Attractlet attractlet model · § 7.4

The Lorenz Attractor

The most famous strange attractor: a butterfly that almost every start falls onto and none ever settles on, held up by the heat in its equations and the solver that runs them.

In 1963 Edward Lorenz cut a model of convection, a fluid layer heated from below, down to three equations. Run them and almost every starting point, from anywhere, falls onto the same butterfly-shaped set and loops around it without ever settling. Nudge a point off and it falls back, though not to the same place on it. By surface measurement this passes for an attractor, and in the dynamical-systems sense it is the textbook one. But nothing in the picture produces the motion. The solver advances the equations, and the heating term ρ drives them. Turn the heat down and the butterfly is gone.

Three Equations, One Butterfly

side view · x across, z up

still moving,

settled at rest,

settled on a steady roll,

twin separation,

solver clock t,

solverrunning

settling

The motion is the solver’s, and the drive is ρ. Each step is computed and applied from outside, and the t in the equations is the solver’s clock, not something the butterfly keeps. Stop the solver and every point freezes. Turn ρ below 1 and every start relaxes to rest at the origin. Between 1 and about 24, starts settle onto one of two steady convection rolls, though above about 20 that can take a long time. Near 28 they fall onto the butterfly. White rings: the resting state and, above ρ = 1, the two steady rolls. Trails fade for the eye; the points move only by the equations.

The Equations, and What the Heat Does

two numbers fixed, one handed to you

dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz
σ = 10, β = 8/3, ρ = the heating slider

Lorenz kept three variables from a model of a fluid layer heated from below: roughly, how fast the fluid turns over (x) and how the temperature is arranged across it (y and z). ρ is how hard the layer is heated. Below ρ = 1 the only resting state is no motion at all. Above it two more appear, one steady roll turning each way, at

C± = ( ±√(β(ρ − 1)), ±√(β(ρ − 1)), ρ − 1 )

The rolls stop holding their neighbours at ρ ≈ 24.74, which works out from σ and β as σ(σ+β+3)/(σ−β−1) = 24.737. Above that, nothing steady is left to settle on. Each row below is 1,200 starts from the panel’s own scatter, run with its own step and its own rule for “settled”:

ρ	settled by t = 40	by 200	by 400	onto
0.5	1,200	1,200	1,200	rest
10	1,200	1,200	1,200	rolls
14	1,200	1,200	1,200	rolls
20	864	1,200	1,200	rolls
23	3	728	946	rolls
24.5	0	1	14	rolls
28	0	0	0	none

Below 24.74 the rolls are still there to be found, but the closer ρ gets to it the longer starts wander first. At 24.5, fourteen of 1,200 had found a roll by t = 400. At 28, none do.

The Basin, Measured

this card’s claim, earned

Almost every start arrives. The exceptions, the three resting states and the thin sets that lead exactly into them, take up no volume. Scatter new starts draws from a box far wider than the wings, and at ρ = 28 all 1,200 starts in the table above were still moving at t = 400, none of them stuck anywhere.

A nudged point comes back to the butterfly. Nudge the points pushes each point 4 to 12 units in a random direction. That was done to 2,000 points already on the butterfly, and each was then measured against a dense map of it (125,000 points):

time after the nudge	within 0.5 of the butterfly
0	6.1%
0.5	94.7%
5	97.8%
10	99.0%
20	99.5%
never nudged	99.75%

Most are back within half a unit of time. The slow few were pushed into the eye of a wing, next to one of the two steady rolls. At ρ = 28 a roll no longer holds anything, but it lets go gently: a point near it spirals outward at a rate of 0.094 per unit of time, doubling its distance about every 7.4, before it rejoins the wing.

What the basin does not give back is position. A nudged point returns to the butterfly, not to where it was on it. Returning to a set and returning to a state are different things, and the next section puts a number on how different.

Released Twins, and a Published Number

a result the panel measures rather than looks up

Release twins starts two points a millionth of a unit apart. On the butterfly the gap grows by a factor of e per unit of time at an average rate called the largest Lyapunov exponent. J. C. Sprott gives the three exponents for Lorenz’s original settings as “(0.906, 0, -14.572)”, computed with a Runge-Kutta step of 0.001 over more than a million units of time.

The panel was measured with its own step of 0.004, over eight independent runs of 2,000 units each:

source	largest exponent
Sprott	0.906
panel, 8 runs	0.8985 to 0.9067
panel, mean	0.903

Released from 200 different places on the butterfly, twins took a median of t = 16.9 on the solver clock to drift 10 units apart, and 90% did it between 15.2 and 19.4. A difference of one part in a million is gone before the solver clock reaches 20.

Why This Is an Attractlet

the kernel’s own classification

The kernel’s Attractlet Model (Formal or Symbolic) class explicitly includes “ODE and PDE formalisms” (ATTRACTLET, § 7.4). The Lorenz system is three ordinary differential equations. Whatever it does, it does inside the formalism, and the formalism runs only while a solver runs it.

It is supplied twice over. The solver supplies every step. Inside the equations, ρ, the heating, supplies the drive: set it below 1 and every start relaxes to rest. The butterfly exists only while both are handed in.

It also shows why the framework keeps time out of an attractor’s logic. That logic contains no external time parameter (Time and the Arrow of Sequence). The Lorenz equations are written against one: t, the solver’s clock, handed in from outside.

None of the measurements above moves the classification. § 7.4: “No attractlet may be promoted to attractor status by complexity, longevity, apparent stability, mathematical elegance, or external coupling strength.” The butterfly is complex, elegant and apparently stable, and none of that counts.

What This Panel Is Not

the limits, stated before anyone finds them

A side view. The picture shows x across and z up. The third variable, y, is hidden, so paths that look as if they cross do not.

A fixed-step solver. Classical Runge-Kutta with a step of 0.004. Because nearby paths separate so fast, no computed path stays on the exact path of the equations for long: at the measured rate, rounding error in the sixteenth digit grows to the size of the butterfly in roughly 40 units of time. What survives is the butterfly’s shape and its averages, which is why the exponent above can be checked against Sprott’s and an individual path cannot.

“Settled” is a rule. A point counts as settled after 45 display ticks moving slower than 0.5 within 1 unit of a resting state. A stricter or looser rule would move the numbers in the first table.

Three equations are not a fluid. Lorenz kept three variables from a much larger description of convection. The butterfly belongs to the three equations; nothing here claims a real heated fluid layer does the same.

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.
Sprott, J. C. “Lyapunov Exponent and Dimension of the Lorenz Attractor,” University of Wisconsin, sprott.physics.wisc.edu/chaos/lorenzle.htm (fetched and read 2026-09-16). Source of the quoted exponents and their method.
“The Lorenz system: An introduction to chaos,” University of Toronto, MAT332 course notes, math.toronto.edu/kzhang/teaching/courses/mat332-2022/_8-lorenz-system/ (fetched and read 2026-09-16). Source of the steady rolls C±, the change at ρ = 1 and the value ρ ≈ 24.74.
Every other number on this page was measured on the panel’s own equations, step and settling rule, on 2026-09-16.

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