Computing the exact number of periodic orbits for planar flows

TL;DR

This paper investigates the computability of counting periodic orbits in polynomial planar flows, revealing that the problem is noncomputable in general but computable for stable systems, and identifies limitations in bounding the number of orbits.

Contribution

It establishes the noncomputability of the exact number of periodic orbits in general polynomial planar flows and shows computability for structurally stable systems.

Findings

The problem is noncomputable for general polynomial planar flows.

It is computable for structurally stable systems on the unit disk.

Some systems lack a computable upper bound on the number of periodic orbits.

Abstract

In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the problem is noncomputable on the one hand and, on the other hand, computable uniformly on the set of all structurally stable systems defined on the unit disk. We also prove that there is a family of polynomial planar systems which does not have a computable sharp upper bound on the number of its periodic orbits.