Mass action systems: two criteria for Hopf bifurcation without Hurwitz

TL;DR

This paper introduces two new criteria for predicting oscillations in mass action systems that avoid complex determinant calculations, utilizing linear algebra concepts like D-stability and P-matrices.

Contribution

It presents two novel, complementary criteria for Hopf bifurcation in mass action systems based on linear algebra, bypassing Hurwitz determinant computations.

Findings

Criteria successfully predict oscillations in example systems.

One criterion relates to positive feedback, the other to negative feedback.

For fully-open networks, Hopf bifurcation capacity equals eigenvalue capacity.

Abstract

We state two sufficient criteria for periodic oscillations in mass action systems. Neither criterion requires a computation of the Hurwitz determinants. Instead, both criteria exploit the linear algebra concepts of $D$-stability and $P$-matrices. The criteria are complementary: the first is based on a stable matrix that is not a $P^-$ matrix, while the second is based on a $P^-$ matrix that is not stable. In analogy, a qualitatively different interpretation follows: the first criterion relates to positive feedback in the network, while the second concerns negative feedback. We present examples that showcase the applicability of both criteria. As a final independent remark, we prove that for the special case of fully-open networks, the capacity for Hopf bifurcation is just equivalent to the capacity for a steady-state with a complex pair of eigenvalues with positive-real part.