Geodesic
On the uniqueness problem for a central invariant manifold
Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 1, pp. 59-73 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a system of autonomous nonlinear ordinary differential equations for which the existence conditions for an invariant manifold are satisfied in the case where this manifold is central. It is well known that the theorem on the existence of a central invariant manifold cannot be supplemented with the statement of its uniqueness. We obtain sufficient conditions that guarantee the uniqueness of the central invariant manifold.
Keywords: central invariant manifold, uniqueness, asymptotic stability, Lyapunov function.
Mots-clés : existence 
@article{TMF_2024_220_1_a4,
     author = {A. N. Kulikov},
     title = {On the~uniqueness problem for a~central invariant manifold},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {59--73},
     year = {2024},
     volume = {220},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2024_220_1_a4/}
}
TY  - JOUR
AU  - A. N. Kulikov
TI  - On the uniqueness problem for a central invariant manifold
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2024
SP  - 59
EP  - 73
VL  - 220
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2024_220_1_a4/
LA  - ru
ID  - TMF_2024_220_1_a4
ER  - 
%0 Journal Article
%A A. N. Kulikov
%T On the uniqueness problem for a central invariant manifold
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2024
%P 59-73
%V 220
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2024_220_1_a4/
%G ru
%F TMF_2024_220_1_a4
A. N. Kulikov. On the uniqueness problem for a central invariant manifold. Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 1, pp. 59-73. http://geodesic.mathdoc.fr/item/TMF_2024_220_1_a4/

[1] V. A. Pliss, “Printsip svedeniya v teorii ustoichivosti dvizheniya”, Izv. AN SSSR. Ser. matem., 28:6 (1964), 1297–1324 | MR | Zbl

[2] Al. Kelley, “The stable, center-stable, center, center-unstable, unstable mamifolds”, J. Differ. Equ., 3:4 (1967), 546–570 | DOI | MR

[3] J. E. Marsden, M. McCraken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer, New York, 1976 | DOI | MR

[4] A. M. Lyapunov, Obschaya zadacha ob ustoichivosti dvizheniya, Gostekhizdat, M.–L., 1950 | MR | MR | Zbl

[5] I. G. Malkin, Teoriya ustoichivosti dvizheniya, Nauka, M., 1966 | MR | Zbl

[6] S. Lefshets, Geometricheskaya teoriya differentsialnykh uravnenii, IL, M., 1961 | MR | MR | Zbl

[7] Zh. Palis, V. Melu, Geometricheskaya teoriya dinamicheskikh sistem, Mir, M., 1986 | DOI | MR | MR | Zbl

[8] F. Khartman, Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | MR | Zbl | Zbl

[9] A. N. Kulikov, “O gladkikh invariantnykh mnogoobraziyakh polugruppy nelineinykh operatorov v banakhovom prostranstve”, Issledovaniya po ustoichivosti i teorii kolebanii, Sbornik nauchnykh trudov, Izd-vo YarGU, Yaroslavl, 1976, 114–129 | MR

[10] A. N. Kulikov, “Inertsialnye invariantnye mnogoobraziya nelineinoi polugruppy operatorov v gilbertovom prostranstve”, Materialy Vserossiiskoi nauchnoi konferentsii “Differentsialnye uravneniya i ikh prilozheniya”, posvyaschennoi 85-letiyu professora M. T. Terekhina (Ryazanskii gosudarstvennyi universitet im. S. A. Esenina, Ryazan, 17–18 maya 2019 g. Chast 2), Itogi nauki i tekhn. Ser. Sovrem. matem. i ee pril. Temat. obz., 186, VINITI, M., 2020, 57–66 | DOI

[11] V. I. Arnold, Dopolnitelnye glavy obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[12] A. N. Kulikov, “Odno zamechanie o svoistvakh dvumernykh invariantnykh mnogoobrazii”, Issledovanie po ustoichivosti i teorii kolebanii, Mezhvuz. sb., Izd-vo YarGU, Yaroslavl, 1978, 77–80

[13] A. N. Kulikov, “O edinstvennosti invariantnykh mnogoobrazii”, Issledovanie po ustoichivosti i teorii kolebanii, Mezhvuz. sb., Izd-vo YarGU, Yaroslavl, 1979, 81–85

[14] M. W. Hirsch, C. C. Pugh, M. Shub, “The local theory of normally hyperbolic, invariant, compact manifolds”, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, Berlin, 1977, 39–53 | DOI | MR

[15] A. N. Kulikov, D. A. Kulikov, A. V. Sekatskaya, Invariantnye mnogoobraziya: nekotorye prilozheniya k teorii dinamicheskikh sistem, Izd-vo “Filigran”, Yaroslavl, 2022

[16] E. A. Koddington, N. Levinson, Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958 | MR | Zbl

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

[18] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. Chua, Metody kachestvennoi teorii v nelineinoi dinamike, Izd-vo IKI, M.–Izhevsk, 2004 | MR | Zbl