Mathematical Sciences: Dynamics and Symmetry

数学科学：动力学和对称性

基本信息

批准号：
9403624
负责人：
Martin Golubitsky
金额：
$ 22.78万
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：
1994
资助国家：
美国
起止时间：
1994-07-01 至 1997-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403624&HistoricalAwards=false
关键词：
Mathematical Sciences Dynamics Symmetry

项目摘要

9403624 Golubitsky In this project we study both the theory and application of symmetric dynamical systems. In many applications, symmetry is introduced either by the supposition of a regular geometry or by the interchangability of parts in a complex system. It has been well established that the existence of regular patterns in experiments and in computer simulations are often due to symmetry. Only recently has it been shown that the existence of intermittency in the dynamics and patterns in the time-average of solutions or experiments can also be forced by symmetry. It is the purpose of this proposal to investigate these and related phenomena. In particular we will study how symmetry and chaotic dynamics intertwine in dynamical systems and how symmetry affects the types of solutions found in systems of coupled cells. In this project we will study iteration of symmetric maps (symmetric chaos), the investigation of coupled systems of ODEs where each subsystem or cell has its own internal symmetry, the stability of heteroclinic cycles (which occur naturally in symmetric systems of differential equations), and the effect of forced symmetry breaking (where perturbations of a symmetric equation to one with less symmetry are considered). Applications to patterns on average in systems of PDEs and experiments (particularly in the Taylor-Couette system and the Faraday surface wave experiment) will be considered as will the existence of intermittency (heteroclinic cycles) in an experiment modeling convection in a porous media. In addition, the dynamics of a number of differential equations with symmetry will be studied by direct computer simulation and the results compared with current theory.

9403624 Golubitsky在这个项目中，我们研究了对称动力学系统的理论和应用。在许多应用中，对称性是通过常规几何形状的假设或复杂系统中零件的互换性引入的。已经很好地确定，实验和计算机模拟中的常规模式通常是由于对称性引起的。直到最近，还可以通过对称性强迫解决方案或实验时间平均的动力学和模式中的间歇性存在。该提案的目的是研究这些现象和相关现象。特别是我们将研究对称和混乱动力学如何在动力学系统中交织在一起，以及对称性如何影响耦合细胞系统中发现的溶液类型。在这个项目中，我们将研究对称地图（对称混乱）的迭代，对每个子系统或细胞具有其自身内部对称性的ODES耦合系统的研究，杂节循环的稳定性（它们在微分方程式的对称系统中自然发生），以及强制对称性断裂的效果（其中考虑了对称方程对对称性较少的对称方程的扰动）。在PDE和实验系统中的平均应用（尤其是在Taylor-Couette系统和Faraday Surface Wave实验中），将被视为在多孔培养基中的实验建模对流中的间歇性（杂节循环）的存在。另外，将通过直接计算机模拟和与当前理论相比的结果研究许多具有对称性的微分方程的动力学。