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dc.contributor.authorDelshams Valdés, Amadeu
dc.contributor.authorGonchenko, Marina
dc.contributor.authorGutiérrez Serrés, Pere
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtiques
dc.identifier.citationDelshams, 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.description.abstractThe 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:
dc.format.extent44 p.
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística
dc.subject.lcshHamiltonian systems
dc.titleExponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
dc.subject.lemacSistemes hamiltonians
dc.subject.lemacSistemes dinàmics
dc.contributor.groupUniversitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC
dc.description.peerreviewedPeer Reviewed
dc.subject.amsClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
dc.subject.amsClassificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
dc.rights.accessOpen Access
dc.description.versionPostprint (published version)
upcommons.citation.authorDelshams, A., Gonchenko, M., Gutiérrez, P.
upcommons.citation.publicationNameSIAM journal on applied dynamical systems

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