Shift-and-invert iteration for purely imaginary eigenvalues with application to the detection of Hopf Bifurcations in large scale problems

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 of a sequence of large sparse eigenvalue problems. This is not only an expensive operation, but the computation of right-most eigenvalues is often not reliable for the commonly used methods for large sparse matrices. In the literature a method has been proposed that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilises the Kronecker product and involves the solution of matrices of squared dimension, which is impractical for large scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearisation process). 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. The power of our method is tested on three numerical examples, one of which is a discretised PDE with two space dimensions.