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Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
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.
Àrees temàtiques de la UPC::Matemàtiques i estadística
splitting of separatrices
Melnikov integrals
quadratic and cubic frequencies
Sistemes dinàmics diferenciables
Attribution-NonCommercial-NoDerivs 3.0 Spain
Artículo - Borrador

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