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Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
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 Matemàtica Aplicada I |
dc.date.accessioned | 2013-09-18T12:06:15Z |
dc.date.available | 2013-09-18T12:06:15Z |
dc.date.created | 2013 |
dc.date.issued | 2013 |
dc.identifier.citation | Delshams, A.; Gonchenko, M.; Gutiérrez, P. "Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies". 2013. |
dc.identifier.uri | http://hdl.handle.net/2117/20156 |
dc.description.abstract | We study the splitting of invariant manifolds of whiskered t ori with two or three frequencies in nearly-integrable Hamiltonian systems. We consider 2-dimensional tori with a frequency vector ω = (1 , Ω) where Ω is a quadratic irrational number, or 3-dimensional tori with a frequency v ector ω = (1 , Ω , Ω 2 ) where Ω is a cubic irrational number. Applying the Poincar ́e–Melnikov method, we find exponentia lly small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associa ted to the invariant torus, showing that such estimates depend strongly on the arithmetic properties of the frequen cies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfille d in 24 cases, which allows us to provide asymptotic estimate s in a simple way. In the cubic case, we focus our attention to th e case in which Ω is the so-called cubic golden number (the real root of x 3 + x − 1 = 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubi c cases. |
dc.format.extent | 17 p. |
dc.language.iso | eng |
dc.relation.ispartofseries | [prepr201310DelGG] |
dc.rights | Attribution-NonCommercial-NoDerivs 3.0 Spain |
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.other | splitting of separatrices |
dc.subject.other | Melnikov integrals |
dc.subject.other | quadratic and cubic frequencies |
dc.title | Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies |
dc.type | External research report |
dc.subject.lemac | Sistemes dinàmics diferenciables |
dc.contributor.group | Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
dc.relation.publisherversion | https://www.ma1.upc.edu/recerca/preprints/preprints-2013/Fitxers/prepr201301gutierrez.pdf |
dc.rights.access | Open Access |
local.identifier.drac | 12762496 |
dc.description.version | Preprint |
local.citation.author | Delshams, A.; Gonchenko, M.; Gutiérrez, P. |
local.citation.publicationName | Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies |
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