> This includes answering “what the @#!% is a Lyapunov function and why the %@ should I care???”
I'm glad that Lyapunov functions are quite central to to Marc's and Manoj's blogs, since they were the only thing I did sort of understand beforehand!
I think I'd start an explanation with a ball rolling around in a bowl. Add some syrup to stop the the ball having enough kinetic energy to worry about. Then the gravitational potential (or just the height) of the ball can serve as a Lyapunov function. Without the syrup, things become more complicated, but at least you expect the potential plus kinetic energy of the ball to monotonically decrease. Without the bowl, who knows where the ball might go.
It seems quite intuitive that:
1. Lyapunov functions are great things if you can get them because they give a guide to the global properties of the system.
2. You don't expect to find them in general, and even when one exists, it may be difficult to find.
3. Free energy or entropy, or analogues, are a good place to start looking.