Abstract
The thesis studies two problems. The first is a problem with 0(2) symmetry which gives rise to a branch with Z2 symmetry. The Jacobian along the Z2-symmetric branch is always singular, due to the group orbit of solutions. The situation when a pair of complex eigenvalues cross the imaginary axis is considered and canonical coordinates are used to remove the degeneracy from the system so that the standard theory can be applied. This method is used in order to understand the type of solutions that occur. The method cannot be applied to partial differential equations and so a method involving a phase condition is employed which has also been used by Aston, Spence and Wu. Two examples have been considered. The first is a simple example on C3 and the second example is the Kuramoto Sivashinsky equation for which a multiple Hopf bifurcation on a D3 symmetric branch has been studied. A Hopf bifurcation on a so called strange fixed point has also been considered and results compared with the numerical results of Hyman and Nicolaenko. The second problem involves mode interactions between a steady state bifurcation and a Hopf bifurcation in problems with Z2 symmetry. Two different extended systems have been considered and it is shown that the mode interaction corresponds to a bifurcation of these systems. Finally a method for predicting the existence of a tertiary Hopf bifurcation arising from a mode interaction, using only the bifurcation diagrams, has also been considered. Initially a problem with Z2 ⊗ Z2 symmetry, which involves a steady state/steady state mode interaction, has been considered and then similar results are obtained for Hopf/steady state mode interactions.