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Title: The solution of two-parameter eigenvalue problems arising from the determination of Hopf bifurcations of dynamical systems
Authors: Meerbergen, Karl
Spence, Alastair
Elman, Howard
Wu, Minghao #
Issue Date: Apr-2012
Conference: MOPNET 6 location:Bath, UK date:2-3 April 2012
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 for a sequence of large sparse eigenvalue problems.
We discuss 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. The proposed method is based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem. 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. We show numerical examples from Navies-Stokes equations and show a connection with the implicitly restarted Krylov method and the rational Krylov method.
Publication status: published
KU Leuven publication type: IMa
Appears in Collections:Numerical Analysis and Applied Mathematics Section
# (joint) last author

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