Introduction to the Numerical Solution of Symmetry- Breaking Bifurcation Problems

Abstract

This paper gives an introduction to systematic methods for obtaining reliable numerical solutions of symmetry-breaking bifurcation problems. The symmetry properties of a nonlinear equation are described by an equivariance property on which all the theory is based. Fixed point spaces are described since they are invariant under the nonlinear operator. The isotypic decomposition is carefully developed as the isotypic components are invariant under the linearisation of the nonlinear operator and this is employed in the efficient detection and computation of bifurcation points. The theory is illustrated throughout with an example and numerical results are presented.