Bifurcation of solutions of separable parameterized equations into lines

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 $(N+n) imes N$ 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