Geodesic
On a Problem of V.~V.~Nemytskii
Matematičeskie zametki, Tome 113 (2023) no. 2, pp. 182-196.

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We study trajectories in a neighborhood of attractors and weak attractors of dynamical systems on a metric space. The properties of elliptic and weakly elliptic points of compact invariant sets are studied. A solution of the generalized problem of V. V. Nemytskii concerning the existence of compact invariant sets of weakly elliptic type for the case of asymptotically compact dynamical systems is given.
Keywords: dynamical system, attraction
Mots-clés : invariant set, elliptic point. 
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B. S. Kalitin. On a Problem of V.~V.~Nemytskii. Matematičeskie zametki, Tome 113 (2023) no. 2, pp. 182-196. http://geodesic.mathdoc.fr/item/MZM_2023_113_2_a2/

[1] V. V. Nemytskii, V. V. Stepanov, Kachestvennaya teoriya differentsialnykh uravnenii, GITTL, M.–L., 1949 | MR

[2] V. V. Nemytskii, “Topologicheskaya klassifikatsiya osobykh tochek mnogomernykh sistem.”, Tezisy kratkikh nauchnykh soobschenii “Mezhdunarodnyi kongress matematikov”, 1S1. Sektsiya 6, M., 1966, 40

[3] V. V. Nemytskii, “Topologicheskaya klassifikatsiya osobykh tochek i obobschennye funktsii Lyapunova”, Differents. uravneniya, 3:3 (1967), 359–370 | MR | Zbl

[4] L. A. Chelysheva, “Topologicheskaya klassifikatsiya zamknutykh invariantnykh mnozhestv”, Matem. issledovaniya, 3, no. 1, Shtiintsa, Kishinev, 1968, 184–197

[5] N. N. Ladis, “Otsutstvie dvustoronnikh tochek prityazheniya”, Differents. uravneniya, 4:6 (1968), 1157 | MR | Zbl

[6] M. S. Izman, “Topologicheskaya klassifikatsiya zamknutykh invariantnykh mnozhestv”, Matem. issledovaniya, 3, no. 3, RIO AN MSSR, Kishinev, 1968, 60–78 | MR

[7] T. Saito, “Isolated minimal sets”, Funkc. Ekvac., 11:3 (1969), 155–167 | MR

[8] K. S. Sibirskii, Vvedenie v topologicheskuyu dinamiku, RIO AN MSSR, Kishinev, 1970 | MR

[9] N. P. Bhatia, G. Szegö, Stability Theory of Dynamical Systems, Springer, Berlin, 1970 | MR

[10] S. H. Saperstone, Semidynamical Systems in Infinite Dimentional Space, Springer, Berlin, 1981 | MR

[11] V. I. Zubov, Ustoichivost dvizheniya (metody Lyapunova i ikh primenenie), Vysshaya shkola, M., 1984 | MR

[12] B. S. Kalitin, “Ustoichivost po Lyapunovu i orbitalnaya ustoichivost dinamicheskikh sistem”, Differents. uravneniya, 40:8 (2004), 1033–1042 | MR

[13] B. S. Kalitin, “O strukture okrestnosti ustoichivykh kompaktnykh invariantnykh mnozhestv”, Differents. uravneniya, 41:8 (2005), 1062–1073 | MR

[14] B. S. Kalitin, “Neustoichivost zamknutykh invariantnykh mnozhestv poludinamicheskikh sistem. Metod znakopostoyannykh funktsii Lyapunova”, Matem. zametki, 85:3 (2009), 382–394 | DOI | MR | Zbl

[15] B. S. Kalitin, Kachestvennaya teoriya ustoichivosti dvizheniya dinamicheskikh sistem, BGU, Minsk, 2002

[16] N. P. Bhatia, “Attraction and non-saddle sets in dynamical systems”, J. Differential Equations, 8 (1970), 229–249 | DOI | MR

[17] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge, Cambridge Univ. Press., 1991 | MR

[18] B. S. Kalitin, “Kharakteristika kompaktnykh slabo prityagivayuschikh polozhitelno invariantnykh mnozhestv”, Differents. uravneniya, 24:6 (1988), 1085

[19] B. S. Kalitin, “O strukture okrestnosti slabo prityagivayuschikh kompaktnykh mnozhestv”, Differents. uravneniya, 30:4 (1994), 565–574 | MR

[20] B. S. Kalitin, “Kachestvennaya kharakteristika traektorii v okrestnosti prityagivayuschego kompaktnogo mnozhestva”, Differents. uravneniya, 34:7 (1998), 886–893 | MR

[21] B. S. Kalitin, “O svoistvakh okrestnosti attraktora dinamicheskoi sistemy”, Matem. zametki, 109:5 (2021), 734–746 | DOI

[22] B. S. Kalitin, Ustoichivost dinamicheskikh sistem (Kachestvennaya teoriya), LAP Lambert Academic Publishing, Saarbryuken, 2012

[23] B. S. Kalitine, “About asymptotic stability in semi-dynamical systems”, Vestnik BGU. Ser. 1, 2016, no. 1, 114–119

[24] J. H. Arredondo, P. Seibert, “On a characterization of asymptotic stability”, National Congress of the Mexican Mathematical Society, Aportaciones Mat. Comun., 29, Soc. Mat. Mexicana, México, 2001, 11–16 | MR

[25] N. N. Ladis, “Topologicheskaya ekvivalentnost nekotorykh differentsialnykh sistem”, Differents. uravneniya, 8:6 (1972), 1116–1119 | MR | Zbl

[26] I. G. Petrovskaya, “$TM$-ekvivalentnost i klassifikatsiya nekompaktnykh invariantnykh mnozhestv dinamicheskikh sistem”, Differents. uravneniya, 17:2 (1981), 233–240 | MR | Zbl

[27] B. A. Scherbakov, “Klassy ustoichivykh po Lyapunovu dinamicheskikh sistem vozmuschennykh dvizhenii”, Differents. uravneniya, 17:12 (1981), 2195–2200 | MR | Zbl

[28] B. S. Kalitin, “Klassifikatsiya kompaktnykh invariantnykh mnozhestv”, Dokl. AN BSSR, 31:11 (1987), 971–974 | MR

[29] B. S. Kalitin, “Invariantnost svoistv ustoichivosti pri gomomorfizme dinamicheskikh sistem”, Dokl. AN BSSR, 31:11 (1987), 869–871

[30] A. F. Filippov, “Klassifikatsiya kompaktnykh invariantnykh mnozhestv dinamicheskikh sistem”, Izv. RAN. Ser. matem., 57:6 (1993), 130–140 | MR | Zbl