Excluding purely imaginary eigenvalues from reaction networks

The spectrum of the Jacobian matrix G of a dynamical system plays a central role in the stability and bifurcation analysis of equilibria. In particular, a complex pair of purely imaginary eigenvalues of G is a necessary condition for Hopf bifurcation and consequent oscillatory behavior.

Reaction networks give rise to parametric systems of equations, and thus to parametric families of Jacobian matrices. In this talk I share our work-in-progress about characterizing the networks for which there is no choice of parameters such that G possesses purely imaginary eigenvalues.