Summary: Quasi-periodic two-frequency perturbations are studied in a system which is close to a nonlinear two-dimensional Hamiltonian one. The example of Duffing equation with a saddle and two separatix loops is considered. Several problems are studied: dynamical behavior in a neighborhood of a resonance level of the unperturbed system, conditions for the existence of resonance quasi-periodic solutions (two-dimensional resonance tori), global behavior of solutions inside domains separated from the unperturbed separatrix. In a neighborhood of the unperturbed separatrix the problem of relative position of stable an unstable separatrix manifolds is studied, conditions for the existence of doubly asymptotic solutions are found.