Singularities of Trajectories of Analytic and Algebraic Vector Fields – STAAVF

This project focuses on the geometric behavior of trajectories of analytic vector fields, that is the solutions of ordinary differential equations with analytic coefficients.

Ordinary differential equations arise in many different areas of science and much study has been devoted to understanding their solutions through different methods and approaches. In the case where the equation is linear, it can be solved explicitly by analytical methods. It is well known that most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions, obtained by numerical methods, do not give satisfactory information on the qualitative behavior of true solutions, such as stability, limit sets, limit cycles, and (non)-oscillation. The trajectories of real analytic vector fields are transcendental in general. However their geometry is often “tame”, though the precise mathematical definition of tameness has to be established.

The understanding of the qualitative geometric behavior of trajectories of analytic vector fields is the primary objective of our project. We want to determine in which cases the solutions are tame and to make precise the meaning of tameness in each case.

In our study we will use a large variety of methods, namely: Singularity Theory, resolution of singularities, classification of real analytic function germs, stratifications and conormal geometry, ridge and valley lines, semi-algebraic and o-minimal geometry, quasi-analytic classes, (pseudo)abelian integrals, formal series and asymptotic analysis, resurgent methods and resummation processes. In this project we will combine the most recent advances of these techniques. Further development of these methods is an important inseparable part of this project.