Mathematical Sciences: Non-Linear Dynamics of Cylindrically Symmetric Systems

非平凡对称群的存在可以深刻地 影响一个物理系统的动力学。 特别是， 最近认识到，不稳定性或分叉 在这样的系统中观察到的典型地表现出高度非通用性 当对称性破坏发生时。 这个项目是为了研究 非线性数学模型的详细调查， 具体的物理系统。 特别强调的是 对称群0（2）和S0（2），因为它们与 的应用，以及由多个 模式相互作用的不稳定性。 要应用的技术 等变分歧理论、中心流形 简化和范式分析。 目标是获得一个 理解对称性破缺的过渡， 混沌状态以及对称性对 过渡到混沌。 对于特定的流体系统， 对称性人们期望得到的结果可以在 数值模拟和实验室实验。

The presence of a nontrivial symmetry group can profoundly affect the dynamics of a physical system. In particular, it has been recently appreciated that the instabilities or bifurcations observed in such systems typically exhibit highly nongeneric features when symmetry breaking occurs. This project is to study the nonlinear mathematical models with detailed investigation of specific physical systems. Particular emphasis is placed on the symmetry groups 0(2) and S0(2) because of their relevance to applications, and on the dynamics arising from multiple instabilities of mode interactions. The technique to be applied are those of equivariant bifurcation theory, center manifold reduction, and normal form analysis. The goal is to obtain an understanding of symmetry breaking transitions between pre- chaotic states as well as the impact of symmetry on the transition to chaos. For specific fluid systems with circular symmetry one expects to obtain results which may be tested in numerical and laboratory experiments.