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:

The original Jurassic Park was terrible, but fun. The new movies are so bad in comparison... 

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.

Credit: Bassam Jalgha

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:



Comments and questions?