Lorenz Attractor
Draw the Lorenz attractor: the path of a point that follows the three Lorenz equations, dx/dt = σ(y − x), dy/dt = x(ρ − z) − y and dz/dt = xy − βz. The tool works the path out step by step with the classic fourth-order Runge–Kutta method (RK4), up to 60,000 steps, from any start point and with any σ, ρ and β. Look at it from the side, the top, the end or turned and tilted in 3-D, colour the line by time or by speed, mark the fixed points, and add a second run started a tiny distance away to see how far apart the two paths drift. Download a PDF poster, a 300 DPI PNG or an SVG with the path as vector lines. 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.
Equations
View and colour
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Runs in your browser with our own code: nothing you type is uploaded or saved. Every point is worked out by our own fourth-order Runge–Kutta integrator from the equations shown. You may use what you make here, including commercially, with no credit needed.
How to draw the Lorenz attractor
- Pick a preset, or type σ, ρ and β (a fraction such as 8/3 is fine).
- Set the step h, the number of steps and the start point.
- Choose the view and the colours; turn on the fixed points or the second run if you like.
- Choose Fitted to one page or Real size and the paper.
- Download the PDF and print at 100% (Actual size), or the PNG or SVG.
What you can use it for
- Posters and wall art of the Lorenz attractor
- Maths and physics lessons on differential equations and chaos
- Showing sensitive dependence with two nearby starts
- Plotter, laser and vinyl line art from the SVG
- Book covers, cards and tattoo line studies
- Trying other values of σ, ρ and β
The 16 presets
| Preset | σ | ρ | β | h | Steps | View | x range | z range |
|---|---|---|---|---|---|---|---|---|
| σ = 10, ρ = 28, β = 8/3, side view (x and z), 10,000 steps | 10 | 28 | 8/3 | 0.01 | 10,000 | side (x, z) | −18.2 to 19.6 | 1 to 47.8 |
| σ = 10, ρ = 28, β = 8/3, top view (x and y) | 10 | 28 | 8/3 | 0.01 | 10,000 | top (x, y) | −18.2 to 19.6 | 1 to 47.8 |
| σ = 10, ρ = 28, β = 8/3, end view (y and z) | 10 | 28 | 8/3 | 0.01 | 10,000 | end (y, z) | −18.2 to 19.6 | 1 to 47.8 |
| σ = 10, ρ = 28, β = 8/3, turned 40°, tilted 25° (3-D view) | 10 | 28 | 8/3 | 0.01 | 10,000 | turned 40°, tilted 25° | −18.2 to 19.6 | 1 to 47.8 |
| Coloured by time, 20,000 steps | 10 | 28 | 8/3 | 0.01 | 20,000 | side (x, z) | −18.2 to 19.6 | 1 to 47.8 |
| Coloured by speed, 20,000 steps | 10 | 28 | 8/3 | 0.01 | 20,000 | side (x, z) | −18.2 to 19.6 | 1 to 47.8 |
| Two runs started 0.000000001 apart (sensitivity) | 10 | 28 | 8/3 | 0.01 | 4,000 | side (x, z) | −17.4 to 19.6 | 1 to 47.8 |
| With the three fixed points marked | 10 | 28 | 8/3 | 0.01 | 10,000 | side (x, z) | −18.2 to 19.6 | 1 to 47.8 |
| Fine line, 60,000 steps (h = 0.005) | 10 | 28 | 8/3 | 0.005 | 60,000 | side (x, z) | −18.7 to 19.6 | 1 to 47.8 |
| ρ = 14: spirals into a fixed point | 10 | 14 | 8/3 | 0.01 | 6,000 | side (x, z) | 0.4 to 11.3 | 0.9 to 21.4 |
| ρ = 20, started near the origin, after 2,000 skipped steps | 10 | 20 | 8/3 | 0.01 | 6,000 (+2,000 skipped) | side (x, z) | −7.4 to −6.9 | 18.5 to 19.4 |
| ρ = 99.96, 3-D view, after 5,000 skipped steps | 10 | 99.96 | 8/3 | 0.005 | 8,000 (+5,000 skipped) | turned 30°, tilted 20° | −25.8 to 33.6 | 52.8 to 144.7 |
| ρ = 160, after 5,000 skipped steps | 10 | 160 | 8/3 | 0.005 | 8,000 (+5,000 skipped) | side (x, z) | −39.6 to 39.6 | 103.7 to 216.6 |
| σ = 14, ρ = 28, β = 8/3 | 14 | 28 | 8/3 | 0.01 | 15,000 | side (x, z) | −19.4 to 21.1 | 1 to 48.2 |
| Poster: 3-D view coloured by time, 40,000 steps | 10 | 28 | 8/3 | 0.005 | 40,000 | turned -35°, tilted 15° | −18.7 to 19.6 | 1 to 47.8 |
| Real size: side view 150 mm wide | 10 | 28 | 8/3 | 0.01 | 10,000 | side (x, z) | −18.2 to 19.6 | 1 to 47.8 |
Every range is read from the run the tool draws.
RK4 checked on an equation with a known answer
| Step h | Error at t = 1 (dy/dt = −y) | Error ÷ next error |
|---|---|---|
| 0.2 | 5.797e−6 | – |
| 0.1 | 3.332e−7 | 17.4 |
| 0.05 | 1.998e−8 | 16.68 |
| 0.025 | 1.223e−9 | 16.34 |
Halving the step divides the error by about 16 = 2⁴: the integrator is fourth order.
The fixed points (σ = 10, β = 8/3)
| ρ | Fixed points away from the origin | Every equation is zero there |
|---|---|---|
| 2 | (±1.633, ±1.633, 1) | yes |
| 14 | (±5.8878, ±5.8878, 13) | yes |
| 20 | (±7.1181, ±7.1181, 19) | yes |
| 24 | (±7.8316, ±7.8316, 23) | yes |
| 28 | (±8.4853, ±8.4853, 27) | yes |
| 99.96 | (±16.2448, ±16.2448, 98.96) | yes |
| 160 | (±20.5913, ±20.5913, 159) | yes |
Two runs started a tiny distance apart
| Second run starts this much further in x | Steps until 1 or more apart | Time t | Largest distance in 6,000 steps |
|---|---|---|---|
| 1e-12 | 3,973 | 39.73 | 50.6 |
| 1e-9 | 3,348 | 33.48 | 52.53 |
| 0.000001 | 2,456 | 24.56 | 49.97 |
| 0.001 | 1,834 | 18.34 | 50.03 |
σ = 10, ρ = 28, β = 8/3, h = 0.01, start (1, 1, 1). Measured between the two runs, step by step.
Where the classic run goes
| Step h | Steps | x | y | z |
|---|---|---|---|---|
| 0.01 | 10,000 | −18.21 to 19.56 | −24.56 to 27.18 | 0.96 to 47.83 |
| 0.005 | 20,000 | −18.67 to 19.56 | −25.45 to 27.18 | 0.96 to 47.83 |
| 0.01 | 60,000 | −18.56 to 19.56 | −25.22 to 27.18 | 0.96 to 47.83 |
The smallest and largest x, y and z reached from (1, 1, 1) with σ = 10, ρ = 28, β = 8/3, read from the runs.
What prints
| Download | What is in it |
|---|---|
| The path fitted to one page or at real size, vector lines, with the values used and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The path at the printed size as vector lines, one per colour band, with the fixed-point circles if shown |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| The Lorenz equations solved and drawn | Lorenz Attractor (this page) |
| Branching patterns grown by a field simulation | Lichtenberg Figure |
| Mandelbrot, Julia and line fractals | Fractal Generator |
| Curves drawn by rolling circles | Spirograph Generator |
| Contour-line patterns | Topographic Map Generator |
Other tools: the Fractal Generator renders Mandelbrot, Julia and line fractals, the Spirograph Generator draws rolling-circle curves and the Topographic Map Generator draws contour patterns.
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Frequently asked questions
What is the Lorenz attractor?
The shape traced by the solutions of the Lorenz system, three equations that Edward Lorenz developed in 1963 as a simplified model of atmospheric convection (Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences, volume 20, pages 130–141). With the values Lorenz used, σ = 10, ρ = 28 and β = 8/3, the path never settles down and loops round two lobes; the shape is often said to look like a butterfly.
How does the tool solve the equations?
With the classic fourth-order Runge–Kutta method (RK4) and a fixed step h: each step takes four slopes of the equations and averages them. Our tests run the same integrator on dy/dt = −y, whose exact answer is known, and halving h divides the error by about 16, as a fourth-order method should (table below).
What are σ, ρ and β?
The three numbers in the equations: σ (sigma) sets how fast x follows y, ρ (rho) feeds z back into y, and β (beta) damps z. The tool accepts any σ from 0 to 100, ρ from 0 to 500 and β from 0 to 20.
What are the fixed points?
Points where the path would stand still. The origin is one; for ρ above 1 there are two more, (±√(β(ρ − 1)), ±√(β(ρ − 1)), ρ − 1), at the centres of the two lobes. Our tests check that all three make every equation zero. Turn on Mark the fixed points to see them as circles.
What does the second run show?
Two paths that start almost at the same place: the second starts 0.000000001 further along x. The tool measures their distance at every step. With σ = 10, ρ = 28, β = 8/3 and h = 0.01 the two are 1 or more apart after 3,348 steps (t = 33.48), as the table shows; nothing is assumed, it is read from the two runs.
Why does ρ = 14 look different?
With ρ = 14 the path spirals in towards one of the fixed points instead of looping between two lobes: after 6,000 steps it is less than 0.0001 away from it (measured from the run). Values of ρ above about 24.74 make the fixed points repel the path, as the Lorenz system article on Wikipedia notes.
What do Steps skipped first do?
They are worked out but not drawn, so the picture starts after the path has had time to settle onto its long-term shape.
How big can it go?
Up to 60,000 drawn steps (plus up to 20,000 skipped), with h from 0.001 to 0.02; 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 point and branch is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.