Thus far, the analysis has been restricted to bifurcations involving only one parameter, and these are known as codimension-1 bifurcations. This example illustrates what can happen when two parameters are varied, allowing codimension-2 bifurcations.

The following two-parameter system of differential equations may be used to model a simple laser:

˙ x = x(y − 1), ˙ y = α + βy − xy.

Find and classify the critical points and sketch the phase portraits. Illustrate the different types of behavior in the (α, β) plane and determine whether or not any bifurcations occur.