Separable parameterized equations, rank deficiency, Golub-Pereyra variable projection method, bordered matrix, singular value decomposition, Newton's method
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  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.
Electronic Journal of Differential Equations
Required Publisher's Statement
Published by the Department of Mathematics, Texas State University
Ypma, Tjalling and Shen, Yun-Qiu, "Bifurcation of Solutions of Separable Parameterized Equations into Lines" (2010). Mathematics. 95.
Subjects - Topical (LCSH)
Seperable algebras; Bifurcation theory
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