Combinatorial approaches to Hopf bifurcations in systems of interacting elements

TL;DR

This paper introduces combinatorial methods using digraphs and additive compound matrices to determine the presence of Hopf bifurcations in parameterized dynamical systems, especially interaction networks.

Contribution

It develops a novel combinatorial framework involving DSR^[2] graphs to analyze eigenvalues and Hopf bifurcations in systems of interacting elements.

Findings

Cycle structure conditions prevent nonreal eigenvalues with positive real part.

The methods apply to chemical reaction networks and other interaction systems.

Provides necessary conditions for Hopf bifurcations in parameterized systems.

Abstract

We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR^[2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed "interaction networks", of which networks of chemical reactions are a special case.