**UoP maths seminar (2013-14)**

**Combinatorial approaches to Hopf bifurcation in systems of interacting elements.**

*Speaker*: Murad Banaji

*Venue*: Wednesday 30th October 2013, 2pm, LG2.04a (refreshments provided)

*Abstract*: A necessary condition for Hopf bifurcation in a family of vector fields is the passage of a pair of nonreal eigenvalues of their Jacobian matrices through the imaginary axis. However, in some cases, structural features of a dynamical system forbid this occurring and Hopf bifurcation can be ruled out. For example all nonreal eigenvalues may be forced to have nonnegative real part, or nonzero real part. I'll present some recent work in this direction providing necessary conditions for Hopf bifurcation. The results are applicable to arbitrary dynamical systems, but prove most useful in the study of networks of interacting elements, with chemical reaction networks as a special case. The techniques depend on the spectral properties of compound matrices and form part of a suite of results relating "structure" and spectrum in matrices.