Edward Lorenz was a meteorologist who, around the time I was born, noticed
that weather models were highly sensitive to tiny perturbations in their initial
conditions – he discovered the so-called butterfly effect
. We now
look back on this as the first serious study of what are now termed chaotic
systems
– and we now know that dynamical systems whose models using
first-order differential equations aren't linear (i.e. the rate at which one of
the variables is changing isn't simply a sum of the various variables each
scaled by a constant, but involves other functions of those variables) commonly
are chaotic if they have more than three free variables. In any case, back when
I was still a child, Lorenz came up with the following simple little system that
exhibits chaotic behaviour:
in which s, r and b are some constants and U, V, W are quantities that
vary with time, t. I learned of this (and the following is influenced
by) Sabine
Hossenfelder's
video How can climate
be predictable if weather is chaotic?
, which I can happily recommend to
anyone interested in the subject. The point of the video is that, while small
perturbations in the input lead to subsequent histories that are routinely far
apart, one can make clear statistical predictions about each solution
spending quite definite proportions of its time behaving in each of two common
behaviours, and likewise statistically characterise what any solution is
typically doing the rest of the time. So the details are hard to pin down but
the general character of (nearly) all solutions can be quite plainly described,
all the same.
That aside, I've created this page in the vague thought that I might some day actually study this system of equations in more detail. I haven't done that yet, but at least I've got the equations written down somewhere I can easily find them.
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