Use of the parametric representation method in revealing the root structure and hopf bifurcation

Abstract

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =uo +u1x +g(x), whereuo andu1 are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the “stable region” on the parameter plane, where the real parts of the roots off are negative. The D-curve (along which the discriminant off is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.