Singularities of dynamical systems and their unfoldings

动力系统的奇点及其展开

• 批准号: RGPIN-2016-03862
• 负责人: Rousseau, Christiane
• 金额: $ 2.29万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614678
• 关键词: Singularities; dynamical; systems; their; unfoldings

The research project is focused on the study of singularities of dynamical systems, both ordinary differential equations (ODE) and difference equations, and how these singularities organize the dynamics. In the case of ODE, the project comprises two parts: 1) a first part on the bifurcation theory of planar vector fields; 2) a second part on the study of unfoldings of singularities of ODE of low codimension in real or complex finite-dimensional space. In the case of difference equations, the research project is focused on the study of unfoldings of finite codimension resonant singularities in dimension 1 and 2.Bifurcation theory is a tool to discover the phase portraits of planar vector fields, through embedding a single vector field in a family of vector fields depending on a finite number of parameters and analyzing the bifurcations in the family. The bifurcations of highest codimension in the family are the most important: they organize the bifurcation diagram and the dynamics. The main motivation for this part of the project is to advance the progress on a solution on the finiteness part of Hilbert’s 16th problem through the Dumortier-Roussarie-Rousseau program for proving the existence of a uniform bound for the number of limit cycles of a polynomial quadratic vector field. The program consists in proving that 121 graphics have finite cyclicity inside the family of quadratic vector fields. Significant progress on the program is expected from new techniques developed recently, and more are in the process of being developed. I am also interested in applying bifurcation tools to some predator-prey models in mathematical biology, mainly in students’ theses. The core part of the project deals with the study of equilibrium positions of analytic dynamical systems (either ODE, linear differential equations or difference equations) depending on parameters, more precisely with the problem of analytic classification of germs of families of dynamical systems unfolding a dynamical system with a singularity of finite codimension: when are two germs of analytic families of dynamical systems equivalent modulo an analytic change of parameters, and possibly a reparameterization of time? There are many geometric obstructions to such equivalences. The project is focused on 1) identifying a complete modulus of analytic classification for such a germ of family, and 2) identifying the moduli space for such germs. One motivation is to understand the geometric meaning of the analytic obstructions to equivalence and to describe these obstructions. The types of singularities to be studied are parabolic points of 1-dimensional germs of diffeomorphisms, saddle-nodes of germs of 2-dimensional diffeomorphisms, resonant saddles or saddle-nodes of 2-dimensional complex foliations and resonant singularities of linear differential systems.

该研究项目的重点是研究动力系统的奇点，包括常微分方程（ODE）和差分方程，以及这些奇点如何组织动力学。在常微分方程的情况下，该项目包括两个部分：1）第一部分的分歧理论的平面向量场; 2）第二部分的研究展开的奇点的常微分方程的低余维在真实的或复杂的有限维空间。在差分方程的情形下，研究了1维和2维有限余维共振奇点的开折问题。分歧理论是一种发现平面向量场相图的工具，通过将单个向量场嵌入到依赖于有限个参数的向量场族中，并分析族中的分歧。族中最高余维的分支是最重要的：它们组织了分支图和动力学。本项目这一部分的主要动机是通过证明多项式二次向量场极限环个数存在一致界的Dumortier-Eisarie-Rousseau程序，推进希尔伯特第16问题有限性部分的解决方案。该程序包括证明121图形有有限的循环内家庭的二次向量场。最近开发的新技术有望使该方案取得重大进展，更多的技术正在开发中。我也有兴趣将分歧工具应用于数学生物学中的一些捕食者-食饵模型，主要是在学生的论文中。该项目的核心部分涉及分析动力系统的平衡位置的研究（ODE，线性微分方程或差分方程）取决于参数，更确切地说，与动力系统家族的芽的解析分类问题展开具有有限余维奇异性的动力系统：什么时候动力系统的解析族的两个芽等价于模参数的解析变化，并且可能是时间的重新参数化？这种等价性有许多几何障碍。该项目的重点是1）确定一个完整的模的分析分类这样一个芽的家庭，和2）确定模空间这样的芽。一个动机是理解几何意义的分析障碍等价和描述这些障碍。研究的奇异性类型有：1维芽的抛物点、2维芽的鞍点、2维复叶理的共振鞍点或鞍点以及线性微分系统的共振奇异性。