Experiments 17
Which Way It Turns
A Malkus waterwheel of leaky cups under a tap obeys the Lorenz equations. Adjust the flow rate to move the wheel between rest, steady rotation, periodic oscillation, and chaos.
Rust · WASM · Runge–Kutta · Canvas · SVGView source
Tumbling. Watch for the moment it changes direction.
The wheel
Drag it to spin it. Tap a cup to splash water in.
Its path
The shape it keeps tracing is its attractor. Tap it to start a few wheels there.
Copy makes a second wheel from this one’s numbers rounded to three places, the way Edward Lorenz retyped a printout in 1961.
Twins are 100 wheels that each start a hair away from this one. Do they stay together? You can also tap the path to start a few wheels anywhere.
Drag the wheel to spin it, or tap a cup to splash water in. The shape its path keeps tracing is its attractor. Tap the path to start a few wheels there.
Notes
The Malkus waterwheel and the Lorenz system
Willem Malkus and Lou Howard constructed a leaky waterwheel at MIT as a mechanical realization of Edward Lorenz’s equations. Expanding the rim’s water distribution in a Fourier series (Strogatz, Nonlinear Dynamics and Chaos, §9.1), only the first harmonic a₁ sin θ + b₁ cos θ exerts torque:
ȧ₁ = ω b₁ − K a₁
ḃ₁ = −ω a₁ − K b₁ + q₁
ω̇ = (−ν ω + π g r a₁) / I
With leak rate K, damping ν, moment of inertia I, and inflow q₁, rescaling time by 1/K and setting x = ω/K, y = (πgr/Kν) a₁, and z = ρ − (πgr/Kν) b₁ yields the Lorenz system ẋ = σ(y − x), ẏ = ρx − y − xz, ż = xy − βz with σ = 10, ρ = πgr q₁/(K²ν), and β = 1.
Regimes across flow rate ρ
- ρ < 1: the origin is globally stable and the wheel comes to rest.
- 1 < ρ < 8.18: pitchfork bifurcation into two stable fixed points corresponding to steady clockwise or counterclockwise rotation.
- 8.18 < ρ < 17.5: past the homoclinic bifurcation at 8.18, basin boundaries fold and a strange attractor coexists with the stable rotations above ρ ≈ 14.9.
- ρ > 17.5: subcritical Hopf bifurcation at
σ(σ + β + 3)/(σ − β − 1) = 17.5destabilizes steady rotation into chaotic reversals, interrupted by periodic limit-cycle windows around 26.75–27.75, 40–41.5, and 326–362.
Simulation and sensitivity
attractor.rs integrates the system in WebAssembly using fourth-order Runge–Kutta (dt = 0.005), alongside a Swarm of perturbed trajectories and a three-decimal rounded twin modeled on Lorenz’s 1961 Royal McBee LGP-30 printout rerun. Warwick Tucker proved in 2002 via validated numerics that the Lorenz attractor is genuinely chaotic rather than a floating-point artifact (see No Goals).
Sources
- Edward N. Lorenz, “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences 20 (1963): 130–141.
- Steven H. Strogatz, Nonlinear Dynamics and Chaos, §9.1.
- Warwick Tucker, “A Rigorous ODE Solver and Smale’s 14th Problem,” Foundations of Computational Mathematics 2 (2002): 53–117.
- Malkus waterwheel and Lorenz system on Wikipedia.