Geodesic
Normal Forms near Two-Dimensional Resonance Tori for the Multidimensional Anharmonic Oscillator
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 701-713.

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

We study the problem of constructing a normal form near a two-dimensional invariant isotropic torus for a multidimensional anharmonic oscillator. We construct a fourth-order normal form with respect to harmonic oscillator type variables. We show that, in the presence of resonances, the dependence of the normal form on the action type variables becomes nonpolynomial. 
@article{MZM_2004_76_5_a5,
     author = {S. Yu. Dobrokhotov and M. A. Poteryakhin},
     title = {Normal {Forms} near {Two-Dimensional} {Resonance} {Tori} for the {Multidimensional} {Anharmonic} {Oscillator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {701--713},
     publisher = {mathdoc},
     volume = {76},
     number = {5},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a5/}
}
TY  - JOUR
AU  - S. Yu. Dobrokhotov
AU  - M. A. Poteryakhin
TI  - Normal Forms near Two-Dimensional Resonance Tori for the Multidimensional Anharmonic Oscillator
JO  - Matematičeskie zametki
PY  - 2004
SP  - 701
EP  - 713
VL  - 76
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a5/
LA  - ru
ID  - MZM_2004_76_5_a5
ER  - 
%0 Journal Article
%A S. Yu. Dobrokhotov
%A M. A. Poteryakhin
%T Normal Forms near Two-Dimensional Resonance Tori for the Multidimensional Anharmonic Oscillator
%J Matematičeskie zametki
%D 2004
%P 701-713
%V 76
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a5/
%G ru
%F MZM_2004_76_5_a5
S. Yu. Dobrokhotov; M. A. Poteryakhin. Normal Forms near Two-Dimensional Resonance Tori for the Multidimensional Anharmonic Oscillator. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 701-713. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a5/

[1] Arnold V. I., “Malye znamenateli i problemy ustoichivosti v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6 (1963), 91–192 | MR | Zbl

[2] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974

[3] Broer H. W., Huitema G. B., Sevryuk M. B., Quasi-Periodic Motions in Families of Dynamical Systems, Lecture Note in Math., 1645, Springer, Berlin, 1996 | MR

[4] Bryuno A. D., Ogranichennaya zadacha trekh tel: ploskie periodicheskie orbity, Nauka, M., 1990 | Zbl

[5] Dzhakalya G. E. O., Metody teorii vozmuschenii dlya nelineinykh sistem, Nauka, M., 1979

[6] Jorba A., Llave R. de la, Zou M., “Linstedt series for lower dimensional tori of hamiltonian systems with three or more degrees of freedom”, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci. (S'Agaro, Spain, June 1995), ed. C. Simo, 1995

[7] Rüssmann H., “Invariant tori in non-degenerate nearly integrable Hamiltonian systems”, Regular and Chaotic Dynamics, 6:2 (2001), 119–204 | DOI | MR | Zbl

[8] Belov V. V., Dobrokhotov O. S., Dobrokhotov S. Yu., “Izotropnye tory, kompleksnyi rostok i indeks Maslova, normalnye formy i kvazimody v mnogomernykh spektralnykh zadachakh”, Matem. zametki, 69:4 (2001), 483–514 | MR | Zbl

[9] Belov V. V., Dobrokhotov S. Yu., Maksimov V. A., “Yavnye formuly dlya obobschennykh peremennykh deistvie–ugol v okrestnosti izotropnogo tora i ikh primenenie”, TMF, 135:3 (2003), 378–408 | MR

[10] Yakubovich V. A., Starzhinskii V. M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami, Nauka, M., 1972

[11] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody v teorii kolebanii, Nauka, M., 1974

[12] Karasev M. V., Kvantovaya reduktsiya na orbity algebr, Preprint ITF-87-157R, In-t teor. fiz. AN USSR, Kiev, 1988

[13] Arnold V. I., Givental A. B., Simplekticheskaya geometriya, RKhD, Izhevsk, 2000

[14] Dobrokhotov S. Yu., Shafarevich A. I, “Quantum selection of isotropic partially integrable Hamiltonian systems in quasiclassical approximation”, Russian J. Math. Phys., 5:2 (1998), 267–272 | MR

[15] Dobrokhotov S. Yu., Olive V. M., “Zamknutye traektorii i dvumernye tory v kvantovoi probleme dlya trekhmernogo angarmonicheskogo ostsillyatora”, Tr. MMO, 58, URSS, M., 1997, 3–87 | MR

[16] Dobrokhotov S. Yu., Olive Martinez V., Shafarevich A. I, “Closed trajectories and two-dimensional tori in the quantum problem for a three-dimensional resonant anharmonic oscillator”, Russian J. Math. Phys., 3:1 (1995), 133–138 | MR | Zbl

[17] Karasev M. V., “Novye globalnye asimptotiki i anomalii v zadachakh kvantovaniya adiabaticheskikh invariantov”, Funktsion. analiz i ego prilozh., 24:2 (1990), 24–36 | MR | Zbl