Long-time analysis of extended RKN integrators for Hamiltonian systems with a solution-dependent high frequency

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.