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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

[1] Bogolyubov H. H., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[2] Chirikov B. V., “Prokhozhdenie nelineinoi kolebatelnoi sistemy cherez rezonans”, Dokl. AN SSSR, 125:5 (1959), 1015–1018 | MR | Zbl

[3] Kalyakin L. A., “Asimptotiki reshenii uravnenii glavnogo rezonansa”, Teor. i matem. fiz., 137:1 (2003), 142–152 | MR | Zbl

[4] Kolomenskii A. A., Lebedev A. N., Teoriya tsiklicheskikh uskoritelei, Fizmatlit, M., 1962

[5] Meerson B., Friedland L., “Strong autoresonance excitation of Rydberg atoms: the Rydberg accelerator”, Phys. Rev. A, 41 (1990), 5233–5236 | DOI

[6] Kalyakin L. A., “Avtorezonans v dinamicheskoi sisteme”, Sovrem. matem. i ee prilozh., 5, 2003, 79–108

[7] Kolomenskii A. A., Fizicheskie osnovy metodov uskoreniya zaryazhennykh chastits, Izd-vo MGU, M., 1980

[8] Bryuno A. D., Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Nauka, M., 1998 | MR | Zbl

[9] Kozlov V. V., Furta S. D., Asimptotiki reshenii silno nelineinykh sistem differentsialnykh uravnenii, Izd-vo MGU, M., 1996 | MR | Zbl

[10] Barenblatt G. I., Podobie, avtomodelnost, promezhutochnye asimptotiki, Gidrometeoizdat, L., 1978 | MR

[11] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[12] Kuznetsov A. N., “O suschestvovanii vkhodyaschikh v osobuyu tochku reshenii avtonomnoi sistemy, obladayuschei formalnym resheniem”, Funkts. analiz i ego prilozh., 23:4 (1989), 63–74 | MR | Zbl

[13] Kalyakin L. A., “Justification of asymptotic expansions for the principal resonance equations”, Proc. Steklov Inst. Math., Suppl. 1, M., 2003, S108–S122 | MR | Zbl

[14] Fedoryuk M. B., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[15] Fedoryuk M. V., Metod perevala, Nauka, M., 1977 | MR

[16] Dobrohotov S. Yu., Maslov V. P., “Multiphase asymptotics of nonlinear partial differevtial equations with a small parameter”, Sovet. Sci. Rev., 3 (1988), 271–311

[17] Kuzmak G. E., “Asimptoticheskie resheniya nelineinykh differentsialnykh uravnenii s peremennymi koeffitsientami”, Prikl. matem. i mekhan., 23:3 (1951), 519–526

[18] Bourland F. J., Haberman R., “The modulated phase shift for strongly nonlinear, slowly varyning and weasly damped oscillators”, SIAM J. Appl. Math., 48:3 (1988), 737–748 | DOI | MR | Zbl

[19] Neishtadt A. I., “Prokhozhdenie cherez separatrisu v rezonansnoi zadache s medlenno izmenyayuschimsya parametrom”, Prikl. matem. i mekhan., 39:4 (1975), 621–632 | MR

[20] Kiselev O. M., Glebov S. G., “An asymptotic solution slowly crossing the separatrix near a saddle-centre bifurcation point”, Nonlinearity, 16 (2003), 327–362 | DOI | MR | Zbl