Singularities of dynamical systems and their unfoldings

本研究项目的重点是研究动力系统的奇异性，包括常微分方程和差分方程，以及这些奇异性如何组织动力学。在ODE的情况下，项目包括两个部分：1)第一部分是平面矢量场的分岔理论；第二部分是关于低余维ODE在实或复有限维空间中奇点展开的研究。在差分方程的情况下，研究项目的重点是研究1维和2维有限协维共振奇点的展开。**分岔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有限数量参数的向量场族中，并分析该族中的分岔。家族中最高协维数的分岔是最重要的，它组织了分岔图和动力学。这个项目的主要动机是通过dumtier - roussarie - rousseau程序来证明多项式二次向量场极限环数的一致界的存在，从而推进Hilbert第16问题有限部分的解的进展。该程序包括证明121图形在二次向量场族内具有有限循环性。该计划有望从最近开发的新技术中取得重大进展，更多的技术正在开发中。我也对将分岔工具应用于数学生物学中的一些捕食者-猎物模型很感兴趣，主要是在学生的论文中。**项目的核心部分是研究解析动力系统（ODE、线性微分方程或差分方程）随参数的平衡位置，更准确地说，是研究动力系统族的细菌的解析分类问题，展开具有有限余维奇点的动力系统。什么时候动力系统解析族的两个胚等价模是参数的解析变化，可能是时间的重新参数化？这种等价存在许多几何障碍。本课题的重点是：1)识别该类胚芽的解析分类的完全模；2)识别该类胚芽的模空间。一个动机是理解解析障碍对等价的几何意义，并描述这些障碍。研究奇异点的类型有：一维微分同态胚芽的抛物点、二维微分同态胚芽的鞍节点、二维复叶的共振鞍或鞍节点以及线性微分系统的共振奇异点。**

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. **