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Эта публикация цитируется в 13 научных статьях (всего в 13 статьях)
Arnold Diffusion for a Complete Family of Perturbations
Amadeu Delshams, Rodrigo G. Schaefer Department de Matemàtiques,
Universitat Politècnica de Catalunya,
Av. Diagonal 647, 08028 Barcelona
Аннотация:
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 $H(p,q,I,\varphi,s) = p^{2}/2+\cos q -1 +I^{2}/2 + h(q,\varphi,s;\varepsilon)$ — proving that for any small periodic perturbation of the form
$h(q,\varphi,s;\varepsilon) = \varepsilon\cos q\left( a_{00} + a_{10}\cos\varphi + a_{01}\cos s \right)$
($a_{10}a_{01} \neq 0$) 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 ($I^*\thicksim\pi/2\mu$, $\mu = a_{10}/a_{01}$), 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 $\mu$).
The bifurcations of the scattering map are also studied as a function of $\mu$.
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.
Ключевые слова:
Arnold diffusion, normally hyperbolic invariant manifolds, scattering maps.
Поступила в редакцию: 17.09.2015 Принята в печать: 20.12.2015
Образец цитирования:
Amadeu Delshams, Rodrigo G. Schaefer, “Arnold Diffusion for a Complete Family of Perturbations”, Regul. Chaotic Dyn., 22:1 (2017), 78–108
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd244 https://www.mathnet.ru/rus/rcd/v22/i1/p78
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