Эта публикация цитируется в
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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.
MSC: 37J40 Поступила в редакцию: 17.09.2015
Принята в печать: 20.12.2015
Язык публикации: английский
DOI:
10.1134/S1560354717010051