We now consider a detailed example to help reinforce the barrage of mathematical definitions and concepts in the previous sections. Our example is taken from nonlinear optics and is known as the laser rate equation [10],
In this model u is the laser intensity and z is the population inversion.
The parameters 
and 
are damping constants.
When certain lasers are turned on they tend to settle down to a
constant intensity light output (constant u) after a series
of damped oscillations (ringing) around the stable steady state solution.
This behavior is predicted by the laser rate equation.
To calculate the stability of the steady states we need to know
the derivative
of 
:
Matrix