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

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.

Figures (4)

• Figure 1: Polycycle with one turn.
• Figure 2: Splitting up a polycycle with one turn.
• Figure 3: Local surgery.
• Figure 4: Cauchy--Heine transform.

Theorems & Definitions (42)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Corollary 1
• proof : sketch
• Definition 5
• Definition 6
• Definition 7