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

K Meerbergen, Alastair Spence

Research output: Contribution to journalArticle

19 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)1982-1999
Number of pages18
JournalSIAM Journal On Matrix Analysis and Applications (SIMAX)
Volume31
Issue number4
Early online date6 May 2010
DOIs
Publication statusPublished - 2010

Fingerprint

Inverse Iteration
Large-scale Problems
Hopf Bifurcation
Eigenvalue
Guess
Eigenvalue Problem
Lyapunov Equation
Kronecker Product
Inverse Method
Computing
Iteration Method
Linearization
Two Parameters
Efficient Algorithms
Dynamical system
Numerical Examples

Keywords

  • inverse iteration
  • purely imaginary eigenvalues
  • Lyapunov equation
  • eigenvalue problem

Cite this

@article{e613f4cd886c47a7ad8f94d5df64c8f3,
title = "Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large-scale problems",
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.",
keywords = "inverse iteration, purely imaginary eigenvalues, Lyapunov equation, eigenvalue problem",
author = "K Meerbergen and Alastair Spence",
year = "2010",
doi = "10.1137/080742890",
language = "English",
volume = "31",
pages = "1982--1999",
journal = "SIAM Journal On Matrix Analysis and Applications (SIMAX)",
issn = "0895-4798",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

TY - JOUR

T1 - Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large-scale problems

AU - Meerbergen, K

AU - Spence, Alastair

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - inverse iteration

KW - purely imaginary eigenvalues

KW - Lyapunov equation

KW - eigenvalue problem

UR - http://www.scopus.com/inward/record.url?scp=77956053711&partnerID=8YFLogxK

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

U2 - 10.1137/080742890

DO - 10.1137/080742890

M3 - Article

VL - 31

SP - 1982

EP - 1999

JO - SIAM Journal On Matrix Analysis and Applications (SIMAX)

JF - SIAM Journal On Matrix Analysis and Applications (SIMAX)

SN - 0895-4798

IS - 4

ER -