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One of the most characteristic features of relativistic particle dynamics at cyclotron resonances is the stepwise dependence of energy and momentum on time [V. A. Buts and A. G. Zagorodny, Phys. Plasmas 28(2), 022311 (2021)]. The reason for these steps is the phase synchronization of charged particles and waves. To describe other features, we introduce new dependent variables in this paper. These variables allow us to find a convenient representation of the relativistic resonance that fits the numerical results and thus establish new expressions for the stationary phase, as well as analytically describe effects that were previously visible only in numerical experiment. In particular, the temporal width of the steps, the dynamics of particles on the steps, and, most importantly, the moments of random jumps of particles from one step to another are determined. Analysis of the obtained results opens the possibility to find the conditions for the unlimited acceleration of electrons by transverse electromagnetic waves in a vacuum without an external magnetic field.