Numerical bifurcation of separable parameterized equations

Shen, Yun-Qiu ; Ypma, Tjalling J.

Electronic transactions on numerical analysis, Tome 34 (2009).

Résumé

Summary: Many applications give rise to separable parameterized equations, which have the form $A(y, \mu )z + b(y, \mu ) = 0$, where z $\in $RN , y $\in $Rn, $\mu \in $Rs, and the (N + n) $\times N$ matrix $A(y, \mu )$ and (N + n) vector $b(y, \mu )$ are C2-Lipschitzian in $(y, \mu ) \in \Omega \subset $Rn $\times $Rs. We present a technique which reduces the original equation to the form f $(y, \mu ) = 0$, where f : $\Omega \rightarrow $Rn is C2-Lipschitzian in $(y, \mu )$. This reduces the dimension of the space within which the bifurcation relation occurs. We derive expressions required to implement methods to solve the reduced equation. Numerical examples illustrate the use of the technique.

Détail

Classification : 65P30, 65H10, 37G10, 34C23
Keywords: separable parameterized equations, singular value decomposition, static bifurcation points, extended systems, Newton's method, LU factorization, curve switching and tracking