#### Document Type

Conference Proceeding

#### Publication Date

2010

#### Abstract

Many applications give rise to separable parameterized equations of the form *A*(*y,µ*)*z *+ *b*(*y, **µ*) = 0, where *y **∈ *R^{n}, *z **∈ *R^{N }and the parameter *µ **∈ *R; here *A*(*y,µ*) is an (*N *+ *n*) *× **N *matrix and *b*(*y, **µ*) *∈ *R^{N +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.

#### Publication Title

Electronic Journal of Differential Equations

#### Volume

19

#### First Page

254

#### Last Page

255

#### Required Publisher's Statement

Published by the Department of Mathematics, Texas State University

#### Recommended Citation

Ypma, Tjalling J. and Shen, Yun-Qiu, "Bifurcation of Solutions of Separable Parameterized Equations into Lines" (2010). *Mathematics.* Paper 95.

http://cedar.wwu.edu/math_facpubs/95

## Comments

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

This is an open access journal.