# Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large-scale problems

### Reference:

Meerbergen, K. and Spence, A., 2010. Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large-scale problems. *SIAM Journal On Matrix Analysis and Applications (SIMAX)*, 31 (4), pp. 1982-1999.

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### Official URL:

http://dx.doi.org/10.1137/080742890

### Abstract

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, Gueron, and Harris-Warrick [SIAM J. Numer. Anal., 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 utilizes 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 linearization 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 four numerical examples.

### Details

Item Type | Articles |

Creators | Meerbergen, K.and Spence, A. |

DOI | 10.1137/080742890 |

Uncontrolled Keywords | inverse iteration, purely imaginary eigenvalues, lyapunov equation, eigenvalue problem |

Departments | Faculty of Science > Mathematical Sciences |

Refereed | Yes |

Status | Published |

ID Code | 19893 |

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