ANALYSIS AND COMPUTATION OF SYMMETRY-BREAKING BIFURCATION AND SCALING LAWS USING GROUP THEORETIC METHODS

Group theoretic methods are used to analyse symmetry-breaking bifurcation for nonlinear equations defined on a real Hilbert space. An important result is the decomposition of the Hilbert space into orthogonal isotypic components, since the Jacobian of the nonlinear operator can be decomposed on the isotypic components. This decomposition is exploited in the detection and computation of bifurcation points. Then scaling laws that arise in many problems are considered, and a natural context is developed for the existence of a scaling law based on the symmetry of the problem. The effect of the scaling law on the bifurcation theory is explored. This theory is applied to the gravity wave problem. Also shown is the way in which the theory can extend to boundary value problems, where the natural group equivariance of the equations is destroyed by the boundary conditions.