Intermediate asymptotics for solutions to the degenerate principal resonance equations

The system of two first-order differential equations that arises in averaging nonlinear systems over fast single-frequency oscillations is investigated. The averaging is performed in the neighborhood of the critical free frequency of a nonlinear system. In this case, the original equations differ from the principal resonance equations in the general case. The main result is the construction of the asymptotics of a two-parameter family of solutions in the neighborhood of a solution with unboundedly increasing amplitude. The results, in particular, provide a key to understanding the particle acceleration process in relativistic accelerators near the critical free frequency.