Lyapunov Inverse Iteration for Identifying Hopf Bifurcations in Models of Incompressible Flow
Lyapunov Inverse Iteration for Identifying Hopf Bifurcations in Models of Incompressible Flow
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Date
2011-03-07
Authors
Elman, Howard C.
Meerbergen, Karl
Spence, Alastair
Wu, Minghao
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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 [Meerbergen
and 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 and 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.