Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor

Algaba, Antonio; García, Cristóbal; Giné, Jaume

doi:10.1016/j.cnsns.2018.09.018

Cited by 8 publications

(16 citation statements)

References 21 publications

(20 reference statements)

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“…The next theorem is the main result of [9] that solves the case B) of Proposition 1.1, see also [10] where the characterization is through the existence of an inverse integrating factor.…”

Section: Remarkmentioning

confidence: 92%

Analytic integrability around a nilpotent singularity: The non-generic case

Algaba¹,

Díaz²,

García³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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“…The next theorem is the main result of [9] that solves the case B) of Proposition 1.1, see also [10] where the characterization is through the existence of an inverse integrating factor.…”

Section: Remarkmentioning

confidence: 92%

Analytic integrability around a nilpotent singularity: The non-generic case

Algaba¹,

Díaz²,

García³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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“…Moreover F has the inverse integrating factor V = h(1 − 3a 90 x 3 y). Hence by [7 xy 2 ) T , that it is R x reversible and therefore the origin is a center.…”

Section: Applicationsmentioning

confidence: 99%

The center problem for Z2-symmetric nilpotent vector fields

Algaba

García

Giné

et al. 2018

Journal of Mathematical Analysis and Applications

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We say that a polynomial differential systemẋ = P (x, y),ẏ = Q(x, y) having the origin as a singular point is Z 2-symmetric if P (−x, −y) = −P (x, y) and Q(−x, −y) = −Q(x, y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C ∞ first integral. But up to know there are no characterized these two kinks of nilpotent centers. Here we prove that the origin of any Z 2-symmetric is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x, y) = y 2 + • • •, where the dots denote terms of degree higher than two. 2010 Mathematics Subject classification. 34C05, 34A34, 34C14. Keywords and phrases. Z 2-symmetric differential systems, center problem, nilpotent singularity.

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“…We define the linear operator L : P we remark that either ker L = {0} when k ≡ 0 mod 6 or dim(ker L) = 1 if k ≡ 0 mod 6. In particular, when k is odd, ker L = {0} and consequently dim(range L) = (k) + 1 ≤ dim P (1,3) k+2 where the inequality comes from the fact that dim P…”

Section: Proof Letmentioning

confidence: 99%

Center cyclicity for some nilpotent singularities including the ℤ2-equivariant class

García

2020

Commun. Contemp. Math.

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This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum [Formula: see text]-quasihomogeneous weighted degree, being [Formula: see text] the Andreev number of the singularity. These families strictly include the case [Formula: see text] and also the [Formula: see text]-equivariant families. In some cases for such families we solve, under additional assumptions, the local Hilbert 16th problem giving global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family. Several examples are given.

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Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor

Abstract: Està subjecte a una llicència de Reconeixement-NoComercial-SenseObraDerivada 4.0 de Creative Commons

Cited by 8 publications

References 21 publications

Analytic integrability around a nilpotent singularity: The non-generic case

Analytic integrability around a nilpotent singularity: The non-generic case

The center problem for Z2-symmetric nilpotent vector fields

Center cyclicity for some nilpotent singularities including the ℤ2-equivariant class

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