Lorenz's Water Wheel

In Lorenz's water wheel, equally spaced buckets hang in a circular array. Water pours into the top bucket and leaks out of each bucket at a fixed rate. The wheel behaves chaotically for certain choices of parameters, showing unpredictable changes in the direction of rotation. This behavior of this system is analogous to that of a Lorenz attractor.

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DETAILS

The equations of motion are:
where:
is the radius of the wheel
is the leakage rate
is the flow rate into the top bucket
is the rotational damping rate
is the moment of inertia of the wheel
is gravitational acceleration
The system is modeled as a ring; by assumption, the amount of water in a section (represented by a bucket) of the ring is proportional to , where is the angle of the bucket moving with angular velocity for initial conditions , , and .
Reference
[1] T. Tél and M. Gruiz, Chaotic Dynamics, An Introduction Based on Classical Mechanics, New York: Cambridge University Press, 2006.
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