The detection of a Hopf bifurcation in a large scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues for a sequence of large sparse eigenvalue problems.
We discuss a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilises a certain sum of Kronecker products and involves
the solution of matrices of squared dimension. The proposed method is based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem. The problem is formulated as an inexact inverse iteration method that 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. We show numerical examples from Navies-Stokes equations and show a connection with the implicitly restarted Krylov method and the rational Krylov method.