On a Theorem of Lyapunov–Poincaré in Higher Dimensions

León, V.; Scárdua, Bruno

doi:10.1007/s40598-021-00183-x

Cited by 2 publications

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References 5 publications

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“…There are some equivalent statements in terms of differential one-forms ( [5]). A quite geometrical proof was given by Moussu ([7]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On real center singularities of complex vector fields on surfaces

Scárdua

León

2022

Preprint

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One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with ``many'' periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincar\'e-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.

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“…There are some equivalent statements in terms of differential one-forms ( [5]). A quite geometrical proof was given by Moussu ([7]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The remaining part follows as usual. □ Proof of Theorem C. We proceed as in [5]. Assume that F admits a holomorphic first integral of the form f g with f, g ∈ O 2 , f (0) = g(0) = 0, f and g (being germs reduced and irreducible and) in general position.…”

Section: Complexification Of Foliations Proof Of Theorems a And Cmentioning

confidence: 99%

On real center singularities of complex vector fields on surfaces

Scárdua

León

2022

Preprint

View full text Add to dashboard Cite

show abstract

On real center singularities of complex vector fields on surfaces

León,

Scárdua

2023

Dynamical Systems

View full text Add to dashboard Cite

One of the various versions of the classical Lyapunov-Poincar? center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations ([7]). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two.In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.

show abstract

On a Theorem of Lyapunov–Poincaré in Higher Dimensions

Cited by 2 publications

References 5 publications

On real center singularities of complex vector fields on surfaces

On real center singularities of complex vector fields on surfaces

On real center singularities of complex vector fields on surfaces

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