Malkus Chaotic Waterwheel

The Setup

A wheel (radius \(R\), moment of inertia \(I\)) rotates freely about a horizontal axis. Around its rim are \(N\) leaky cups. Water flows in from above, peaking at the top (angle \(\theta=0\)). Each cup drains through a hole at the bottom at rate proportional to its water mass.

Two things tend to happen. (i) As the wheel turns, a cup that fills near the top swings outward, its weight driving the wheel further - positive feedback. (ii) Leakage and viscous damping \(\nu\) bleed the rotation away - negative feedback. The competition between these produces extraordinarily rich dynamics: steady rotation, periodic oscillation, and - at high enough inflow - chaotic reversals indistinguishable from the Lorenz attractor.

Equations of Motion

Let \(m(\theta,t)\) be the mass density of water at angle \(\theta\) around the rim. Conservation of mass in a rotating frame gives

where \(Q(\theta)\) is the inflow (water falling in) and \(K\) is the leakage rate. The wheel's angular momentum equation is

The gravity torque integral picks up only the \(\sin\theta\) (first-sine) Fourier component of the mass distribution.

Fourier Reduction - How Lorenz Emerges

Expand the mass density in a Fourier series around the rim:

Take an inflow profile that is purely first-harmonic in cosine: \(Q(\theta) = q_1\cos\theta\) (water peaks at the top, \(\theta=0\)). Substituting into the PDE and projecting onto \(\cos\theta\) and \(\sin\theta\) decouples the first-harmonic modes \((a_1,b_1)\) from all higher modes:

Three ODEs in three unknowns \((\omega,b_1,a_1)\). Rescaling time and state gives exactly the Lorenz system:

The Lorenz parameter \(\beta\) is not free here - it is pinned to 1 by the waterwheel geometry. This is why Malkus's model is said to realise Lorenz exactly on a planar cross-section of parameter space.

What You'll See

• At low inflow \(q_1\): damping wins. The wheel either stays still (if initial \(\omega=0\)) or settles into steady one-directional rotation (symmetric pitchfork of stable fixed points \(C^\pm\), corresponding to Lorenz equilibria).
• At intermediate inflow: a Hopf bifurcation gives birth to a periodic orbit - the wheel oscillates without crossing zero angular velocity.
• At high inflow: chaos. The wheel reverses direction at unpredictable times, and the point \((\omega,b_1,a_1)\) traces the famous Lorenz butterfly in phase space - two lobes around \(C^+\) and \(C^-\).

Numerical Method

This simulation uses the discrete version of the Malkus wheel: \(N\) buckets with individual masses \(m_i(t)\), positions \(\theta_i(t) = \theta_i(0) + \varphi(t)\) where \(\dot\varphi=\omega\). The equations are

Note the \(\max(0,\cos\theta_i)\) inflow profile - physically realistic (water only lands in cups near the top), though it contains higher harmonics. For sufficiently large \(N\) these higher harmonics are damped out by leakage faster than the fundamental, and the effective dynamics in \((\omega,b_1,a_1)\)-space remain Lorenz-like. Integration uses a fixed-step explicit 4th-order Runge–Kutta scheme with \(\Delta t=0.01\).

The Fourier coefficients displayed in real time are

Sensitive Dependence on Initial Conditions

Toggle Sensitivity (twin trajectory) in the controls to launch a second trajectory whose initial angular velocity differs from the first by \(\varepsilon\). The waterwheel animation is replaced by a log-scale plot of the separation

In the steady or periodic regime the two trajectories converge and the separation decays. In the chaotic regime they diverge exponentially,

where \(\lambda_1>0\) is the maximal Lyapunov exponent. On the log plot this appears as a straight line whose slope is \(\lambda_1\); for Malkus / Lorenz with \(\sigma\!\approx\!10\), \(\rho\!\approx\!28\) the value is \(\lambda_1\approx 0.9\). After roughly \(1/\lambda_1\) time units the separation saturates at the diameter of the attractor - the two systems are then uncorrelated and prediction is lost. A dashed reference line \(\varepsilon e^{0.9 t}\) is overlaid for comparison.

The waterwheel is hidden in this mode because two near-identical wheels would not visibly differ until divergence has already occurred. Use very small \(\varepsilon\) (e.g. \(10^{-9}\)) to make the predictability horizon stretch farther into the simulation.

How to Use

• Click a preset (Steady / Periodic / Chaos / Intense Chaos) to load parameters that demonstrate each regime. The wheel is reset and animation starts.
• Use the parameter sliders to adjust inflow \(q_1\), leakage \(K\), damping \(\nu\), and moment of inertia \(I\). Changes take effect immediately without resetting the state.
• The Play / Pause button controls animation. Reset zeroes the state and starts over.
• Watch the right-hand plots: the top shows \(\omega(t)\) over time - a good indicator of the regime. The bottom 3D plot traces the orbit in \((\omega,b_1,a_1)\) space, where the Lorenz butterfly emerges in the chaotic regime.