Geodesic
Orbital stability of a small periodic solution of an autonomous system of differential equations
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 3-12.

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

We consider a normal autonomous system of differential equations with a small parameter, which has a critical linear approximation at zero value of the parameter. We introduce the concept of orbital stability with respect to the parameter; according to this concept, the closeness of the right semitrajectories is achieved not only due to the proximity of initial values of solutions, but also due to the smallness of the parameter. We examine the problem of branching of a stable periodic solution with a period close to the period of solutions of the corresponding linear homogeneous system. Sufficient conditions for the solvability of the problem are established. Our reasonings are based on the properties of the first homogeneous nonlinear approximation of the monodromy operator.
Keywords: autonomous system of differential equations, small parameter, qualitative theory, small periodic solution, orbital stability, parameter stability, monodromy operator. 
@article{INTO_2020_185_a0,
     author = {V. V. Abramov},
     title = {Orbital stability of a small periodic solution of an autonomous system of differential equations},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--12},
     publisher = {mathdoc},
     volume = {185},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_185_a0/}
}
TY  - JOUR
AU  - V. V. Abramov
TI  - Orbital stability of a small periodic solution of an autonomous system of differential equations
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 3
EP  - 12
VL  - 185
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_185_a0/
LA  - ru
ID  - INTO_2020_185_a0
ER  - 
%0 Journal Article
%A V. V. Abramov
%T Orbital stability of a small periodic solution of an autonomous system of differential equations
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 3-12
%V 185
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_185_a0/
%G ru
%F INTO_2020_185_a0
V. V. Abramov. Orbital stability of a small periodic solution of an autonomous system of differential equations. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 3-12. http://geodesic.mathdoc.fr/item/INTO_2020_185_a0/

[9] Abramov V. V., “Nenulevoe periodicheskoe reshenie nelineinoi sistemy differentsialnykh uravnenii”, Differ. uravn., 33:11 (1997), 1572

[10] Abramov V. V., “K voprosu ob ustoichivosti resheniya po parametru”, Izv. TulGU. Ser. Differ. uravn. Prikl. zadachi., 1 (2005), 3–8

[11] Abramov V. V., “Ustoichivost nulevogo resheniya periodicheskoi sistemy differentsialnykh uravnenii s malym parametrom”, Zh. Srednevolzh. mat. o-va., 12:4 (2010), 49–54 | MR | Zbl

[12] Abramov V. V., “Ustoichivost malogo periodicheskogo resheniya”, Vestn. RAEN., 13:4 (2013), 3–5

[13] Abramov V. V., “Dvustoronnyaya ustoichivost malogo periodicheskogo resheniya”, Vestn. RAEN., 14:5 (2014), 6–9

[14] Demidovich B. P., “Ob odnom analoge teoremy Andronova––Vitta”, Dokl. AN SSSR., 176:5 (1967), 994–996 | Zbl

[15] Dunaeva O. V., Shestakov A. A., “O ponyatiyakh orbitalnoi ustoichivosti i fazovoi ustoichivosti dvizhenii dinamicheskoi sistemy”, Dokl. SSSR., 335:3 (1997), 33–341

[16] Zubov V. I., Teoriya kolebanii, Vysshaya shkola, M., 1979

[17] Krasnoselskii M. A., Kuznetsov N. A., Yumagulov M. G., “Operatornyi metod analiza ustoichivosti tsiklov pri bifurkatsii Khopfa”, Avtomat. telemekh., 12 (1996), 15–24

[18] Malkin I. G., Nekotorye zadachi teorii nelineinykh kolebanii, GITTL, M., 1956

[19] Mamonov S. S., Kharlamova A. O., “Vynuzhdennaya sinkhronizatsiya sistem fazovoi sinkhronizatsii s zapazdyvaniem”, Vestn. RGRTU., 62 (2017), 26–35

[20] Mamonov S. S., Kharlamova A. O., Ionova I. V., “Kolebatelno-vraschatelnye tsikly fazovoi sistemy differentsialnykh uravnenii”, Vestn. RAEN., 18:4 (2018), 51–57

[21] Khapaev M. M., Usrednenie v teorii ustoichivosti, Nauka, M., 1986 | MR

[22] Khessard B., Kazarinov N., Ven I., Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Mir, M., 1985