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

9410115西尔伯研究员将发展等变分叉理论，应用于流体动力系统和全局耦合振荡器阵列中的对称破坏分叉。本论文的第一部分主要研究了流体力学系统的平移对称性破缺问题。平移对称性被控制方程的吸引解自发破坏的情况和通过摄动方程从外部破坏的情况都将被考虑。在前一种情况下，我们将研究对称性破缺态在可能的不稳定性演化中的潜在平移对称性的表现。在外对称破缺的情况下，研究的重点是旋转Rayleigh-Benard对流的特定物理系统；对于这个系统，已经确定了破缺平移对称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对$S_n$等变Hopf分支问题的群论/动力系统分析。这个分支问题与n个全局耦合的完全相同的极限环振子的动力学有关。这项研究是由最近对约瑟夫森结串联阵列的解析和数值研究推动的。全局耦合是振荡器耦合的一个重要的极限情况；研究项目的这一方面应该有助于我们理解在物理、化学和生物系统的建模中普遍存在的振荡器阵列。***

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. ***