Communications in Mathematical Sciences

Volume 12 (2014)

Number 6

Combinatorial approaches to Hopf bifurcations in systems of interacting elements

Pages: 1101 – 1133



David Angeli (Department of Electrical and Electronic Engineering, Imperial College London, United Kingdom)

Murad Banaji (Department of Mathematics, University of Portsmouth, Portsmouth, Hampshire, United Kingdom)

Casian Pantea (Department of Mathematics, West Virginia University, Morgantown, West Virginia, U.S.A.)


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.


Hopf bifurcation, compound matrices, interaction networks

2010 Mathematics Subject Classification

05C90, 15A18, 15A75, 34C23, 37C27

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