dc.contributor.author Delshams Valdés, Amadeu dc.contributor.author Gonchenko, Marina dc.contributor.author Gutiérrez Serrés, Pere dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtiques dc.date.accessioned 2016-09-20T10:43:14Z dc.date.available 2016-09-20T10:43:14Z dc.date.issued 2016-06 dc.identifier.citation Delshams, A., Gonchenko, M., Gutiérrez, P. Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio. "SIAM journal on applied dynamical systems", Juny 2016, vol. 15, núm. 2, p. 981-1024. dc.identifier.issn 1536-0040 dc.identifier.uri http://hdl.handle.net/2117/90067 dc.description.abstract The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega),$ where the frequency ratio $\Omega$ is a quadratic irrational number. Applying the Poincaré--Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in $\varepsilon$, with the functions in the exponents being periodic with respect to $\ln\varepsilon$, and can be explicitly constructed from the continued fraction of $\Omega$. In this way, we emphasize the strong dependence of our results on the arithmetic properties of $\Omega$. In particular, for quadratic ratios $\Omega$ with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios, respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of $\varepsilon$, and the transversality can be established for a majority of values of $\varepsilon$, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur. Read More: http://epubs.siam.org/doi/10.1137/15M1032776 dc.format.extent 44 p. dc.language.iso eng dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/ dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística dc.subject.lcsh Hamiltonian systems dc.title Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio dc.type Article dc.subject.lemac Sistemes hamiltonians dc.subject.lemac Sistemes dinàmics dc.contributor.group Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC dc.identifier.doi 10.1137/15M1032776 dc.description.peerreviewed Peer Reviewed dc.subject.ams Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems dc.subject.ams Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics dc.rights.access Open Access drac.iddocument 18766316 dc.description.version Postprint (published version) upcommons.citation.author Delshams, A., Gonchenko, M., Gutiérrez, P. upcommons.citation.published true upcommons.citation.publicationName SIAM journal on applied dynamical systems upcommons.citation.volume 15 upcommons.citation.number 2 upcommons.citation.startingPage 981 upcommons.citation.endingPage 1024
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