The detection of a Hopf bifurcation in a large scale dynamical system that depends on a physical parameter often consists of attempting to compute pure imaginary eigenvalues of large sparse matrices.

In this talk we describe a new method that utilises a certain sum of Kronecker products but in practice requires the solution of a sequence of Lyapunov equations with low rank right hand sides. It is this last fact that makes the method feasible for large systems.