💧 Lorenz Waterwheel
Malkus waterwheel & Lorenz attractor
Presets
Parameters
State
x
0.00
y
0.00
z
0.00
ω (rotation)
0.00
Chaotic
Lorenz Equations: The Malkus waterwheel (1972) is a physical realization of Edward Lorenz's 1963 equations. Water drips into buckets on a wheel at constant rate; buckets leak proportional to fill. For certain parameters the rotation direction reverses unpredictably — a perfect mechanical analogy of atmospheric convection.

About this simulation

This simulation models the Malkus waterwheel, a mechanical system whose equations of motion are mathematically identical to Edward Lorenz's 1963 chaos equations. Twelve buckets are mounted around a wheel; water drips in at a constant rate at the top and each bucket leaks out at a rate proportional to how full it is. Gravity acting on the unevenly distributed water produces a torque that spins the wheel, while friction (implicit in the σ, r, b parameters) opposes it — and for the right parameter values this tug-of-war never settles down, instead flipping rotation direction unpredictably forever. The same x, y, z state driving the wheel's angle and bucket fill also traces the classic butterfly-shaped Lorenz attractor in the phase-portrait panel.

🔬 What it shows

The wheel's rotation angle is driven directly by the x variable of the Lorenz equations (dx/dt = σ(y−x), dy/dt = x(r−z)−y, dz/dt = xy−bz), integrated with RK4. Water pours into whichever bucket is currently at the top and drains from every bucket in proportion to its fill level, so the visible sloshing water mirrors the underlying chaotic state. The green/red arc around the wheel and the ω readout show the current spin direction and speed, while the phase-portrait and x(t) strip on the right plot the same trajectory as a strange attractor.

🎮 How to use

Choose a preset (Chaotic, Stable, Edge) or drag the σ, r, b and Speed sliders yourself to change the dynamics. Use Pause to freeze the wheel and attractor at any instant for inspection, and Reset to restart from a fresh near-zero state (with a tiny random perturbation) and clear the trail — useful for seeing how a barely different start can lead to a wildly different path.

💡 Did you know?

The physical Malkus waterwheel was built in the 1970s specifically to demonstrate that Lorenz's equations, first derived from a simplified model of atmospheric convection, aren't just an abstract curiosity — a real spinning wheel with leaky buckets reverses direction just as chaotically as the weather does, using the exact same three-variable math.

Frequently asked questions

What is the Lorenz waterwheel, physically?

It is a wheel with a ring of buckets, each with a small hole. Water drips onto the bucket currently at the top from a fixed nozzle, and every bucket continuously leaks water in proportion to how full it is. Gravity pulls harder on the heavier, wetter side of the wheel, creating a torque; if the inflow is fast enough relative to the leak rate and friction, the wheel accelerates, but the shifting weight can also make it slow, stop and reverse.

Why do the equations match the Lorenz system exactly?

Malkus and Howard showed in the 1970s that if you write Newton's law for the wheel's angular velocity together with a Fourier decomposition of the water distribution around the rim, the first three coefficients obey equations of the same form as dx/dt = σ(y−x), dy/dt = x(r−z)−y, dz/dt = xy−bz. Here x is proportional to the wheel's rotation rate, y and z describe how the water mass is distributed around the wheel, and σ, r, b are combinations of the physical drip rate, leak rate, moment of inertia and friction.

Why does the wheel suddenly reverse direction?

Reversal happens because the water distribution can become asymmetric enough that the heavier side crosses over the top before it has drained, flipping which side gravity favours. In the chaotic regime (the default preset, r = 28) this happens at seemingly random intervals because the trajectory is bouncing between the two "wings" of the Lorenz attractor, and small differences in exactly when it crosses determine whether it flips early or late.

What do the σ, r and b sliders actually control?

σ (sigma) sets how quickly the wheel's rotation responds to the imbalance in water distribution — physically related to the ratio of momentum damping to water redistribution. r (the Rayleigh-like parameter) scales with the drip/leak rate and determines how much driving energy is available; low r settles to a fixed point or gentle spiral, while r above roughly 24.7 (with the default b) produces the chaotic reversing regime. b relates the leak rate to the wheel's damping and shapes how quickly water redistributes as the wheel turns.

Is this simulation just showing the Lorenz attractor with extra steps?

In terms of the underlying numbers, yes — the same RK4-integrated x, y, z trajectory drives both the wheel animation and the phase portrait, so the two panels are two views of one chaotic system. The value of the waterwheel view is intuition: it turns an abstract three-variable ODE into a tangible mechanism (dripping water, leaking buckets, a spinning wheel) so the cause of the unpredictable direction reversals is easier to see than in the abstract x-y-z plot alone.