Realization problems for limit cycles of planar polynomial vector fields

TL;DR

This paper constructs explicit planar polynomial vector fields that realize any finite configuration of closed curves as limit cycles, with prescribed properties, demonstrating the flexibility of polynomial vector fields in modeling complex limit cycle arrangements.

Contribution

It provides a method to explicitly realize any finite configuration of limit cycles with prescribed periods, multiplicities, and stabilities in polynomial vector fields, under natural compatibility conditions.

Findings

Constructed polynomial vector fields realize any finite limit cycle configuration.

The vector fields are Darboux integrable with polynomial inverse integrating factors.

The realization respects stability and multiplicity compatibility conditions.

Abstract

We show that for any finite configuration of closed curves $\Gamma\subset \mathbb{R}^2$, one can construct an explicit planar polynomial vector field that realizes $\Gamma$, up to homeomorphism, as the set of its limit cycles with prescribed periods, multiplicities and stabilities. The only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities. The constructed vector fields are Darboux integrable and admit a polynomial inverse integrating factor.