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Asymptotic expansions of characteristic orbits of planar real analytic vector fields

Jun Zhang

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.

Asymptotic expansions of characteristic orbits of planar real analytic vector fields

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.
Paper Structure (5 sections, 4 theorems, 45 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Proof of the main result
  3. Desingularization
  4. Asymptotic expansion of the inverse function
  5. Proof of Theorem \ref{['thm:O']}

Key Result

Theorem 1.1

Suppose that $y=y(x)$ for small $x>0$ is a characteristic orbit of vector field equ:XY. Then either

Figures (1)

  • Figure 1: Desingularization process along characteristic orbit $\Gamma$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Lemma 2.3