columns | 651 of 1,000 r values chaotic |
| Real size: diagram 150 mm wide | Diagram | r 2.8–4, 700 columns, x 0–1 | 234 columns with more than 64 values |
Counts are worked out by the tool from the drawing it makes; start value x0 = 0.2.
Period and Lyapunov exponent at some values of r
| r | Period found | Cycle | Lyapunov exponent |
|---|---|---|---|
| 0.8 | 1 | 0 | −0.223 |
| 2.5 | 1 | 0.6 | −0.693 |
| 2.8 | 1 | 0.6429 | −0.223 |
| 3.2 | 2 | 0.513 / 0.7995 | −0.916 |
| 3.449 | 2 | 0.4401 / 0.8499 | −0.001 |
| 3.5 | 4 | 0.3828 / 0.5009 / 0.8269 / 0.875 | −0.873 |
| 3.55 | 8 | 8 values | −0.1 |
| 3.566 | 16 | 16 values | −0.074 |
| 3.7 | none up to 64 | – | 0.354 |
| 3.83 | 3 | 0.1561 / 0.5047 / 0.9574 | −0.37 |
| 3.9 | none up to 64 | – | 0.498 |
| 4 | none up to 64 | – | 0.693 |
Period: the smallest p up to 64 with the orbit back within 0.000000001 after 20,000 iterations from x0 = 0.2. Exponent: averaged over 100,000 iterations after 1,000 dropped.
Doubling points and the Feigenbaum ratio
| n | Cycle length 2^n | r where 1/2 is on the cycle | Ratio of successive gaps |
|---|---|---|---|
| 0 | 1 | 2 | – |
| 1 | 2 | 3.2360679775 | 4.70894 |
| 2 | 4 | 3.4985616993 | 4.68077 |
| 3 | 8 | 3.5546408628 | 4.66296 |
| 4 | 16 | 3.5666673799 | 4.6684 |
| 5 | 32 | 3.5692435316 | 4.66895 |
| 6 | 64 | 3.5697952937 | 4.66916 |
| 7 | 128 | 3.5699134654 | 4.66919 |
| 8 | 256 | 3.5699387742 | – |
Solved by Newton’s method in r. The ratios approach δ = 4.6692…; n = 1 is 1 + √5.
What prints
| Download | What is in it |
|---|---|
| The diagram, cobweb or exponent plot fitted to one page or at real size, vector shapes and lines, with axes, the values used and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The plot at the printed size: one filled path per shade, lines and labels |
| CSV | Diagram: per column the number of distinct kept values and their range; cobweb: the orbit; exponent: r and the exponent |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| The logistic map: diagram, cobweb, exponent | Bifurcation Diagram (this page) |
| A chaotic system in three dimensions | Lorenz Attractor |
| A chaotic swing | Double Pendulum |
| Mandelbrot and Julia sets | Fractal Generator |
Other tools: the Fractal Generator renders Mandelbrot and Julia sets, Graph Paper prints grids for cobweb plots by hand and the Spirograph Generator draws rolling-circle curves.
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Frequently asked questions
What is a bifurcation diagram?
A picture of how a system’s long-run behaviour changes with a parameter. Here the parameter is the growth rate r of the logistic map; for each r the tool drops the first iterations and plots the values the orbit keeps visiting, so one line means a fixed point, two lines a cycle of period 2, and a smeared band chaos.
What is the logistic map?
The rule x → r x (1 − x) for x between 0 and 1, a simple model of a population with a limit. It was popularised in a 1976 paper by the biologist Robert May, Simple mathematical models with very complicated dynamics (Nature 261), as the Wikipedia article on the logistic map notes (read 13 November 2026).
What happens as r grows?
Below r = 1 the orbit dies out to 0. From 1 to 3 it settles on one value, 1 − 1/r. At r = 3 that point splits into a cycle of 2, at r = 1 + √6 = 3.449490 into a cycle of 4, and the doublings come faster and faster until chaos sets in at about 3.56995 (Wikipedia). Inside the chaos there are windows of order; the widest, with period 3, opens at r = 1 + √8 = 3.828427. The tool computes r1, r2 and the window start itself, and they match those values.
How are r2 and the period-3 window computed?
r2 is where the cycle of 2 stops attracting: the tool finds the cycle’s two points and searches for the r where the product of the slopes there is −1. The window start is where the third iterate of the map first touches the line y = x; the tool solves both conditions together with Newton’s method. Our tests check both against 1 + √6 and 1 + √8.
What is the Feigenbaum constant?
The ratio of the lengths of successive period-doubling steps approaches δ = 4.6692…, the same for a whole family of maps; Mitchell Feigenbaum found it in 1975 and published it in 1978 (Wikipedia, Feigenbaum constants). The table on this page computes the doubling points where 1/2 is on the cycle and their ratios; they approach 4.6692 (4.66919 at the last step).
What is a cobweb plot?
A way to follow one orbit by hand: draw the curve y = r x (1 − x) and the line y = x, go up from x0 to the curve, across to the line, up or down to the curve again, and so on. A spiral that closes in means the orbit settles; a square that repeats means a cycle; a tangle means chaos. The tool also reports the period it finds (up to 64) and the Lyapunov exponent.
What is the Lyapunov exponent?
The average of ln|r (1 − 2x)| along the orbit: how fast nearby starts separate. Negative means the orbit settles, positive means chaos. For the cycle of 2 at r = 3.2 it is ln(0.16) / 2 = -0.916, and at r = 4 it is ln 2 = 0.693 (Wikipedia); the tool computes -0.916 and 0.693.
What does the shading mean?
Each column of the diagram counts how many kept values fall in each row. Rows visited by 5% or more of the kept values print darkest, then 1%, then 0.2%, then any visit at all. One colour turns the shading off.
Will another browser give the same picture?
Yes for the diagram and cobweb: the map uses only multiplication and subtraction, which every JavaScript engine rounds the same way, so even chaotic orbits come out bit for bit the same in our tests. The Lyapunov exponent also uses a logarithm, so the last decimal places can differ very slightly.
How big can it go?
100 to 1,000 columns, up to 5,000 iterations dropped and 2,000 kept, 100 to 800 rows, and at most 4,000,000 iterations per drawing; cobweb plots up to 500 steps. Every preset draws in under 2 seconds in our tests.
Can it print at real size?
Yes, when the drawing fits on one sheet: choose Real size and set the printed width. Every printed point is within 0.2 mm of its true place in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar in the footer lets you confirm the scale. Bigger drawings print fitted to the page.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every value is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.