We can make some general statements about the prescence of steady states in this CSTR model.

If the parameter b > 20, then we know that there are two limit point bifurcations with respect to a. The steady states exist in either solitude or in groups of three (and only at the limit point bifurcation points, in pairs), depending on the value of a. When three are present, a source, saddle, and a sink simultaneously exist.

If the parameter b = 20, we know that there are no limit point bifurcations. There is only one steady state for arbitrary a, and it is always a sink.

If the paraemter b < 20, we know that there are no limit point bifurcations. There is only one steady state for arbitrary a, and it is always a sink.

We can generalize as we did because there are no transcritical bifurcations in this model; the stability of a given steady state will not change with a change in a.

We found that Hopf bifurcation points do exist in this model. We originally found a critical minimum value of b approximately equal to 11.89 that guarantees the existence of a Hopf bifurcation point. However, in this analysis we forgot to include the criterion that det(Jacobian(*u*,*v*)) > 0 for a Hopf bifurcation point to exist. We noticed a problem when the xrk plot with b = 11.9 did not show any periodic steady states ("bananas"). This may be the cause of the problem.

Date: October 15, 1998

Time: 12:45 PM