Asymptotic expansions of characteristic orbits of planar real analytic vector fields

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.

💡 Research Summary

The paper studies characteristic orbits of planar real‑analytic vector fields near an isolated singularity. A characteristic orbit is an invariant curve that approaches the singular point in a fixed direction rather than spiralling. The classical Newton‑Puiseux theorem guarantees that each branch of a real‑analytic plane curve admits a convergent fractional power series (Puiseux series) independent of the choice of analytic coordinates. The author seeks an analogous coordinate‑free asymptotic expansion for characteristic orbits of vector fields, which are more delicate because the dynamics may introduce logarithmic terms.

The main result, Theorem 1.1, states that if (y=y(x)) (for small (x>0)) parametrizes a characteristic orbit of a vector field (V=X(x,y)\partial_x+Y(x,y)\partial_y), then (y(x)) belongs to one of three families:

1. A convergent Puiseux series (y(x)\in\mathbb R