Some General Bifurcation Techniques

Abstract

The problem $H(y) = H(x,\lambda ) = \theta $, where $H:\mathbb{R}^{n + 1} \to \mathbb{R}^n $ is considered. Numerical techniques for locating bifurcation points $\bar y$ and for following arcs leading from $\bar y$ are presented. These techniques are valid for primary and secondary bifurcation points, and at multiple bifurcation points, regardless of whether there is a change in the sign of the determinant or the Jacobi matrix $H_x $ at $\bar y$ ; they can also possibly be used when arcs intersect tangentially. The techniques do not require computation of second partial derivatives, although Jacobi matrices are computed using finite differences in neighborhoods of bifurcation points. Details for incorporation into a derivative-free arc-following method, developed in a previous work, are given. Computational results for five test examples appear. Directions for further investigations and improvements are listed. The stepsize control and Jacobi matrix update techniques may be improved for large, sparse problems, when first partial derivatives are easy to compute, or in the absence of bifurcation points.