Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow

Howard C. Elman, Karl Meerbergen, Alastair Spence, Minghao Wu

Research output: Contribution to journalArticle

Abstract

The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [K. Meerbergen and A. Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982--1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.


Read More: http://epubs.siam.org/doi/abs/10.1137/110827600
LanguageEnglish
PagesA1584-A1606
Number of pages23
JournalSIAM Journal on Scientific Computing
Volume34
Issue number3
Early online date11 Jun 2012
DOIs
StatusPublished - 2012

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Inverse Iteration
Lyapunov Equation
Hopf bifurcation
Incompressible flow
Fluid dynamics
Incompressible Flow
Hopf Bifurcation
Lyapunov
Identification (control systems)
Dynamical systems
Eigenvalue
Iteration
Eigenvalue Distribution
Generalized Eigenvalue Problem
Inverse Method
Iterative Solution
Iteration Method
Fluid Dynamics
Test Problems
Dynamical system

Cite this

Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow. / Elman, Howard C.; Meerbergen, Karl; Spence, Alastair; Wu, Minghao.

In: SIAM Journal on Scientific Computing, Vol. 34, No. 3, 2012, p. A1584-A1606.

Research output: Contribution to journalArticle

Elman, Howard C. ; Meerbergen, Karl ; Spence, Alastair ; Wu, Minghao. / Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow. In: SIAM Journal on Scientific Computing. 2012 ; Vol. 34, No. 3. pp. A1584-A1606
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