Nine strange attractors, side by side
Nine chaotic systems, three equations each, with the parameters Chaos Studies uses. Every name opens a page with a live simulation, the history and what to look for.
Lorenz
restlessRössler
meditativeAizawa
coiledBurke-Shaw
compactHalvorsen
architecturalChen
turbulentNose-Hoover
patientSprott B
energeticDadras
woven| Attractor | Equations | Parameters | Divergence ∇·f | First described |
|---|---|---|---|---|
| Lorenzrestless | ẋ = σ(y − x) ẏ = x(ρ − z) − y ż = xy − βz | σ = 10, ρ = 28, β = 8/3 | −(σ + 1 + β) ≈ −13.67 | Edward Lorenz, 1963 |
| Rösslermeditative | ẋ = −y − z ẏ = x + ay ż = b + z(x − c) | a = 0.2, b = 0.2, c = 5.7 | a + x − c (varies) | Otto Rössler, 1976 |
| Aizawacoiled | ẋ = (z − b)x − dy ẏ = dx + (z − b)y ż = c + az − z³/3 − (x² + y²)(1 + ez) + fzx³ | a = 0.95, b = 0.7, c = 0.6, d = 3.5, e = 0.25, f = 0.1 | varies | attributed to Yoji Aizawa |
| Burke-Shawcompact | ẋ = −s(x + y) ẏ = −y − sxz ż = sxy + v | s = 10, v = 4.272 | −(s + 1) = −11 | Bill Burke and Robert Shaw, 1981 |
| Halvorsenarchitectural | ẋ = −ax − 4y − 4z − y² ẏ = −ay − 4z − 4x − z² ż = −az − 4x − 4y − x² | a = 1.89 | −3a ≈ −5.67 | named after Arne Dehli Halvorsen |
| Chenturbulent | ẋ = a(y − x) ẏ = (c − a)x − xz + cy ż = xy − bz | a = 40, b = 3, c = 28 | −a + c − b = −15 | Guanrong Chen and Tetsushi Ueta, 1999 |
| Nose-Hooverpatient | ẋ = y ẏ = −x + yz ż = 1 − y² | no free parameters | z, zero on average | Shuichi Nosé, 1984; William Hoover, 1985 |
| Sprott Benergetic | ẋ = yz ẏ = x − y ż = 1 − xy | no free parameters | −1 | J. C. Sprott, 1994 |
| Dadraswoven | ẋ = y − ax + byz ẏ = cy − xz + z ż = dxy − ez | a = 3, b = 2.7, c = 1.7, d = 2, e = 9 | −a + c − e = −10.3 | Sara Dadras and Hamid Reza Momeni, 2009 |
Divergence is the rate at which the flow shrinks a small volume of starting points. Where it is negative everywhere, as for Lorenz, space collapses onto a thin set: the attractor. Nosé-Hoover is the exception, with a divergence that averages to zero.
Hold them in your hands
Chaos Studies draws all nine in full 3D with a spatial soundtrack. $1.99 once for iPhone, iPad and Mac, $2.99 on Playdate.
