The relationship between the Malkus waterwheel and the Lorenz system

This is not a math tutorial. See Why am I writing about math?

There is a correlation between the structure and movement of the Malkus waterwheel and the Lorenz differential equations.

NOTE: when you run the Lorenz equations on a computer, you’re working with dimensionless numbers. For a physical waterwheel, x has units of angular velocity (radians per second), y and z have units of mass times length (kg times meters). The numerical values depend on wheel size, bucket size, flow rate, friction, etc.

The Lorenz equations

Where $\sigma$, $\rho$, and $\beta$ are fixed parameters that define aspects of the system, and $x$, $y$, $z$ are set to some initial values.

In (imprecise) words:

• the (instantaneous) rate of change of x with regard to time is the value of sigma times (x - y)
• the (instantaneous) rate of change of y with regard to time is x times (rho - z) minus y
• the (instantaneous) rate of change of z with regard to time is x times y minus beta times z

How the Lorenz equation parameters relate to the Malkus waterwheel

• $\sigma$ (sigma): the damping coefficient (the friction in the wheel)
• $\rho$ (rho): water inflow rate
• $\beta$: water outflow (leakage from buckets)

Typical parameter values

• $\sigma = 10.0$
• $\rho = 28.0$
• $\beta = 8.0/3.0 \approx 2.67$

How the Lorenz equation variables x,y, and z relate to the Malkus waterwheel

The Lorenz equations

$x$ is the angular velocity of the wheel. It represents how fast the wheel is spinning, and what direction it is spinning in. When $x$ is positive, the wheel is spinning one way (I’ll assume clockwise), when $x$ is negative, the wheel is spinning in the opposite direction. When $x$ is large, the wheel is spinning quickly (the angular velocity is high). When $x$ is small, the wheel is spinning slowly (the angular velocity is low).

The Lorenz equations

$y$ is the asymmetry in water distribution (left-right). I’m understanding this to mean how water is distributed between the left and right sides of the wheel. If more water is on the left side ($y$ is negative), the wheel will spin counter-clockwise. If more water is on the right side ($y$ is positive) the wheel will spin clockwise. Changes to

The Lorenz equations

$z$ is the vertical distribution of water (up-down). It represents how much water has accumulated in the lower buckets. Water accumulating in the lower buckets affects the wheel’s moment of inertia and stability. (reference needed.)

Typical ranges of x, y, and z

• $x$: -2 to +2 radians/second (for a small waterwheel)
• $y$ (a dimensionless measure of water distribution), in the range -5, +5
• $z$ (vertical water mass), in the range 0, +50, typically (27-50) A large $z$ indicates less water in the lower buckets. $z = 0$ would mean that all the water is in the lower buckets. The system would be fairly static at this point. When $z$ is high, the system is dynamic.

Relating the Lorenz equation variables and parameters to the equations

The interlocking nature of the equations makes it difficult to think about the system as a whole.

Starting with $dz/dt = xy - \beta{z}$:

• the $xy$ term pushes $z$ upward. The $\beta{z}$ represents decay. $z$ wants to decay (the buckets are leaky). $xy$ assuming non-tiny values, wants to push the draining buckets back to the faucet.

$dx/dt = \sigma(y - x)$:

This says that $x$ is trying to chase $y$, it wants to match $y$, but lags behind. Remember that $\sigma$ (friction) typically has values around 10.0:

• if $y$ is greater than $x$, $y - x$ will be a positive number. If $y < x$, $y - x$ will be a negative number. Think about this in terms of water in the buckets. The horizontal distribution of water ($y$) will pull the movement ($x$) towards the side that has more water.

$dy/dt = x(\rho - z) - y$, remember that $\rho$ is the water inflow rate with a typical value of 28:

• $x(\rho - z)$ is the driving term:
  • when $z$ is small (when it’s less than $\rho$) the driving term is positive, causing $y$ to increase
  • when $z$ is large (the system is in a dynamic state), $y$ will decrease (?)
  • as $z$ rises, the driving term has less influence on $y$
• $-y$ is the damping term, it pulls $y$ back down

How data recorded from the waterwheel might map to a Lorenz system

I think another way of putting this is how the waterwheel could be represented in phase space.

For the Lorenz attractor to waterwheel mapping:

• $x \leftrightarrow \omega$: $x$ maps to the angular velocity of the wheel
• $y \leftrightarrow$ first cosine mode of water distribution
• $z \leftrightarrow$ first sine mode of water distribution

How the y and z values could be measured for a moment in time

For $y$, measure how much water is on the right side versus the left side of the wheel. This could be done by weighing each bucket’s water and scaling it by its horizontal distance from the either the vertical or horizontal axis. $y$ is therefore a measure of the system’s horizontal imbalance.

For $z$, measure the vertical distribution of the water. I think this could be approached in a similar way the $y$ measurement. I’m wondering about the role of gravity in all this.

Essentially, capture weighted averages of where the water is concentrated at a moment in time.

What does first cosine mode and first sine mode mean?

In the context of the waterwheel, I think the terms first cosine mode and first sine mode are a way of describing the spacial distribution of the water. (I had the idea that the terms were related to Fourier decomposition, but that doesn’t make sense as the waveforms generated by the wheel aren’t periodic.) Somewhat confusingly, using the terms first cosine mode and first sine mode to describe the distribution of the water is describing the water using a spacial Fourier series — the distribution of water at different angles around the wheel.

The distinction is that a spacial Fourier decomposition describes the distribution around the circumference at one moment in time; temporal Fourier decomposition describes how things change over time. It can’t (I think) be applied to a chaotic system — it’s for describing period functions.

At any moment in time, water is distributed around the circular wheel — each bucket is at some angle from a bucket that’s (arbitrarily (?)) given the angle $\theta = 0$. The first cosine mode captures how water is distributed along the vertical axis, the first sine mode captures how water is distributed along the vertical axis. Given a system with 4 buckets, the formulas would be something like:

y(t) ~ m₁(t)cos(0) + m₂(t)cos(π/2) + m₃(t)cos(π) + m₄(t)cos(3π/2)
z(t) ~ m₁(t)sin(0) + m₂(t)sin(π/2) + m₃(t)sin(π) + m₄(t)sin(3π/2)

But, this only makes sense to me if the angles are measured from a fixed reference that’s relative to the hub of the wheel. If the hub is taken as the origin of a 2D plane, where angle $\theta = 0$ is the equivalent of x = spoke length, y = 0 on a Cartesian plane, the system’s $y$ value (not to be confused with the x,y Cartesian points) will be most influenced by buckets that are on the y axis of the imagined Cartesian plane ($\cos(0) = 1$, $\cos(\pi) = -1$, $\cos(\pi/2) = 0$, $\cos(3\pi/2 = 0$)

The details of this aren’t super relevant to me at the moment.