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Regul. Chaotic Dyn., 2014, Volume 19, Issue 6, Pages 663–680 (Mi rcd190)  

This article is cited in 3 scientific papers (total in 3 papers)

Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio

Amadeu Delshamsa, Marina Gonchenkob, Pere Gutiérreza

a Dep. de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain
b Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, D-10623 Berlin, Germany

Abstract: We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincaré – Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the transversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.

Keywords: transverse homoclinic orbits, splitting of separatrices, Melnikov integrals, silver ratio

Funding Agency Grant Number
Ministerio de Economía y Competitividad de España MTM2012-31714
Russian Science Foundation 14-41-00044
Deutsche Forschungsgemeinschaft TRR 109
Generalitat de Catalunya 2014SGR504
This work has been partially supported by the Spanish MINECO-FEDER Grant MTM2012-31714, the Catalan Grant 2014SGR504, and the Russian Scientific Foundation Grant 14-41-00044. The author MG has also been supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.


DOI: https://doi.org/10.1134/S1560354714060057

References: PDF file   HTML file

Bibliographic databases:

MSC: 37J40, 70H08
Received: 16.09.2014
Accepted:29.09.2014
Language:

Citation: Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez, “Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio”, Regul. Chaotic Dyn., 19:6 (2014), 663–680

Citation in format AMSBIB
\Bibitem{DelGonGut14}
\by Amadeu~Delshams, Marina~Gonchenko, Pere~Guti\'errez
\paper Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 6
\pages 663--680
\mathnet{http://mi.mathnet.ru/rcd190}
\crossref{https://doi.org/10.1134/S1560354714060057}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3284607}
\zmath{https://zbmath.org/?q=an:06507825}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000345996200005}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Hamed Norouzi, Davood Younesian, “Chaos Control for the Plates Subjected to Subsonic Flow”, Regul. Chaotic Dyn., 21:4 (2016), 437–454  mathnet  crossref
    2. A. Delshams, M. Gonchenko, P. Gutierrez, “Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio”, SIAM J. Appl. Dyn. Syst., 15:2 (2016), 981–1024  crossref  mathscinet  zmath  isi  scopus
    3. C. Simó, A. Vieiro, E. Fontich, “On the “Hidden” Harmonics Associated to Best Approximants Due to Quasi-periodicity in Splitting Phenomena”, Regul. Chaotic Dyn., 23:6 (2018), 638–653  mathnet  crossref
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