Dulac's Theorem Revisited

Abstract

Dulac's theorem states that the number of limit cycles of any given polynomial vector field on the plane is finite. After compactifying the phase plane to a sphere and some well known arguments one only has to prove that limit cycles cannot accumulate onto elementary graphics which we will call polycycles. Dulac in his proof unfortunately made an unproved statement by inferring the triviality of the return map of a polycycle from the triviality of its asymptotic expansion. Ilyashenko in (Russ Math Surv 40(6):1-49, 1985. https://doi.org/10.1070/rm1985v040n06abeh003701) produced a clever counter example, clearly showing why Dulac's arguments failed and additionally he showed that Dulac's theorem is valid for hyperbolic polycycles, i.e. polycycles with only hyperbolic equilibria. It is a corner stone that has been completely understood. Afterwards Ilyashenko published his own full proof of Dulac's theorem in Ilyashenko (in: Translations of mathematical monographs, American Mathematical Society, Providence, 1991). We provide evidence that the approach of Ilyashenko (1991) to the proof of Dulac's theorem has a gap. Although the asymptotics of Ilyashenko (1991) capture far more than the asymptotics of Dulac, we prove that the arguments for why the asymptotics in Ilyashenko (1991) are not themselves oscillatory is insufficient. We give an explicit counterexample and we draw confines to which Ilyashenko's result may be restricted in order to keep the validity.