Calculating Bifurcation Diagrams for Rational Differential Equation Systems Using the Resultant

The resultant is a well-known tool in number theory and algebraic geometry used to find simultaneous zeros of two polynomials. Here we show how to use the resultant for the analysis of polynomial or rational systems of ordinary differentials to find saddle-node, Hopf, Takens–Bogdanov (codimension two), and codimension three bifurcations. In the process, we find new features of classical dynamical systems including the Goodwin feedback oscillator, the Oregonator, and the Brusselator, among others. In addition, we prove a new theorem (Theorem 4A) relating the resultant and the calculation of Hopf bifurcation curves.