Mathematical Sciences: Symmetry-Breaking Bifurcations and Pattern Formation

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9404266 SILBER The goal of the project is to develop equivariant bifurcation theory for applications to pattern formation and for applications to globally coupled oscillator arrays. The proposed research will be carried out within the framework of finite-dimensional center manifold normal forms of bifurcation problems. The symmetries of these bifurcation problems will be exploited in their analysis and in the interpretation of the results. Special attention will also be given to developing geometric methods that exploit integrable Hamiltonian limits of the bifurcation problems. Symmetries play a central role in the evolution of pattern-forming instabilities in hydrodynamic systems. The primary goal of the proposed research is to develop mathematical methods that exploit their presence. The research on pattern formation will aid the interpretation of large-scale numerical studies of three-dimensional magnetoconvection, and laboratory studies of electroconvection in nematic liquid crystals. The research on globally coupled oscillators will contribute to our understanding of the dynamics of series arrays of Josephson junctions.