Mathematical Sciences: Symmetry Breaking in Spatially-Extended Systems and in Globally-Coupled Oscillator Arrays

数学科学：空间扩展系统和全局耦合振荡器阵列中的对称性破缺

• 批准号: 9410115
• 负责人: Mary Silber
• 金额: $ 1.8万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9410115&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetry; Breaking; Spatially

9410115 Silber The investigator will develop equivariant bifurcation theory for applications to symmetry-breaking bifurcations in hydrodynamic systems and in globally coupled oscillator arrays. The first part of the research focuses on translation symmetry breaking in hydrodynamic systems that are extended in more than one space dimension. Both the situation where translation symmetry is broken spontaneously by an attracting solution of the governing equations, and the situation where it is broken externally by perturbing the equations will be considered. In the former case, the manifestations of the underlying translation symmetry in the possible evolution of instabilities of the symmetry-broken state will be investigated. In the case of external symmetry breaking, the research focuses on the specific physical system of rotating Rayleigh-Benard convection; this is a system for which it has already been established that breaking translation symmetry can have a profound effect on certain instabilities. The second part of the research project consists of a group theoretic/dynamical systems analysis of the $S_n$-equivariant Hopf bifurcation problem. This bifurcation problem is pertinent to the dynamics of n globally-coupled identical limit cycle oscillators. This research is motivated by recent analytic and numerical studies of series arrays of Josephson junctions. Global coupling represents an important limiting case for the coupling of oscillators; this aspect of the research project should contribute to our understanding of oscillator arrays ubiquitous in the modeling of physical, chemical, and biological systems. ***

小行星9410115 研究者将发展应用于流体动力系统和全局耦合振子阵列的等变分岔理论。 第一部分的研究集中在平移对称性破缺的流体动力学系统，扩展到一个以上的空间维。将考虑两种情况，其中平移对称性自发地被破坏的吸引解决方案的控制方程，并在外部扰动方程的情况下，它被打破。在前一种情况下，潜在的平移对称性的表现形式的可能演化的不稳定性的破缺状态将被调查。在外部对称性破缺的情况下，研究的重点是旋转瑞利-贝纳德对流的特定物理系统;这是一个已经建立的系统，破缺平移对称性可以对某些不稳定性产生深远的影响。研究项目的第二部分包括$S_n$-等变Hopf分支问题的群论/动力系统分析。这个分歧问题是有关的n个全球耦合相同的极限环振子的动力学。这项研究的动机是最近的分析和数值研究的约瑟夫森结的串联阵列。全局耦合代表了振荡器耦合的一个重要的限制情况，这方面的研究项目应有助于我们理解振荡器阵列无处不在的物理，化学和生物系统的建模。 ***