Eighth Mississippi State  UAB Conference on Differential Equations
and Computational Simulations.
Electron. J. Diff. Eqns., Conference 19 (2010), pp. 245255.
Bifurcation of solutions of separable parameterized
equations into lines
YunQiu Shen, Tjalling J. Ypma
Abstract:
Many applications give rise to separable parameterized equations
of the form
, where
,
and the parameter
;
here
is an
matrix and
.
Under the assumption that
has full rank we showed in [21] that
bifurcation points can be located by solving a reduced equation
of the form
.
In this paper we extend that method
to the case that
has rank deficiency one at the
bifurcation point. At such a point the solution curve
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;
GolubPereyra variable projection method; bordered matrix;
singular value decomposition; Newton's method.
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YunQiu Shen
Department of Mathematics,
Western Washington University
Bellingham, WA 982259063, USA
email: yunqiu.shen@wwu.edu


Tjalling J. Ypma
Department of Mathematics,
Western Washington University
Bellingham, WA 982259063, USA
email: tjalling.ypma@wwu.edu

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