Reference systems for splitting of separatrices

The investigation of exponentially small splitting of separatrices for high-frequency time-periodic perturbation of a Hamiltonian with one degree of freedom leads to a reference system in the complex phase space. The reference system is independent of a small parameter, e.g., the perturbation period of the original system. The splitting of the invariant manifolds is described for the system v2 2 − (1 + G ′ (τ))eu, which is a reference system for high-frequency perturbations of the pendulum.