FOLLOWING PATHS OF SYMMETRY-BREAKING BIFURCATION POINTS

Abstract

We propose a pseudo-arclength continuation algorithm for computing paths of Z 2-symmetry-breaking bifurcation points for two-parameter nonlinear elliptic problems. The algorithm consists of an Euler predictor step and a solution step composed of a sequence of Newton iterations. This work generalizes the algorithm of Werner and Spence for locating a one-parameter symmetry-breaking bifurcation point by using the approach of Keller and Fier for following a (two-parameter) path of limit points (a "fold"). By repeated use of the bordering algorithm, we solve linear systems whose matrix is the "symmetric" Jacobian or "antisymmetric" Jacobian, thus fully exploiting any (block tridiagonal) structures present. We give numerical results for the steady, axisymmetric flow between rotating coaxial cylinders (Taylor–Couette flow). For finite cylinders, we compute the fold curve and path of symmetry-breaking bifurcation points for small aspect ratios, and illustrate a new method to accurately locale the two Z 2-symmetric codimension one singularities. For infinite cylinders, we show the projections on the (aspect ratio, Reynolds number) plane of the folds and bifurcation point paths in the neighborhood of the two-cell/four-cell neutral curve crossing. We numerically verify a conjecture of Meyer–Spasche and Wagner concerning the connection of two neutral-curve crossings by a path of secondary subharmonic bifurcations.