Borel summation and splitting of separatrices for the Hénon map

Gelfreich, Vassili; Sauzin, David

doi:10.5802/aif.1831

Cited by 54 publications

(83 citation statements)

References 3 publications

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“…It was proved in [11] that in the case of the Hénon map the first component of does not vanish. COROLLARY 2.2.…”

Section: Analytic Invariants 1823mentioning

confidence: 99%

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Analytic invariants associated with a parabolic fixed point in 2

Gelfreich¹,

Naudot²

2008

Ergod. Th. Dynam. Sys.

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It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

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“…It was proved in [11] that in the case of the Hénon map the first component of does not vanish. COROLLARY 2.2.…”

Section: Analytic Invariants 1823mentioning

confidence: 99%

“…In [11] it was proved that this invariant does not vanish in the case of the area-preserving Hénon map with a parabolic fixed point, which can be written in the following form:…”

Section: Introductionmentioning

confidence: 99%

Analytic invariants associated with a parabolic fixed point in 2

Gelfreich¹,

Naudot²

2008

Ergod. Th. Dynam. Sys.

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“…It is worth mentioning that the Poincaré-Melnikov function does not always give the correct prediction for the splitting; see [Tre97], [Gel97b]. As we have pointed out elsewhere, we can encounter this phenomenon in maps; see [Laz03], [DRR98], [GS01]. The methods developed in these works draw heavily on the Hamiltonian character of the system, especially its symplectic structure.…”

Section: Introductionmentioning

confidence: 99%

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Baldomá

Seara

2006

J Nonlinear Sci

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Summary. In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in R 3 . This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

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“…One of these invariants also plays a role in the splitting of separatrices near a saddle-center bifurcation [14]. In particular, for the Hénon map the same study was carried out in [13] where a connection with the resurgent theory of J. Écalle was established. For a more recent treatment on the connection between resurgence and splitting of separatrices the reader is referred to [27].…”

Section: Introductionmentioning

confidence: 87%

Analytic invariants for the 1:-1 resonance

Gaivão¹

2013

Ann. inst. Fourier

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Borel summation and splitting of separatrices for the Hénon map

Cited by 54 publications

References 3 publications

Analytic invariants associated with a parabolic fixed point in 2

Analytic invariants associated with a parabolic fixed point in 2

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Analytic invariants for the 1:-1 resonance

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