Rank deficiencies and bifurcation into affine subspaces for separable parameterized equations

Abstract:

Many applications lead to separable parameterized equations of the form $F(y,\mu ,z) \equiv A(y, \mu )z+b(y, \mu )=0$, where $y \in \mathbb R^n$, $z \in \mathbb R^N$, $A(y, \mu ) \in \mathbb {R}^{(N+n) \times N}$, $b(y, \mu ) \in \mathbb {R}^{N+n}$ and $\mu \in \mathbb R$ is a parameter. Typically $N >>n$. Suppose bifurcation occurs at a solution point $(y^*,\mu ^*,z^*)$ of this equation. If $A(y^*, \mu ^*)$ is rank deficient, then the linear component $z$ bifurcates into an affine subspace at this point. We show how to compute such a point $(y,\mu ,z)$ by reducing the original system to a smaller separable system, while preserving the bifurcation, the rank deficiencies and a non-degeneracy condition. A numerical algorithm for solving the reduced system and examples illustrating the method are provided.