Asymptotic expansions of characteristic orbits of planar real analytic vector fields

TL;DR

This paper extends the Newton-Puiseux Theorem to characteristic orbits of isolated singularities in planar real analytic vector fields, demonstrating that these orbits have a 'power-log' expansion, thus generalizing classical curve expansion results.

Contribution

It introduces a novel generalization of Puiseux expansions to characteristic orbits of singularities in real analytic vector fields, establishing their asymptotic 'power-log' expansions.

Findings

Characteristic orbits admit 'power-log' asymptotic expansions.

Generalization of Newton-Puiseux Theorem to vector field orbits.

Provides a new tool for analyzing singularities in dynamical systems.

Abstract

The well-known Newton-Puiseux Theorem states that each real branch of a planar real analytic curve admits a Puiseux expansion. We generalize this result to characteristic orbit of an isolated singularity of a planar real analytic vector field and prove that each characteristic orbit has a `power-log' expansion.