Thomas Attractor
Flow · (ẋ, ẏ, ż) = f(x, y, z, t)
(sin(y) − 0.2x, sin(z) − 0.2y, sin(x) − 0.2z)
Open in the app The dials and keys named below are the app's.
The system
The three lines are one rule written three times, passed round the variables: a sine of the next coordinate, less a damping 0.2 on this one. Other flow presets exchange the textbook's y and z, because the app is y-up and the classic systems are written z-up; the cycle here is symmetric, so this one needs no swap. Exchanging y and z only reverses the cycle, and the picture is the published attractor mirrored.
A lattice of saddles
A rest point needs sin y = 0.2·x and its two cyclic partners. Since a sine never exceeds 1, every rest point lies inside |x|, |y|, |z| ≤ 1/0.2 = 5, which is exactly the box Span of x, y, z shows. There are twenty-seven of them in that box. Three sit on the diagonal x = y = z, at 0 and ±2.596, where sin x = 0.2·x. At the origin the eigenvalues are 0.8 and −0.7 ± 0.866i: pulled in on a spiral, pushed out along one line.
Every one of the twenty-seven behaves like that. A seed spends its time being caught by one, wound around it and flung toward the next. Thomas called the result labyrinth chaos: a walk through a lattice of doorways rather than a fold. The damping decides how far the walk goes. At 0.2 it is strong enough to keep the seeds to one cell of the lattice, roughly −1.3 < x < 4. The stretching there is slow, a few hundredths per second against about 0.9 for the Lorenz preset, so the threads keep their bundle far longer.
History
René Thomas proposed this system in 1999 as the simplest cyclically symmetric flow, reading it as a particle moving through a three-dimensional lattice of forces against a friction of b.
Try
- Change all three 0.2s to 0.1: less friction, and the seeds break out of their cell to wander the lattice, out past x = ±6 and beyond the box.
- Raise them instead to 0.4: friction wins and every seed drifts to the diagonal rest point at x = y = z = 2.13.
- Top in the deck: the same tangle seen from above. A third of a turn about the diagonal x = y = z carries the figure onto itself, which is the cyclic symmetry of the three expressions.
- Set Seeds to 1 and follow one path through the maze from cell to cell.