Numerical Methods for a Class of Finite Dimensional Bifurcation Problems

For an equation $H(y,t) = 0$, where $H:D \subset R^{n + 1} \to R^n $, let $p:J \subset R^1 \to R^n $ be a primary solution on which a simple bifurcation point $p^ * = p(t^ * )$ with rank $H_y = (p^ * ,t^ * ) = n - 1$ has been detected and a secondary solution is branching off. An iterative process is presented which starts at a point $p^0 = p(t_0 )$ near $p^ * $ and converges to a point on the secondary curve. It is similar in form to methods proposed by H. B. Kelley and others but has considerably lower computational complexity. The process represents a chord iteration with singular iteration matrix and its convergence is derived from a general result for such singular chord iterations. Computational details for the implementation of the method and an informal program are given. Finally, some comments about extensions to the case rank $H_y (p^ * ,t^ * ) < n - 1$ are made.