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

TL;DR

This paper establishes a maximum modulus theorem for certain analytic functions related to Dulac's problem, leading to a dichotomy that either these functions have isolated zeros or are identical, with applications to limit cycle non-accumulation.

Contribution

It introduces a maximum modulus-type theorem for functions with Stokes phenomena, providing a new tool to analyze Dulac's problem and limit cycle behavior.

Findings

Functions either have isolated zeros or are identical.

Non-accumulation of limit cycles around superreal polycycles proven.

Dichotomy theorem extends understanding of Dulac's problem.

Abstract

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.