Geodesic
Conservation and bifurcation of an invariant torus of a vector field
Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 34-44 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider small perturbations with respect to a small parameter $\varepsilon\ge0$ of a smooth vector field in $\mathbb R^{n+m}$ possessing an invariant torus $T_m$. The flow on the torus $T_m$ is assumed to be quasiperiodic with $m$ basic frequencies satisfying certain conditions of Diophantine type; the matrix $\Omega$ of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which $\Omega$ is a nonsingular matrix that can have purely imaginary eigenvalues. 
@article{MZM_1997_61_1_a4,
     author = {Yu. N. Bibikov},
     title = {Conservation and bifurcation of an invariant torus of a~vector field},
     journal = {Matemati\v{c}eskie zametki},
     pages = {34--44},
     year = {1997},
     volume = {61},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_1_a4/}
}
TY  - JOUR
AU  - Yu. N. Bibikov
TI  - Conservation and bifurcation of an invariant torus of a vector field
JO  - Matematičeskie zametki
PY  - 1997
SP  - 34
EP  - 44
VL  - 61
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1997_61_1_a4/
LA  - ru
ID  - MZM_1997_61_1_a4
ER  - 
%0 Journal Article
%A Yu. N. Bibikov
%T Conservation and bifurcation of an invariant torus of a vector field
%J Matematičeskie zametki
%D 1997
%P 34-44
%V 61
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1997_61_1_a4/
%G ru
%F MZM_1997_61_1_a4
Yu. N. Bibikov. Conservation and bifurcation of an invariant torus of a vector field. Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 34-44. http://geodesic.mathdoc.fr/item/MZM_1997_61_1_a4/

[1] Bogolyubov N. N., Mitropolskii Yu. A., Samoilenko A. M., Metod uskorenii skhodimosti v nelineinoi mekhanike, Naukova dumka, Kiev, 1969

[2] Bibikov Yu. N., Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii, Izd-vo LGU, L., 1991

[3] Bibikov Yu. N., “Postroenie invariantnykh torov sistem differentsialnykh uravnenii s malym parametrom”, Tr. LMO, 1 (1991), 26–53 | MR

[4] Hale J. K., “Integral manifolds of perturbed differential systems”, Ann. of Math., 73 (1961), 496–531 | DOI | MR | Zbl

[5] Moser J. K., “Convergent series expansions for quasiperiodic motions”, Math. Ann., 169 (1967), 136–176 | DOI | MR

[6] Bibikov Yu. N., “O suschestvovanii kvaziperiodicheskikh dvizhenii i kvazilineinykh sistem”, PMM, 59:1 (1995), 23–31 | MR

[7] Broer H. W., Huitema G. B., Takens F., “Unfolding of quasiperiodic tori”, Mem. Amer. Soc., 83 (1990), 1–82 | MR

[8] Bibikov Yu. N., “Bifurkatsiya ustoichivogo invariantnogo tora iz sostoyaniya ravnovesiya”, Matem. zametki, 48:1 (1990), 15–19 | MR | Zbl