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.

$\displaystyle \theta(t)$ 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 $\displaystyle \frac{d\theta}{dt}=\omega$ where $\displaystyle \omega$ 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. $\displaystyle \Theta(t)$ is the phase of the stimulus and $\displaystyle \frac{d\Theta}{dt}=\Omega$.

With the introduction of the stimulus firefly, $\displaystyle \frac{d\theta}{dt}$ becomes

$\displaystyle \frac{d\theta}{dt}=\omega+A\sin(\Theta-\theta)$

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.