Mathematical Sciences: Equivariant Bifurcation Theory and Applications

小行星9406144 这是一个研究物理系统中模式的形成和稳定性的建议。由于这种模式往往表现出高度的对称性，在研究中采用的技术旨在充分利用这种对称性。在实际应用中，这些对称性通常不是精确的。因此，拟议工作的一个特别重点是研究系统中小的破坏性"干扰"的影响。因此，所提出的工作是为了使基于安全性的技术适用于现实的系统。该提案的重点是远端壁的影响模式的形式传播波，模式的振荡声激发的球形气泡和形成的三维图案形态。 这是一个研究对称系统中的分岔和斑图形成的建议。这样的系统自然地出现在各种应用中。该建议侧重于发病时的模式形成。在这种情况下，该过程由振幅方程描述。在对称系统中，它们必须遵守假定的对称性。由此产生的方程使人们不仅能够计算可能的模式，而且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激发气泡的振荡模式产生的模式）。此外，为了使技术适用于现实系统中，假设的对称性通常只是近似的建议强调研究小的外部强加的破坏性缺陷。 这样的缺陷可能有重要的定性后果，并可能是负责引入混沌动力学系统，否则将是非混沌。 ***

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. ***