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Lagrangian tori near resonances of near-integrable Hamiltonian systems

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posted on 2015-05-12, 15:56 authored by Anton G. Medvedev, Anatoly NeishtadtAnatoly Neishtadt, Dmitry Treschev
We study families of Lagrangian tori that appear in a neighbourhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the 'integrable' limit ε → 0. Dynamics on these tori are oscillatory in the direction of the resonance phases and rotating with respect to the other (non-resonant) phases. We also show that, if multiplicity of the resonance equals one, generically these tori occupy a set of a large relative measure in the resonant domains in the sense that the relative measure of the remaining 'chaotic' set is of the order $\sqrt \varepsilon$ . Therefore, for small ε > 0 a random initial condition in a $\sqrt \varepsilon$ -neighbourhood of a single resonance occurs inside this set (and therefore generates a quasi-periodic motion) with a probability much larger than in the 'chaotic' set. We present results of numerical simulations and discuss the form of projection of such tori to the action space. At the end of section 4 we discuss the relationship of our results and a conjecture that tori (in a near-integrable Hamiltonian systems) occupy all the phase space except a set of measure ~ε.


The work was supported by grants RFBR 13-01-00251, 05-01-01119, 13-01-12462, State contract no. 8223 Dynamic instability and catastrophe (RF) and RF Program for the State Support of Leading Scienti c Schools NSh-2964.2014.1.



  • Science


  • Mathematical Sciences

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MEDVEDEV, A.G., NEISHTADT, A. and TRESCHEV, D., 2015. Lagrangian tori near resonances of near-integrable Hamiltonian systems. Nonlinearity, 28 (7), pp.2105-2130.


IOP Publishing


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This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

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This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0951-7715/28/7/2105




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