Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 19 (2010), pp. 245-255.

Bifurcation of solutions of separable parameterized equations into lines

Yun-Qiu Shen, Tjalling J. Ypma

Abstract:
Many applications give rise to separable parameterized equations of the form $A(y, \mu)z+b(y, \mu)=0$, where $y \in \mathbb{R}^n$, $z \in \mathbb{R}^N$ and the parameter $\mu \in \mathbb{R}$; here $A(y, \mu)$ is an $(N+n) \times N$ matrix and $b(y, \mu) \in \mathbb{R}^{N+n}$. Under the assumption that $A(y,\mu)$ has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form $f(y, \mu)=0$. In this paper we extend that method to the case that $A(y,\mu)$ has rank deficiency one at the bifurcation point. At such a point the solution curve $(y,\mu,z)$ 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.

Published September 25, 2010.
Math Subject Classifications: 65P30, 65H10, 34C23, 37G10.
Key Words: Separable parameterized equations; bifurcation; rank deficiency; Golub-Pereyra variable projection method; bordered matrix; singular value decomposition; Newton's method.

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Yun-Qiu Shen
Department of Mathematics, Western Washington University
Bellingham, WA 98225-9063, USA
email: yunqiu.shen@wwu.edu
Tjalling J. Ypma
Department of Mathematics, Western Washington University
Bellingham, WA 98225-9063, USA
email: tjalling.ypma@wwu.edu

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