Following Paths of Symmetry-Breaking Bifurcation Points

摘要

We propose a pseudo-arclength continuation algorithm for computing paths of symmetry-breaking bifurcation points for two-parameter nonlinear elliptic problems. The algorithm consists of an Euler predictor step and a corrector 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 exploiting any (block tridiagonal) structures present. We present a new method for locating some high order singularities. We give numerical results for the steady, axisymmetric flow between rotating coaxial cylinders (Taylor-Couette flow). For infinite cylinders we compute the paths of bifurcation points from Couette flow into Taylor vortices. For finite cylinders we compute the fold curve and path of symmetry-breaking bifurcation points for small aspect ratios, and accurately locate the two codimension-one symmetry-breaking singularities. 41 refs. (ERA citation 14:019151)