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In this study we consider the Hamiltonian approach for the construction of amap for a system with nonlinear resonant interaction, including phase trappingand phase bunching effects. We derive basic equations for a single resonanttrajectory analysis and then generalize them into the map in theenergy/pitch-angle space. The main advances of this approach are thepossibility to consider effects of many resonances and to simulate theevolution of the resonant particle ensemble on long time ranges. Forillustrative purposes we consider the system with resonant relativisticelectrons and field-aligned whistler-mode waves. The simulation results showthat the electron phase space density within the resonant region is flattenedwith reduction of gradients. This evolution is much faster than the predictionsof quasi-linear theory. We discuss further applications of the proposedapproach and possible ways for its generalization.

Long-term dynamics driven by resonant wave-particle interactions: from Hamiltonian resonance theory to phase space mapping