  In this paper, we analyse the long-time behaviour of the extended RKN (ERKN)
integrators for solving highly oscillatory Hamiltonian systems with a slowly
varying, solution-dependent high frequency. We prove that a symmetric ERKN
integrator approximately conserves a modified action and a modified total
energy over long time intervals based on the technique of varying-frequency
modulated Fourier expansion. An illustrative numerical experiment is carried
out and the numerical results strongly support the theoretical analysis
presented in this paper. As a byproduct of this work, similar long-time
behaviour is also investigated for an RKN method.
