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Rank deficiencies and bifurcation into affine subspaces for separable parameterized equations


Authors: Yun-Qiu Shen and Tjalling J. Ypma
Journal: Math. Comp. 85 (2016), 271-293
MSC (2010): Primary 65P30, 65H10; Secondary 37G10, 34C23
DOI: https://doi.org/10.1090/mcom/2968
Published electronically: June 2, 2015
MathSciNet review: 3404450
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Abstract: Many applications lead to separable parameterized equations of the form $ F(y,\mu ,z) \equiv A(y, \mu )z+b(y, \mu )=0$, where $ y \in \mathbb{R}^n$, $ z \in \mathbb{R}^N$, $ A(y, \mu ) \in \mathbb{R}^{(N+n) \times N}$, $ b(y, \mu ) \in \mathbb{R}^{N+n}$ and $ \mu \in \mathbb{R}$ is a parameter. Typically $ N >>n$. Suppose bifurcation occurs at a solution point $ (y^*,\mu ^*,z^*)$ of this equation. If $ A(y^*, \mu ^*)$ is rank deficient, then the linear component $ z$ bifurcates into an affine subspace at this point. We show how to compute such a point $ (y,\mu ,z)$ by reducing the original system to a smaller separable system, while preserving the bifurcation, the rank deficiencies and a non-degeneracy condition. A numerical algorithm for solving the reduced system and examples illustrating the method are provided.


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Additional Information

Yun-Qiu Shen
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
Email: yunqiu.shen@wwu.edu

Tjalling J. Ypma
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
Email: tjalling.ypma@wwu.edu

DOI: https://doi.org/10.1090/mcom/2968
Keywords: Separable parameterized equations, bifurcation into affine subspace, rank deficiencies, non-degeneracy condition, bordered matrix, singular value decomposition, LU factorization, Newton's method
Received by editor(s): May 7, 2012
Received by editor(s) in revised form: April 19, 2014
Published electronically: June 2, 2015
Article copyright: © Copyright 2015 American Mathematical Society