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Article
Bifurcation of Solutions of Separable Parameterized Equations into Lines
Electronic Journal of Differential Equations
  • Tjalling Ypma, Western Washington University
  • Yun-Qiu Shen, Western Washington University
Document Type
Conference Proceeding
Publication Date
1-1-2010
Keywords
  • Separable parameterized equations,
  • rank deficiency,
  • Golub-Pereyra variable projection method,
  • bordered matrix,
  • singular value decomposition,
  • Newton's method
Disciplines
Abstract

Many applications give rise to separable parameterized equations of the form A(y,µ)z + b(y, µ) = 0, where y Rn, z RN and the parameter µ R; here A(y,µ) is an (N + n) × N matrix and b(y, µ) RN +n. Under the assumption that A(y, µ) has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form f (y, µ) = 0. In this paper we extend that method to the case that A(y, µ) has rank deficiency one at the bifurcation point. At such a point the solution curve (y, µ, z) branches into infinitely many additional solutions,which form a straight line. A numerical method for reducing the problem to a smaller space and locating such a bifurcation point is given. Applications to equilibrium solutions of nonlinear ordinary equations and solutions of discretized partial differential equations are provided.

Required Publisher's Statement

Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations

This is an open access journal.

Comments

Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations

This is an open access journal.

Subjects - Topical (LCSH)
Seperable algebras; Bifurcation theory
Genre/Form
conference proceedings
Type
Text
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Language
English
Format
application/pdf
Citation Information
Tjalling Ypma and Yun-Qiu Shen. "Bifurcation of Solutions of Separable Parameterized Equations into Lines" Electronic Journal of Differential Equations Vol. 19 (2010) p. 254 - 255
Available at: http://works.bepress.com/tjalling_ypma/11/