I'm trying to figure out this example on one-dimensional flows on a circle using the flash of fireflies.

The information/example is given here:

http://www.paleo.bris.ac.uk/~ggxir/c.../lecture-4.pdf

which starts at page 14, but the equations are given on page 19.

I'm moreover confused about the information given.

is said to be the firefly phase. I'm not sure what this means; I interpret it as the amount of flashes that have gone by since some time t. Next where is a constant. I think this means that the rate at which a firefly flashes is constant.

Next the equation for the stimulus is given in a similar manner. is the phase of the stimulus and .

With the introduction of the stimulus firefly, becomes

Which I don't completely understand why that is. I'm confused on why sine is in the new equation.

Clarification on what these equations represent would be appreciated.

Thank you.