A note on Hilbert 16th Problem

TL;DR

This paper proves that the maximum number of limit cycles in planar polynomial vector fields can be realized by structurally stable systems with hyperbolic cycles and that this maximum increases with degree if finite.

Contribution

It establishes the realizability of the maximum limit cycle count by structurally stable systems and proves its strict monotonicity when finite.

Findings

Maximum limit cycles are realizable by structurally stable vector fields.

The maximum number of limit cycles is a strictly increasing function of degree when finite.

Provides insights into the structure of polynomial vector fields related to Hilbert's 16th problem.

Abstract

Let $\mathcal{H}(n)$ be the maximum number of limit cycles that a planar polynomial vector field of degree $n$ can have. In this paper we prove that $\mathcal{H}(n)$ is realizable by structurally stable vector fields with only hyperbolic limit cycles and that it is a strictly increasing function whenever it is finite.