Geodesic
Intermediate asymptotics for solutions to the degenerate principal resonance equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 83-94

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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. 
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     author = {L. A. Kalyakin},
     title = {Intermediate asymptotics for solutions to the degenerate principal resonance equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {83--94},
     publisher = {mathdoc},
     volume = {46},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a8/}
}
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L. A. Kalyakin. Intermediate asymptotics for solutions to the degenerate principal resonance equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 83-94. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a8/