  In this paper, we study the long-time near conservation of the total and
oscillatory energies for extended RKN (ERKN) integrators when applied to
muti-frequency highly oscillatory Hamiltonian systems. We consider one-stage
explicit symmetric integrators and show their long-time behaviour of numerical
energy conservations by using modulated multi-frequency Fourier expansions.
Numerical experiments are carried out and the numerical results demonstrate the
remarkable long-time near conservation of the energies for the ERKN integrators
and support our theoretical analysis presented in this paper.
