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We consider multiscale Hamiltonian systems in which harmonic oscillators withseveral high frequencies are coupled to a slow system. It is shown that theoscillatory energy is nearly preserved over long times eps^{-N} for arbitraryN>1, where eps^{-1} is the size of the smallest high frequency. The result isuniform in the frequencies and does not require non-resonance conditions.

Energy separation in oscillatory Hamiltonian systems without any non-resonance condition