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

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

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--Hopf} scenario of transition to turbulence in the generalized model ``multiplier-accelerator''},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {39--49},
     publisher = {mathdoc},
     volume = {203},
     year = {2021},
     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
PB  - mathdoc
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
%I mathdoc
%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/