Numerical Study of Bifurcations by Analytic Continuation of a Function Defined by a Power Series

A novel computational approach to the investigation of bifurcations, relying on the use of power series in the bifurcation parameter for a particular solution branch, is presented. The first part of the paper is devoted to the description of a series summation technique based on the assumption that the given series is the local representation of a function algebraic in the independent variable. The procedure leads to a special type of Hermite-Pade approximant. Although no mathematical analysis is presented, the numerical evidence suggests that the error decays faster than exponentially with the number of terms of the series used. The procedure's chief merit is its ability to reveal solution branches of the underlying problem in addition to the one represented by the original series. In the final part of the paper, an algorithm is described for numerically generating the required power series where standard perturbation methods are inadequate. Thus, it is shown how path-following techniques may be combined with the basic procedure for series summation to provide a powerful tool well suited to the numerical analysis of bifurcations in nonlinear problems. Numerical results are presented for a variety of applications, including examples from fluid mechanics.