SIAM Journal on Matrix Analysis and Applications vol:31 issue:4 pages:1982-1999
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. Guckenheimer et. al. (SINUM, 34, (1997) pp. 1-21) proposed 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, 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.