Geodesic
On the stability of the zero solution with respect to a part of variables in linear approximation
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 3, pp. 374-390 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The article presents the sufficient conditions for stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation. the case is considered when the matrix of the linear approximation may contain eigenvalues with zero real parts and the algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the investigated system and its linear approximation. The solutions of such systems starting in a sufficiently small zero neighborhood and the systems themselves possess the same componentwise asymptotic properties in this case. Such solutions' properties are stability and asymptotic stability with respect to some variables, and for systems componentwise local asymptotic equivalence and componentwise local asymptotic equilibrium. Considering the correspondence between the solutions of systems as an operator defined in a Banach space, there is proved that it has at least one fixed point according to the Schauder's principle. The operator allows to construct a mapping that establishes the relationship between the initial points of the investigated system and its linear approximation. Further, a conclusion about the componentwise asymptotic properties of solutions of the nonlinear system is made on the basis of estimates of the fundamental matrix of the linear approximation rows' entries. There is given an example of the investigation of stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system is given, when the linear approximation matrix contains one negative and one zero eigenvalues, and the algebraic and geometric multiplicities of the zero eigenvalue do not coincide.
Keywords: ordinary differential equations, partial stability, local componentwise asymptotic equivalence, Schauder principle. 
@article{VSPUI_2023_19_3_a5,
     author = {P. A. Shamanaev},
     title = {On the stability of the zero solution with respect to a part of variables in linear approximation},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {374--390},
     year = {2023},
     volume = {19},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2023_19_3_a5/}
}
TY  - JOUR
AU  - P. A. Shamanaev
TI  - On the stability of the zero solution with respect to a part of variables in linear approximation
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2023
SP  - 374
EP  - 390
VL  - 19
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2023_19_3_a5/
LA  - ru
ID  - VSPUI_2023_19_3_a5
ER  - 
%0 Journal Article
%A P. A. Shamanaev
%T On the stability of the zero solution with respect to a part of variables in linear approximation
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2023
%P 374-390
%V 19
%N 3
%U http://geodesic.mathdoc.fr/item/VSPUI_2023_19_3_a5/
%G ru
%F VSPUI_2023_19_3_a5
P. A. Shamanaev. On the stability of the zero solution with respect to a part of variables in linear approximation. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 3, pp. 374-390. http://geodesic.mathdoc.fr/item/VSPUI_2023_19_3_a5/

[1] Lyapunov A. M., Study of one of the special cases of the problem of stability of motion, Leningrad State University Press, L., 1963, 116 pp. (In Russian)

[2] Malkin I. G., “On motions stability of Liapounov' sense”, Mat. Sbornik N. S.,, 3(45):1 (1938), 47–101 (In Russian) | MR | Zbl

[3] Rumyantsev V. V., “On motion stability with respect to a part of variables”, Vestnik of Moscow University. Series Mathematics. Mechanics. Astronomy. Physics. Chemistry, 1957, no. 4, 9–16 (In Russian) | MR

[4] Rumyantsev V. V., Oziraner A. S., Stability and stabilization of motion with respect to a part of variables, Nauka Publ, M., 1987, 253 pp. (In Russian) | MR

[5] Vorotnikov V. I., Stability of dynamical systems with respect to a part of variables, Nauka Publ, M., 1991, 288 pp. (In Russian) | MR

[6] Oziraner A. S., “On asymptotic stability and instability with respect to a part of the variables”, Applied Mathematics and Mechanics, 37:4 (1973), 659–665 (In Russian) | MR | Zbl

[7] Prokopiev V. P., “On the stability of motion with respect to a part of variables in the critical case of one zero root”, Applied Mathematics and Mechanics, 39:3 (1975), 422–426 (In Russian) | MR | Zbl

[8] Malkin I. G., Theory of stability of motion, Nauka Publ, M., 1966, 530 pp. (In Russian) | MR

[9] Oziraner A. S., “On stability of motion in critical cases”, Applied Mathematics and Mechanics, 39:3 (1975), 415–421 (In Russian) | MR | Zbl

[10] Shchennikov V. N., “On partial stability in the critical case of 2$k$ purely imaginary roots”, Differential and integral equations: Methods of topological dynamics, Gorkiy State University named after N. I. Lobachevsky, Gorkiy, 1985, 46–50 (In Russian)

[11] Shchennikov V. N., “Investigation of the stability with respect to a part of the variables of differential systems with homogeneous right-hand sides”, Differential Equations, 20:9 (1984), 1645–1649 (In Russian) | MR

[12] Voskresenskiy E. V., Asymptotic methods: theory and applications, Middle Volga Mathematical Society Publ, Saransk, 2000, 300 pp. (In Russian)

[13] Voskresenskiy E. V., Comparison methods in nonlinear analysys, Saransky University Press, Saransk, 1990, 224 pp. (In Russian) | MR

[14] Yazovtseva O. S., “The local component-wise asymptotic equivalence and its application to investigate for stability with respect to a part of variables”, Ogarev-online, 2017, no. 13 (In Russian) (accessed: March 19, 2023) | Zbl

[15] Shamanaev P. A., Yazovtseva O. S., “ The sufficient conditions of local asymptotic equivalence of nonlinear systems of ordinary differential equations and its application for investigation of stability respect to part of variables”, Middle Volga Mathematical Society Journal, 19:1 (2017), 102–115 (In Russian) | Zbl

[16] Shamanaev P. A., Yazovtseva O. S., “The sufficient conditions for polystability of solutions of nonlinear systems of ordinary differential equations”, Middle Volga Mathematical Society Journal, 20:3 (2018), 304–317 (In Russian) | Zbl

[17] Shamanaev P. A., Yazovtseva O. S., “Studying the equilibrium state stability of the biocenosis dynamics system under the conditions of interspecies interaction”, Mordovia University Bulletin Journal, 28:3 (2018), 321–332 (In Russian)

[18] Shamanaev P. A., Yazovtseva O. S., Partial stability of equilibrium positions of dynamical systems, Preprint No 127, Middle Volga Mathematical Society Publ, Saransk, 2018, 20 pp. (In Russian) | Zbl

[19] Aleksandrov A. Yu., Stability of motions of non-autonomous dynamical systems, St. Petersburg University Press, St. Petersburg, 2004, 184 pp. (In Russian)

[20] Aleksandrov A. Yu., Zhabko A. P., “On the asymptotic stability of solutions to nonlinear systems with delay”, Siberian Mathematical Journal, 53:3 (2012), 495–508 (In Russian) | MR | Zbl

[21] Ekimov A. V., Chizhova O. N., Zaranik U. P., “Stability of homogeneous non-stationary systems of differential-difference equations with a linearly increasing delay”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 15:4 (2019), 415–424 (In Russian) | DOI | MR

[22] Bylov B. F., Vinograd R. E., Grobman D. M., Nemytskii V. V., Theory of Lyapunov exponents and its applications to stability problems, Nauka Publ, M., 1966, 576 pp. (In Russian) | MR

[23] Trenogin V. A., Functional analysis, Nauka Publ, M., 1980, 249 pp. (In Russian) | MR