Here's the creek just down the hill from my office.
It's called Boulder Creek. And here's an eddy in that
creek. And as you can see, the water is swirling around.
If you fix your eye on one place, you'll see that the
general pattern of the water is pretty
constant at each place. But,
if you watch those little pieces of foam and leaves,
you'll see that there is a very complicated path
that a single tracer particle on the surface of
the water would follow through that eddy. This
is a classic metaphor for a chaotic attractor.
By the way, take note of that concrete wall
in the back of the shot; that will come back later.
Here I am dropping some leaves in the water.
You couldn't see them very well, but luckily,
this other tracer particle came into the scene.
Keep your eye on that tennis ball. It takes a
very complicated path through the eddy and
it is sensitively dependent on initial conditions.
So a little bit to the left or a little bit to the right,
and the path will be very different, but it keeps
travelling around the same eddy.
If you look closely here, you'll see that there's
not only a tennis ball, but also a small apple
in the creek.
Here, they are very close together, and then they
get spread out a little bit, and then that difference
grows rapidly, causing them to take very different
paths through the eddy. And if you watch for awhile, you
can see the apple and the tennis ball joining
and rejoining and separating and resaperating
as they follow the paths of the water in the eddy.
Here's a different eddy in the same creek, next
to the same rock. Much smaller, but all of
the same ideas.
Bifurcations in the dynamics can occur when
the parameters of the system change; for example,
when there is a lot of water. This is a video
taken from the same bridge where we were
doing those experiments. The concrete wall that
you saw briefly at the beginning of this video
at the flooded creek is the backdrop for the
first shot in this video that you are watching today
for this course.
In September of 2013, Boulder got a year and
a half's worth of rain in 5 days. This is
a reading of the stream flow in the creek
where we were working. That change
in the flow rate completely changed the
topology of the flows in that creek - bifurcations
in the dynamics.
A lot of that water ended up in my crawl space
which necessitated the use of some
of my engineering training.
But back to this plot, I wanted to point out
that this is a log plot. The vertical axis is the
log of the flow and the horizontal axis is time.
We're going to talk a lot about log plots later on.
And I wanted to close by pointing out that the
dynamics you observe may not be everything that
is going on.