The Detection Theorem: The eigenvalues can be detected by plugging the matrix describing the system into the Detect program.
Recall that key to the eigenvalues is that they help us determine some of the system's qualitative properties near an equilibrium point. The most important qualitative property that these eigenvalues allow us to determine is whether an equilibrium point is stable or not. Namely we found the the system is stable if and only if all the lambda are negative. Another quality that the eigenvalues let us determine is whether or not any oscillating behavior occurs near the system's equilibrium point. This property is detected by the presence of a complex number in the list of eigenvalues.
Warning: As we have discussed one should be careful about qualitative claims especially if a wiggle was required in when applying the Big Theorem . This corresponds to the case when their is a repeat or a zero in the list of lambdas. The wiggled solution is then near a bifurcation , in other words a system whose qualitative properties will change after after an arbitrarily small wiggle. For example with regards to the equations dx/dt=x(1-x)-axy and dy/dt=-y+axy we saw that as our parameter a varied from from 1.1 to 2 that the equation experienced a bifurcation . It is a bit difficult to see this from the animation, however at the two extreme a values (a=1.1 and a=2) we can zoom in and can clearly see that a bifurcation must occur. After exploring these equations in detail we found that a bifurcation took place when the parameter a was about equal to 1.207 and that any a near this value will be close to a bifurcation. We gave these types of observations a name...
The Wiggle Test: If there is a zero or repeat in the lambda values then you need to perform a wiggle, and if there is nearly a zero or a repeated lambda value then you are near a bifurcation - and you should mention it!