Abstract

The Lokta-Volterra systems are some of the simplest non-linear ODEs and their behaviour in the real plane has been completely classified.
However, over the complex plane, things are a lot more interesting. Indeed, whilst working on $1:-\lambda$ resonant singularities for polynomial systems with C.Rousseau and P.Mardesic, we found that many of our counter-examples were in fact given by explicit Lokta-Volterra systems.
Since these systems have a lot of intrinsic geometry,it is natural to try and find geometrical reasons to explain their behaviour. In this talk, I would like to give some indication of how we think that this can be done.
Though the work presented here is still in a very preliminary stage, we have obtained some encouraging results, and have also found some intriguing experimental data which still needs to be explained.