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.