Geodesic
The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 3, pp. 295-303.

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

The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.
Keywords: nonlinear ordinary differential equations, Lie algebra, Lie groups, partial stability
Mots-clés : decomposition. 
@article{SVMO_2018_20_3_a3,
     author = {V. I. Nikonov},
     title = {The application of {Lie} algebras and groups to the solution of problems of partial stability of dynamical systems},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {295--303},
     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a3/}
}
TY  - JOUR
AU  - V. I. Nikonov
TI  - The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2018
SP  - 295
EP  - 303
VL  - 20
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a3/
LA  - ru
ID  - SVMO_2018_20_3_a3
ER  - 
%0 Journal Article
%A V. I. Nikonov
%T The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2018
%P 295-303
%V 20
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a3/
%G ru
%F SVMO_2018_20_3_a3
V. I. Nikonov. The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 3, pp. 295-303. http://geodesic.mathdoc.fr/item/SVMO_2018_20_3_a3/

[1] V. I. Vorotnikov, V. V. Rumyantsev, Stability and control with respect to part of the coordinates of the phase vector of dynamical systems: theory, methods and applications, Nauchnyy Mir, Moscow, 2001, 320 pp. (In Russ.) | MR

[2] V. V. Rumyantsev, A. S. Oziraner, Stability and stabilization of motion with respect to a part of the variables, Nauka, Moscow, 1987, 256 pp. (In Russ.)

[3] V. P. Prokopyev, “On stability with respect to some of the variables in the critical case of a single zero root.”, Journal of Applied Mathematics and Mechanics, 39 (1975), 422-426 (In Russ.) | MR | Zbl

[4] V. I. Vorotnikov, Stability of dynamical systems with respect to a part of variables, Nauka, Moscow, 1991, 288 pp. (In Russ.)

[5] V. I. Vorotnikov, “On the stability of dynamical systems with respect to a part of variables”, Journal of Applied Mathematics and Mechanics, 43 (1979), 441-450 (In Russ.) | MR | Zbl

[6] P. Olver, Applications of Lie groups to differential equations, Mir, Moscow, 1989, 639 pp. (In Russ.)

[7] U. A. Mitropolsky, A. K.Lopatin, The group-theoretical approach in asymptotic methods of nonlinear mechanics, Naukova dumka, Kiev, 1987, 270 pp. (In Ukr.) | MR

[8] V. I. Ovsjannikov, Group analysis of differential equations, Nauka, Moscow, 1978, 399 pp. (In Russ.)