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What is the definition of a resonant/invariant torus (in the phase space of a Hamiltonian system)? Are there non-resonant tori?

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Invariant tori in Hamiltonian phase spaces are characterized by two frequencies, one for each dimension: when these frequencies are commensurate ($f_1/f_2$ is rational), curves over these tori close onto themselves and are therefore periodic - these are resonant tori; when the frequencies are not commensurate, the corresponding trajectories are not closed, they are quasi-periodic - and they fill non-resonant tori.

The nomenclature is based on the effect a perturbation have over the Hamiltonian: the resonant tori are more readily affected by the perturbation, through a resonance mechanism, while the non-resonant tori are affect through second-order, nonlinear mechanisms.

You can check Arnold's book to learn more about that.

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