arXiv subjects

Melvin Yeung

Publications and source records attributed to Melvin Yeung.

3 recordsLinked to original sources

arXiv2024

A Maximum Modulus Theorem for functions admitting Stokes phenomena, and specific cases of Dulac's Theorem

We study large classes of real-valued analytic functions that naturally emerge in the understanding of Dulac's problem, which addresses the finiteness of limit cycles in planar differential equations. Building on a Maximum Modulus-type result we got, our main statement essentially follows. Namely, for any function belonging to these classes, the following dichotomy holds: either it has isolated zeros or it coincides with the identity. As an application, we prove that the non-accumulation of limit cycles holds around a specific class of the so-called superreal polycycles.

math.DS↗

arXiv2024

Natural levels in return maps of elementary polycycles

We will provide a proof of a known specific case of Dulac's Theorem in the style of Ilyashenko. From this we derive a quasi-analyticity result for some return maps of polycycles and we give a Structural Theorem for the formal asymptotics of such a polycycle.

math.DS↗

arXiv2024

On the monograph "Finiteness Theorems for limit cycles" and a special case of alternant cycles

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.

math.DS↗