Lagrangian tori near resonances of near-integrable Hamiltonian systems

2013/11/01 by Medvedev, Anton, Neishtadt, Anatoly, Treschev, Dmitry · 1 citation
#37J40 #70H08 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1311.0132

Abstract

In this paper we study families of Lagrangian tori that appear in a neighborhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the "integrable" limit ε→ 0. Dynamics on these tori is 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 large relative measure in the resonant domains in the sense that the relative measure of the remaining "chaotic" set is of order √ε. Therefore for small ε > 0 a random initial condition in a √ε-neighborhood 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.

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• Singular KAM theory for convex Hamiltonian systems

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