  In this study we consider the Hamiltonian approach for the construction of a
map for a system with nonlinear resonant interaction, including phase trapping
and phase bunching effects. We derive basic equations for a single resonant
trajectory analysis and then generalize them into the map in the
energy/pitch-angle space. The main advances of this approach are the
possibility to consider effects of many resonances and to simulate the
evolution of the resonant particle ensemble on long time ranges. For
illustrative purposes we consider the system with resonant relativistic
electrons and field-aligned whistler-mode waves. The simulation results show
that the electron phase space density within the resonant region is flattened
with reduction of gradients. This evolution is much faster than the predictions
of quasi-linear theory. We discuss further applications of the proposed
approach and possible ways for its generalization.
