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Steady-state mode interactions in the presence of O(2)-symmetry. (English) Zbl 0661.58022
This paper studies the multiple bifurcation phenomenon of steady-state mode interactions in the presence of O(2)-symmetry. For such problems the flow on the centre manifold is determined by a vector field in \({\mathbb{R}}^ 4\) or \({\mathbb{C}}^ 2\) that is equivariant under an action of O(2). The action is related to the wave numbers of the unstable modes. The unfolded normal forms for these equivariant bifurcation problems admit primary bifurcations to single-mode solutions, secondary bifurcations to mixed-mode solutions and, in some instances, tertiary bifurcations to traveling and standing waves. The bifurcation behaviour depends crucially on the wave numbers. For small wave numbers, the mixed- mode solutions encounter subordinate saddle-node bifurcations.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
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