Mathematical Sciences: Symmetry-Breaking Bifurcations and Pattern Formation

小行星9404266 该项目的目标是发展等变分岔理论的应用模式的形成和全球耦合振荡器阵列的应用。所提出的研究将在有限维中心流形规范形的分歧问题的框架内进行。这些分歧问题的对称性将被利用在他们的分析和结果的解释。也将特别注意开发几何方法，利用可积哈密顿极限的分歧问题。 对称性在流体动力学系统的模式形成不稳定性的演化中起着核心作用。拟议研究的主要目标是开发利用其存在的数学方法。图案形成的研究将有助于解释大规模的三维磁对流的数值研究，和实验室研究的电对流的液晶。对全局耦合振荡器的研究将有助于我们对约瑟夫森结串联阵列动力学的理解。

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.