Geodesic
On transition processes for the Van der Pol equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 6, pp. 968-979 
A. M. Ter-Krikorov. On transition processes for the Van der Pol equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 6, pp. 968-979. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a3/
@article{ZVMMF_2007_47_6_a3,
     author = {A. M. Ter-Krikorov},
     title = {On transition processes for the {Van} der {Pol} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {968--979},
     year = {2007},
     volume = {47},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a3/}
}
TY  - JOUR
AU  - A. M. Ter-Krikorov
TI  - On transition processes for the Van der Pol equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2007
SP  - 968
EP  - 979
VL  - 47
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a3/
LA  - ru
ID  - ZVMMF_2007_47_6_a3
ER  - 
%0 Journal Article
%A A. M. Ter-Krikorov
%T On transition processes for the Van der Pol equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2007
%P 968-979
%V 47
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a3/
%G ru
%F ZVMMF_2007_47_6_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

For the classical Van der Pol equation at $-\infty, solutions describing the transition from the state of unstable equilibrium to the stable limit cycle are studied. Formal series in powers of a small parameter are constructed. It is shown that the coefficients of the series are periodic functions of a relatively fast independent variable, and an exact description of the coefficient dependence on the slow independent variable is given. It is proved that, for sufficiently small values of the parameter, the exact solution exists in the same functional class as the terms of the formal series, beginning with the second term, and that the formal series are asymptotic with respect to the small parameter for the exact solution.

[1] Andronov A. A., Vitt A. A., Khaikin S. E., Teoriya kolebanii, Fizmatgiz, M., 1959

[2] Lefshets S., Geometricheskaya teoriya differentsialnykh uravnenii, Izd-vo inostr. lit., M., 1961 | MR

[3] Marsden Dzh., Mak-Kraken M., Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Mir, M., 1980 | MR | Zbl

[4] Khessard B., Kazarinov N., Ven I., Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Mir, M., 1985 | MR

[5] Gukenkheimer Dzh., Kholms F., Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, In-t kompyuternykh issledovanii, Moskva, Izhevsk, 2002

[6] Bogolyubov H. H., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Fizmatgiz, M., 1958 | MR

[7] Bautin H. H., Leontovich E. A., Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti, Nauka, M., 1990 | Zbl

[8] Belolipetskii A. A., Ter-Krikorov A. M., “O fundamentalnykh resheniyakh nelineinogo uravneniya teploprovodnosti”, Zh. vychisl. matem. i matem. fiz., 24:6 (1984), 850–863 | MR

[9] Belolipetskii A. A., Ter-Krikorov A. M., “Postroenie fundamentalnykh reshenii abstraktnogo nelineinogo parabolicheskogo uravneniya v okrestnosti tochki bifurkatsii”, Matem. sb., 128:3 (1985), 306–320 | MR

[10] Trenogin V. A., Funktsionalnyi analiz, Fizmatlit, M., 2002