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Rodrigo Schaefer

Departament de Matemàtica Aplicada I ETSEIB-UPC Av
Universitat Politècnica de Catalunya


Delshams A., Schaefer R. G.
Arnold Diffusion for a Complete Family of Perturbations
2017, vol. 22, no. 1, pp.  78-108
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 in the so-called Scattering map. This work has the following structure: In a 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 the 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.
Keywords: Arnold diffusion, normally hyperbolic invariant manifolds, scattering maps
Citation: Delshams A., Schaefer R. G.,  Arnold Diffusion for a Complete Family of Perturbations, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 78-108

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