Normal forms and bifurcations of vector fields

矢量场的范式和分岔

• 批准号: 9420-2010
• 负责人: Rousseau, Christiane
• 金额: $ 2.19万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=521275
• 关键词: Normal; forms; bifurcations; vector; fields

The whole project is centered on fundamental problems in analytic dynamics, especially low dimensional dynamical systems depending on parameters: 1-dimensional diffeomorphisms, planar vector fields, either real vector fields or complex foliations, and linear differential systems. Dynamical systems are systems which depend on the time. It is very difficult to find explicit solutions of a given system. Poincaré was the founder of geometric methods to study and understand the solutions of dynamical systems. These methods allow drawing qualitative conclusions, even without having explicit formulas for the solutions. In order to better understand dynamical systems, it is important to attack them at their weak points: the singularities in phase space and the bifurcation values in parameter space. The research project is centered precisely on these matters:

整个项目集中在分析动力学的基本问题上，特别是依赖于参数的低维动力系统：一维微分同态，平面矢量场，实矢量场或复叶状，以及线性微分系统。动力系统是依赖于时间的系统。给定系统的显式解是非常困难的。庞加莱是研究和理解动力系统解的几何方法的创始人。这些方法可以得出定性的结论，即使没有解的明确公式。为了更好地理解动力系统，重要的是要攻击它们的弱点：相空间的奇异性和参数空间的分岔值。该研究项目正是围绕这些问题展开的：