Monodromic singularities without curves of zero angular speed

Isaac A. García; Jaume Giné; Isaac A. García; Jaume Giné

doi:10.3934/math.2025069

AIMS Mathematics

2025, Volume 10, Issue 1: 1488-1500. doi: 10.3934/math.2025069

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Monodromic singularities without curves of zero angular speed

Isaac A. García ,
Jaume Giné ^,

Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, Lleida 25001, Spain

Received: 01 October 2024 Revised: 16 December 2024 Accepted: 15 January 2025 Published: 22 January 2025
MSC : 34Cxx, 37G15, 37G10

We consider planar analytic vector fields $ \mathcal{X} $ having a monodromic singular point with Poincaré map $ \Pi $. We use the fact that there always exists a real analytic invariant curve $ F = 0 $ of $ \mathcal{X} $ in a neighborhood of that singularity. We find some relations between $ \Pi $ and $ F $ that can be used to determine new conditions that guarantee the analyticity of $ \Pi $ at the singularity. In the special case that $ F $ becomes an inverse integrating factor of $ \mathcal{X} $, we rediscover formulas obtained previously by other methods. Applications to the center-focus problem and also to vector fields with degenerate infinity are given.
- center,
- periodic orbits,
- Poincaré map,
- invariant curve
Citation: Isaac A. García, Jaume Giné. Monodromic singularities without curves of zero angular speed[J]. AIMS Mathematics, 2025, 10(1): 1488-1500. doi: 10.3934/math.2025069

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Abstract

We consider planar analytic vector fields $ \mathcal{X} $ having a monodromic singular point with Poincaré map $ \Pi $. We use the fact that there always exists a real analytic invariant curve $ F = 0 $ of $ \mathcal{X} $ in a neighborhood of that singularity. We find some relations between $ \Pi $ and $ F $ that can be used to determine new conditions that guarantee the analyticity of $ \Pi $ at the singularity. In the special case that $ F $ becomes an inverse integrating factor of $ \mathcal{X} $, we rediscover formulas obtained previously by other methods. Applications to the center-focus problem and also to vector fields with degenerate infinity are given.

References

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