Firefly Dynamics

**Anthonny** · November 1st 2010, 01:13 PM

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.

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

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

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

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.

**Anthonny** · November 4th 2010, 09:14 PM

Sorry to bump this topic, but here is a picture on exactly what the model is described as:

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