Symmetry Adapted Block Diagonalisation in Equivariant Steady State Bifurcation Problems and its Numerical Application

Symmetry Adapted Block Diagonalisation in Equivariant Steady State Bifurcation Problems and its Numerical Application

Peter Stork,Bodo Werner

Peter Stork Bodo Werner

Institut fur Angewandte Mathematik der Universitat Hamburg Bundesstr.55,D-2000 Hamburg 13 Federal Republic of Germany,Institut fur Angewandte Mathematik der Universitat Hamburg Bundesstr.55,D-2000 Hamburg 13 Federal Republic of Germany

(Institut for Angewandte Mathematkcder Universitat HamburgBundesstr. 55, D-2000 Hamburg 13Federal Republic of Germany

摘要 Considering steady state bifurcation of single parameter r-equivariant systemswhere Γ is a compact Lie group acting on X via a faithful orthogonal representation, we simultaneously block-diagonalise all jacobians of g along a solution path with respect to symmetry adapted subspaces and apply bifurcation groups to reduce symmetry breaking bifurcation points to simple bifurcation points. In generical situations this algorithm allows for numerically stable use of continuation methods and steady state bifurcation point detection and computation. The large scale decrease of computing efforts is demonstrated at computing the equilibria of a hexagonal lattice dome with 21 degrees of freedom, we present a great variety of solution arcs with symmetry from full D6 down to Z2 symmetry.

Abstract：Considering steady state bifurcation of single parameter r-equivariant systemswhere Γ is a compact Lie group acting on X via a faithful orthogonal representation, we simultaneously block-diagonalise all jacobians of g along a solution path with respect to symmetry adapted subspaces and apply bifurcation groups to reduce symmetry breaking bifurcation points to simple bifurcation points. In generical situations this algorithm allows for numerically stable use of continuation methods and steady state bifurcation point detection and computation. The large scale decrease of computing efforts is demonstrated at computing the equilibria of a hexagonal lattice dome with 21 degrees of freedom, we present a great variety of solution arcs with symmetry from full D6 down to Z2 symmetry.