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 14 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.

Why This Is an Attractlet

the kernel’s own classification

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

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.

The Shape of the Basin

almost everywhere, but not the same place

At the classic settings the basin is almost everything. From nearly any starting point in the three-dimensional space, trajectories fall onto the butterfly. The exceptions, the three fixed points and the thin sets that lead exactly into them, have zero volume. Press Scatter new starts: they come from far outside the wings and arrive all the same.

What the basin does not give back is position. Press Release twins: two starts a millionth apart drift until they are as far apart as the butterfly allows. 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.

None of this 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.

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.

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