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Original solutions to examinations of the nonlinear systems course of the University of Oxford, using centre manifolds, bifurcation theory and symbolic dynamics.

A supercritical branch emerges on the parameter side where the original state loses stability. In the cubic Landau amplitude equation $\dot{A}=\mu A-g|A{|}^{2}A$ with $g>0$, the nonzero branch $|A{|}^2=\mu /g$ lies at $\mu >0$ and is stable to radial amplitude perturbations. Continuous phase symmetry still leaves a neutral phase direction. The direction of a branch and its stability should both be checked in a general problem of bifurcation theory.