Geodesic
On the stability of integral manifolds of a system of ordinary differential equations in the critical case
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 61-70.

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In this paper, we consider the stability problem for nonzero integral manifolds of a nonlinear, finite-dimensional system of ordinary differential equations whose right-hand side is a periodic vector-valued function with respect to an independent variable containing a parameter. We assume that the system possesses a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the exponential dichotomy property. We find sufficient conditions for the existence of a nonzero integral manifold in a neighborhood of the equilibrium of the system and conditions for its stability or instability. For this purpose, based of the ideas of the Lyapunov method and the method of transform matrices, we construct operators that allow one to reduce the solution of this problem to the search for fixed points.
Keywords: Lyapunov method, method of transform matrices, stability of integral manifolds, system of ordinary differential equations, operator equation. 
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M. I. kuptsov; V. A. Minaev; A. O. Faddeev; S. L. Yablochnikov. On the stability of integral manifolds of a system of ordinary differential equations in the critical case. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 61-70. http://geodesic.mathdoc.fr/item/INTO_2019_168_a7/

[1] Bibikov Yu. N., Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii, Izd-vo LGU, L., 1991

[2] Bogolyubov N. N., O nekotorykh statisticheskikh metodakh v matematicheskoi fizike, Izd-vo AN USSR, Lvov, 1945 | MR

[3] Volkov D. Yu., “Bifurkatsiya invariantnykh torov iz sostoyaniya ravnovesiya pri nalichii nulevykh kharakteristicheskikh chisel”, Vestn. Leningr. un-ta., 1:2 (1988), 102–103

[4] Kuptsov M. I., “Lokalnoe integralnoe mnogoobrazie sistem differentsialnykh uravnenii, zavisyaschikh ot parametra”, Differ. uravn., 35:11 (1999), 1579–1580

[5] Kuptsov M. I., “Suschestvovanie integralnogo mnogoobraziya sistemy differentsialnykh uravnenii”, Differ. uravn., 34:6 (1998), 855 | MR

[6] Kuptsov M. I., Suschestvovanie integralnykh mnogoobrazii i periodicheskikh reshenii sistemy obyknovennykh differentsialnykh uravnenii, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Udmurt. gos. un-t, Izhevsk, 1997

[7] Kuptsov M. I., Terekhin M. T., Tenyaev V. V., “K probleme suschestvovaniya integralnykh mnogoobrazii sistemy differentsialnykh uravnenii, ne razreshennykh otnositelno proizvodnykh”, Zh. Srednevolzh. mat. ob-va., 19:2 (2017), 76–84

[8] Kurbanshoev S. Z., Nusairiev M. A., “Postroenie optimalnykh integralnykh mnogoobrazii dlya nelineinykh differentsialnykh uravnenii”, Dokl. AN Resp. Tadzhikistan., 57:11–12 (2014), 807–812

[9] Mitropolskii Yu. A., Lykova O. B., Integralnye mnogoobraziya v nelineinoi mekhanike, Nauka, M., 1973

[10] Rumyantsev V. V., Oziraner A. S., Ustoichivost i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh, Nauka, M., 1987

[11] Samoilenko A. M., Elementy matematicheskoi teorii mnogochastotnykh kolebanii. Invariantnye tory, Nauka, M., 1987 | MR

[12] Kuptsov M. I., “Local integral manifold of a system of differential equations”, Differ. Equations., 34:7 (1998), 1005–1007 | MR | Zbl

[13] Kuptsov M. I., Minaev V. A., “Stability integral manifold of the differential equations system in critical case”, J. Phys. Conf. Ser., 973 (2018), 012055 | DOI

[14] Prigogine I., “The philosophy of instability”, Futures., 1989, no. 21, 396–400 | DOI

[15] Sobolev V., “Slow integral manifolds and control problems in critical and twice critical cases”, J. Phys. Conf. Ser., 727 (2016), 012017 | DOI | MR