Rössler Attractor
Flow · (ẋ, ẏ, ż) = f(x, y, z, t)
(−z − y, 0.2 + y · (x − 5.7), x + 0.2z)
Open in the app The dials and keys named below are the app's.
The system
These are Rössler's equations of 1976 with his constants a = b = 0.2 and c = 5.7. His y and z are exchanged here, because the app is y-up and the textbook system is z-up, so the height, y, carries what he called z. Only one nonlinear term appears in the whole system, the product y·x in the second line, and it does all the work.
Why it never settles
Ignore the height for a moment. In the floor plane the first and third lines read ẋ = −z, ż = x + 0.2·z: a rotation with a slow outward push. Near the origin the rest point at (0.007, 0.035, −0.035) has eigenvalues 0.097 ± 0.995i, so a seed loops roughly every 6.3 seconds. Its radius grows by a factor of about 1.8 each loop. That is the stretching: a tight band of seeds is drawn out into a long ribbon.
The height supplies the fold. While x < 5.7 the bracket (x − 5.7) is negative and y is held down near the floor. When the widening spiral finally carries a seed past x = 5.7 the bracket turns positive, y grows fast, and the seed is thrown up, carried back toward the axis and dropped near the centre to start again. Stretch, fold, repeat: the recipe for a strange attractor, here with a single fold and nothing else. The stretching is gentler than Lorenz's, so the pack of threads stays recognisable for a good many turns.
History
Otto Rössler published this system in 1976, in a paper in Physics Letters A called An equation for continuous chaos. He put it together by hand as an example of chaos with the least machinery he could manage, one fold and one nonlinear term, rather than as a model of anything physical.
Try
- Change the 5.7 to 2.5: every seed ends on one closed loop. At 3.5 the loop takes two turns to close, at 4 four turns, the period-doubling road to the chaos at 5.7.
- Front in the deck shows the fold end on: a flat band with one lip standing up.
- Set Seeds to 1 and let the single thread lay the band down turn by turn.
- Raise Span of x, y, z to 30: the far rest point sits out at height 28, well off the picture the attractor occupies.