Most people think of chaos as the so-called “butterfly effect”, namely that a butterfly flapping its wings in Beijing can set off a cascading chain of atmospheric events that two weeks later leads to the formation of a catastrophic tornado that obliterates central Zürich. Oh, and Tyrannosaurs:

More mathematically, this is sensitive dependence on initial conditions: in this case the (dynamical) system in question is the weather, and the small change (the butterfly) leads to a large change (the tornado) later down the road.

Before defining this precisely, let's see an example of a dynamical system that displays this "sensitive dependence". The “double pendulum” is what you think it is: take a pendulum and then hang another pendulum on the end of it. This dynamical system is chaotic (defined precisely below) and in particular has sensitive dependence on initial conditions.

Unfortunately the mathematical definition of chaos is rather less glamorous than the “popular science” one. This is actually true of most things in life: adding $\varepsilon$s and $\delta$s rarely make things exciting.

Here are today's notes: