Geodesic
On the possibility of implementing the Landau–Hopf scenario of transition to turbulence in the generalized model “multiplier-accelerator”
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Tome 203 (2021), pp. 39-49 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, we consider two boundary-value problems for the multiplier-accelerator model taking into account spatial effects. We show that, under an appropriate choice of the control parameter, invariant tori of increasing dimensions arise in both boundary-value problems and the invariant torus of the highest dimension is stable. Our results are based on such methods of the theory of dynamical systems with infinite-dimensional phase spaces as the method of integral manifolds, the Poincaré method of normal forms, and F. Takens' plan for implementing the Landau—Hopf scenario as a cascade of Andronov—Hopf bifurcations. For solutions that belong to invariant tori, we obtain asymptotic formulas.
Keywords: Landau–Hopf scenario, normal form, multiplier-accelerator, boundary-value problem.
Mots-clés : stable invariant torus, cascade of bifurcations 
@article{INTO_2021_203_a3,
     author = {A. N. Kulikov and D. A. Kulikov},
     title = {On the possibility of implementing the {Landau{\textendash}Hopf} scenario of transition to turbulence in the generalized model {\textquotedblleft}multiplier-accelerator{\textquotedblright}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {39--49},
     year = {2021},
     volume = {203},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_203_a3/}
}
TY  - JOUR
AU  - A. N. Kulikov
AU  - D. A. Kulikov
TI  - On the possibility of implementing the Landau–Hopf scenario of transition to turbulence in the generalized model “multiplier-accelerator”
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 39
EP  - 49
VL  - 203
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_203_a3/
LA  - ru
ID  - INTO_2021_203_a3
ER  - 
%0 Journal Article
%A A. N. Kulikov
%A D. A. Kulikov
%T On the possibility of implementing the Landau–Hopf scenario of transition to turbulence in the generalized model “multiplier-accelerator”
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 39-49
%V 203
%U http://geodesic.mathdoc.fr/item/INTO_2021_203_a3/
%G ru
%F INTO_2021_203_a3
A. N. Kulikov; D. A. Kulikov. On the possibility of implementing the Landau–Hopf scenario of transition to turbulence in the generalized model “multiplier-accelerator”. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Tome 203 (2021), pp. 39-49. http://geodesic.mathdoc.fr/item/INTO_2021_203_a3/

[1] Demidovich B. P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR

[2] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Razvitie turbulentnosti po Landau v modeli multiplikator-akselerator”, Dokl. RAN., 420:6 (2008), 739–743 | Zbl

[3] Kosareva E. S., Kulikov A. N., “Ob odnoi nelineinoi kraevoi zadache, modeliruyuschei ekonomicheskie protsessy”, Model. anal. inform. sistem., 10:2 (2003), 18–21

[4] Kulikov A. N., Primenenie metoda invariantnykh mnogoobrazii v lokalnykh zadachakh ustoichivosti i teorii kolebanii, Izd-vo YarGU, Yaroslavl, 1983

[5] Kulikov A. N., “Attraktory dvukh kraevykh zadach dlya modifitsirovannogo nelineinogo telegrafnogo uravneniya”, Nelin. dinam., 4:1 (2008), 57–68 | MR

[6] Landau L. D., “K problemam turbulentnosti”, Dokl. AN SSSR., 44:8 (1944), 339–342

[7] Landau L. D., Lifshits E. M., Teoreticheskaya fiziki. T. 6. Gidrodinamika, Nauka, M., 1988 | MR

[8] Sobolevskii P. E., “Ob uravneniyakh parabolicheskogo tipa v banakhovom prostranstve”, Tr. Mosk. mat. o-va., 10 (1967), 297–370

[9] Funktsionalnyi analiz. Spravochnaya matematicheskaya biblioteka, Nauka, M., 1972

[10] Yakubov S. Ya., “Razreshimost zadachi Koshi dlya abstraktnykh kvazilineinykh giperbolicheskikh uravnenii vtorogo poryadka i ikh prilozheniya”, Tr. Mosk. mat. o-va., 23 (1970), 37–60 | Zbl

[11] Broer H. W., Dumortier F., van Strien S. J., Takens F., Structure in Dynamics, Elsevier, 1991 | MR

[12] Hopf E., “A mathematical example displaying the features of turbulence”, Commun. Pure. Appl. Math., 1 (1948), 303–322 | DOI | MR | Zbl

[13] Kolesov A. Yu, Kulikov A. N., Rosov N. Kh., “Invariant tori of a class of point mappings: the annulus principle”, Differ. Equations., 39:5 (2003), 614–639 | DOI | MR

[14] Kolesov A. Yu, Kulikov A. N., Rosov N. Kh., “Invariant tori of a class of point transformations: preservation of an invariant torus under perturbations”, Differ. Equations., 39:6 (2003), 775–790 | DOI | MR | Zbl

[15] Kulikov A. N., “Landau–Hopf scenario of passage to turbulence in some problems of elastic stability theory”, Differ. Equations., 48:9 (2012), 1278–1291 | DOI | MR | Zbl

[16] Marsden J. E., McCraken M., The Hopf Bifurcation and Its Applications, Springer-Verlag, New-York, 1976 | MR | Zbl

[17] Puu T., Nonlinear Economic Dynamics, Springer-Verlag, Berlin, 1997 | MR | Zbl

[18] Segal I., “Nonlinear semigroups”, Ann. Math., 78:2 (1963), 339–364 | DOI | MR | Zbl

[19] Zhang W. B., Synergetic Economics: Time and Change in Nonlinear Economics, Springer-Verlag, Berlin, 1991 | MR | Zbl