Normal forms and bifurcations of vector fields

整个项目集中在分析动力学的基本问题，特别是低维动力系统取决于参数：1维同构，平面向量场，无论是真实的向量场或复杂的叶理，和线性微分系统。动力系统是依赖于时间的系统。很难找到一个给定系统的显式解。庞加莱是创始人的几何方法来研究和理解的解决方案的动力系统。这些方法允许得出定性的结论，即使没有明确的公式的解决方案。为了更好地理解动力系统，重要的是攻击它们的弱点：相空间中的奇点和参数空间中的分叉值。该研究项目正是集中在这些问题： - 更好地理解依赖于参数的动力系统的奇异性， - 研究与定性行为变化相对应的分叉。 奇点的研究是通过规范形式完成的，规范形式代表奇点的等价类的模型。但是，坐标到范式的变化通常是不同的，该项目的一个目的是了解为什么。 至于分叉，该项目集中在二维向量场的分叉和产生极限环的分叉。该项目既考虑在纯数学中的应用，与希尔伯特关于多项式向量场的极限环的最大数量作为度的函数的第16个问题有关，也考虑在应用数学中的应用，研究一些复杂的捕食者-猎物模型。

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: - a better understanding of the singularities of dynamical systems depending on parameters, - a study of the bifurcations corresponding to changes of qualitative behaviour. The study of singularities is done through normal forms, which represent models for equivalence classes of singularities. But the changes of coordinates to normal forms diverge generically and one purpose of the project is to understand why. As for the bifurcations, the project is centered on bifurcations in 2-dimensional vector fields and bifurcations creating limit cycles. The project both considers applications in pure mathematics, in link with Hilbert's 16th problem on the maximum number of limit cycles of a polynomial vector field as a function of the degree, and in applied mathematics, with the study of some sophisticated predator-prey models.