Arnold diffusion for a complete family of perturbations
| Autor: | Amadeu Delshams, Rodrigo G. Schaefer |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
normally hyperbolic invariant manifolds
Degrees of freedom (physics and chemistry) Dynamical Systems (math.DS) 01 natural sciences Instability Hamiltonian system Mathematics (miscellaneous) 0103 physical sciences Arnold diffusion FOS: Mathematics Differentiable dynamical systems Sistemes hamiltonians Mathematics - Dynamical Systems 0101 mathematics Diffusion (business) Hamiltonian systems Mathematical physics Physics Scattering 010102 general mathematics Matemàtiques i estadística [Àrees temàtiques de la UPC] Function (mathematics) scattering maps Sistemes dinàmics diferenciables Action (physics) 37J40 010307 mathematical physics |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) Recercat. Dipósit de la Recerca de Catalunya instname |
| Popis: | In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p 2/2+ cos q − 1 + I 2/2 + h(q, φ, s; e) — proving that for any small periodic perturbation of the form h(q, φ, s; e) = e cos q (a 00 + a 10 cosφ + a 01 cos s) (a 10 a 01 ≠ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ π/2μ, μ = a 10/a 01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any μ). The bifurcations of the scattering map are also studied as a function of μ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map. |
| Databáze: | OpenAIRE |
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