Computation of a simple bifurcation point using one singular value decomposition nearby
A direct method to locate simple bifurcation points of a system of nonlinear parameter dependent equations \(G(x,t)=0\), \(G:D_G\subset \mathbb{R}^n\times \mathbb{R}\to\mathbb{R}^n\) is developed. This method requires a good choice of an initial point \((x^{(0)},t^{(0)})\) only. Auxiliary vectors are defined then automatically on the base of the usage of a singular value decomposition of the Jacobian matrix \(G'(x^{(0)}, t^{(0)})\) near the simple bifurcation point. The suggested approach preserves the advantages of the method of \textit{V. Janovský} [Computing 43, No. 1, 27-36 (1989; Zbl 0695.65033)] by using one LU factorization in each iteration, but also improves it by providing a set of auxiliary vectors which always produces a well-conditioned bordered matrix. The quadratic convergence for the extended system proposed by \textit{G. Pönisch} [Computing 35, 277-294 (1985; Zbl 0569.65041)] is guaranteed.
scientific article; zbMATH DE number 1045486
scientific article