38 research outputs found
Combinatorial approaches to Hopf bifurcations in systems of interacting elements
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.Comment: A number of minor errors and typos corrected, and some results
slightly improve