Mathematical Sciences: Equivariant Bifurcation Theory and Applications

数学科学：等变分岔理论及其应用

• 批准号: 9406144
• 负责人: Edgar Knobloch
• 金额: $ 9万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9406144&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Equivariant; Bifurcation; Theory

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. ***

[406144] Knobloch这是一个研究物理系统中图案的形成和稳定性的建议。由于这种模式通常表现出高度的对称性，因此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，所提出的工作的一个特别重点是研究系统中小的对称性破坏“缺陷”的影响。因此，提出的工作旨在使基于对称性的技术适用于现实系统。该提案侧重于远端壁对传播波形式的模式的影响，声激发球形气泡中的振荡模式以及形态发生中三维模式的形成。这是研究对称系统的分岔和模式形成的一个建议。这样的系统自然出现在各种应用程序中。该建议的重点是在发病时形成模式。在这种情况下，过程用振幅方程来描述。在对称系统中，它们必须尊重假定的对称性。由此得到的方程不仅可以计算出可能的模式，还可以计算出它们的相对稳定性。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气泡的振荡模式产生的模式）。此外，为了使该技术适用于实际系统，其中假设的对称性通常只是近似的，建议强调研究小的外部施加的对称性破坏缺陷。这样的不完美可能会产生重要的定性后果，并可能导致将混沌动力学引入系统，否则系统将是非混沌的。＊＊＊