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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 6, Pages 635–655 (Mi rcd188)

Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation

Vassili Gelfreicha, Lev Lermanb

a Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
b Lobachevsky State University of Nizhny Novgorod, Russia, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia

Abstract: We study the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a non-semisimple double zero one. It is well known that a one-parameter unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of single-round homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.

Keywords: Hamiltonian bifurcation, homoclinic orbit, separatrix splitting, asymptotics beyond all orders

 Funding Agency Grant Number Leverhulme Trust Russian Foundation for Basic Research 14-01-00344 Russian Science Foundation 14-41-00044 Ministry of Education and Science of the Russian Federation 1410 Engineering and Physical Sciences Research Council EP/J003948/1 VG’s research was supported by EPRC (grant EP/J003948/1) and by the Leverhulme Trust research project. LL was supported by RFBR (grant 14-01-00344). Part of this project was supported by the Russian Science Foundation (grant 14-41-00044) and the Ministry of Science and Education of RF under the project 1410 (State Target plan).

DOI: https://doi.org/10.1134/S1560354714060033

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Bibliographic databases:

MSC: 37J20, 37J45, 70K50, 70K70
Accepted:05.10.2014
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Citation: Vassili Gelfreich, Lev Lerman, “Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation”, Regul. Chaotic Dyn., 19:6 (2014), 635–655

Citation in format AMSBIB
\Bibitem{GelLer14}
\by Vassili~Gelfreich, Lev~Lerman
\paper Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 6
\pages 635--655
\mathnet{http://mi.mathnet.ru/rcd188}
\crossref{https://doi.org/10.1134/S1560354714060033}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3284605}
\zmath{https://zbmath.org/?q=an:06507823}

• http://mi.mathnet.ru/eng/rcd188
• http://mi.mathnet.ru/eng/rcd/v19/i6/p635

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. R. H. Goodman, “Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential”, Physica D, 359 (2017), 39–59
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