Singularities of dynamical systems and their unfoldings

该研究项目集中于研究动力系统的奇异性，包括常微分方程(ODE)和差分方程组，以及这些奇异性如何组织动力学。在微分方程组的情况下，本项目包括两部分：1)平面向量场的分叉理论的第一部分；2)实数或复数有限维空间中低余维微分方程组奇点的展开研究。在差分方程组的情况下，研究一维和二维有限余维共振奇点的开折。**分叉理论是一种发现平面向量场相图的工具，它通过将单个矢量场嵌入到依赖于有限个参数的矢量场族中并分析该族中的分叉来实现。在这个族中，余维最高的分支是最重要的：它们组织了分叉图和动力学。该项目这一部分的主要动机是通过Dumortier-Roussarie-Rousseau程序来证明多项式二次向量场的极限环个数的一致界的存在，从而推动在Hilbert的第16个问题的有限部分的解的进展。该程序包括证明121个图在二次向量场族内具有有限循环性。最近开发的新技术有望在该项目上取得重大进展，更多的技术正在开发中。我还对数学生物学中的一些捕食者-食饵模型应用分歧工具感兴趣，主要是在学生的论文中。**该项目的核心部分涉及依赖于参数的解析动力系统(常微分方程组、线性微分方程或差分方程组)的平衡位置的研究，更准确地说，涉及展开具有有限余维奇点的动力系统的动力系统族芽的解析分类问题：何时两个解析族动力系统芽等价于参数的解析改变，并且可能是时间的重新参数化？对于这样的等价性，存在许多几何障碍。该项目的重点是1)确定这样一个科的胚种的一个完整的解析分类模，以及2)确定这样的胚种的模空间。一个动机是理解解析障碍的几何意义，以便等价并描述这些障碍。所研究的奇点类型包括一维微分同胚芽的抛物点、二维微分同胚芽的鞍结点、二维复叶的共振鞍结点和线性微分系统的共振奇点。**

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. **