Divergence Criterion for Generic Planar Systems

The divergence criterion has been shown to be helpful in distinguishing between suband supercritical Hopf bifurcations but its applicability is limited to systems whose divergence is sign definite. A step by step computational procedure which allows one to extend the applicability of the divergence criterion is derived by altering the system to an equivalent one with sign definite divergence. The procedure is based on multiplying the original vector field by a positive quadratic function in a neighborhood of the bifurcating rest point. This procedure is then applied to several examples of planar systems that exhibit the Hopf bifurcation. Specifically, it is demonstrated that only supercritical bifurcations occur in a system modeling specific immune responses with handling time. It is also shown that the Fitzhugh-Nagumo equations and the chemostat equations with substrate inhibition and linear yield coefficient may exhibit both suband supercritical Hopf bifurcations. In both cases, simple analytic criteria for determining the criticality of the bifurcation are presented.