The Theorem of Dulac and real classification of semi hyperbolic saddles.

In this talk we will discuss the Theorem of Dulac, i.e. that a planar polynomial dynamical system has a finite amount of isolated periodic orbits, also called limit cycles.
 
We will discuss a strategy proposed by Yu S. Ilyashenko to prove the Theorem of Dulac. 
Then we will discuss how one could complete this proof in terms of finding real analytic normal forms for semihyperbolic saddles which admit asymptotics beyond all orders.