Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electron. J. Diff. Eqns., Conference 19 (2010), pp. 245-255.
Title: Bifurcation of solutions of separable parameterized
equations into lines
Authors: Yun-Qiu Shen (Western Washington Univ., Bellingham, WA, USA)
Tjalling J. Ypma (Western Washington Univ., Bellingham, WA, USA)
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.