Mathematical Sciences: Bifurcation and Symmetry

数学科学：分岔和对称性

• 批准号: 9101836
• 负责人: Martin Golubitsky
• 金额: $ 22.5万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-09-01 至 1994-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101836&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Bifurcation; Symmetry

The investigators continue studies of the effect that symmetry has on the solutions of differential equations. Three particular topics are the stability of heteroclinic cycles (which is related to the existence of intermittency and bursting phenomena), the symmetry of chaotic attractors (which may be related to some of the distinctive patterns observed in turbulent fluid flow), and the coupling of gauge and spatial symmetries (such as occurs in the Ginzburg-Landau model for superconductivity). They will also attempt to relate their results to specific experiments. They believe that both the bursting phenomena associated with heteroclinic cycles and the symmetry-increasing bifurcations of chaotic attractors have been observed in the Couette-Taylor experiment; part of their work will be to make these connections more precise. Many models of physical phenomena including topics as diverse as the transition from conduction to convection in fluid dynamics and the ways in which animals walk contain symmetries in their formulation. These symmetries are known to influence the kinds of solutions that are found in such models. Most previous work has focused on the kinds of equilibria and periodic solutions that appear. This project undertakes to broaden the class of solutions that are studied to include such topics as chaos and intermittency in symmetric systems. Recent research and experiments indicate that this is both reasonable and tractable.

研究者们继续研究对称性对微分方程解的影响。三个特别的主题是异斜周期的稳定性（这与间歇性和破裂现象的存在有关），混沌吸引子的对称性（这可能与湍流中观察到的一些独特模式有关），以及规范和空间对称性的耦合（例如超导的金兹堡-朗道模型）。他们还将尝试把他们的结果与具体的实验联系起来。他们认为在Couette-Taylor实验中已经观察到与异斜环相关的爆发现象和混沌吸引子的对称性增加分岔；他们的部分工作将是使这些联系更加精确。许多物理现象的模型，包括流体动力学中从传导到对流的转变以及动物行走的方式等各种各样的主题，在其公式中都包含对称性。已知这些对称性会影响在这些模型中找到的解的种类。以前的大部分工作都集中在各种均衡和周期解上。该项目致力于拓宽解决方案的类别，包括对称系统中的混沌和间歇性等主题。最近的研究和实验表明，这是合理的和易于处理的。