Mathematical Sciences: Equivariant Bifurcation Theory and Applications

9406144诺布洛克这是一项研究发生在物理系统中的图案的形成和稳定性特性的提议。因为这种图案通常表现出高度的对称性，所以研究中使用的技术被设计为充分利用这种对称性。在实际应用中，这些对称性通常是不精确的。因此，拟议工作的一个特别重点是研究系统中微小的对称破坏“缺陷”的影响。因此，这项拟议的工作旨在使基于对称的技术适用于现实系统。该提议侧重于远端壁对传播波形式的图案、声激励球形气泡中的振荡图案以及形态发生中的三维图案的形成的影响。这是一项研究对称系统中的分叉和图案形成的建议。这样的系统自然会出现在各种应用中。该提案的重点是在接近发病时形成模式。在这种情况下，这个过程可以用振幅方程来描述。在对称系统中，这些必须尊重假定的对称性。由此得到的方程不仅可以计算可能的模式，还可以计算它们的相对稳定性。该提案的重点是将这种方法扩展到三维模式(例如由图灵不稳定性或声学激发的气泡的振荡模式产生的模式)。此外，为了使该技术适用于假设的对称性通常是近似的现实系统，该方案强调研究外部施加的小的对称破缺缺陷。这样的缺陷可能会有重要的质量后果，并可能导致将混沌动力学引入原本不是混沌的系统。***

9406144 Knobloch This is a proposal to study the formation and stability properties of patterns occurring in physical systems. Because such patterns often exhibit a high degree of symmetry the techniques employed in the study are designed to take full advantage of such symmetries. In practical applications these symmetries are usually not exact. Therefore a particular emphasis of the proposed work is to study the effects of small symmetry-breaking ``imperfections'' in the system. The proposed work is thus designed to make symmetry-based techniques applicable to realistic systems. The proposal focuses on the effects of distant endwalls on patterns in the form of propagating waves, patterns of oscillations in acoustically excited spherical bubbles and the formation of three-dimensional patterns in morphogenesis. This is a proposal to study bifurcations and pattern formation in systems with symmetry. Such systems occur naturally in a variety of applications. The proposal focuses on pattern formation near onset. In this regime the process is described by amplitude equations. In symmetric systems these must respect the assumed symmetries. The resulting equations enable one to compute not only the possible patterns but also their relative stability. The proposal focuses on extending this approach to three-dimensional patterns (such as those created by the Turing instability or the modes of oscillation of an acoustically excited bubble). In addition in order to make the technique applicable to realistic systems in which the assumed symmetries are usually only approximate the proposal emphasizes the study of small externally imposed symmetry-breaking imperfections. Such imperfections may have important qualitative consequences and may be responsible for the introduction of chaotic dynamics into systems that would otherwise be nonchaotic. ***