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

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

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. 
@article{INTO_2019_168_a7,
     author = {M. I. kuptsov and V. A. Minaev and A. O. Faddeev and S. L. Yablochnikov},
     title = {On the stability of integral manifolds of a system of ordinary differential equations in the critical case},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {61--70},
     publisher = {mathdoc},
     volume = {168},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_168_a7/}
}
TY  - JOUR
AU  - M. I. kuptsov
AU  - V. A. Minaev
AU  - A. O. Faddeev
AU  - S. L. Yablochnikov
TI  - On the stability of integral manifolds of a system of ordinary differential equations in the critical case
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 61
EP  - 70
VL  - 168
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_168_a7/
LA  - ru
ID  - INTO_2019_168_a7
ER  - 
%0 Journal Article
%A M. I. kuptsov
%A V. A. Minaev
%A A. O. Faddeev
%A S. L. Yablochnikov
%T On the stability of integral manifolds of a system of ordinary differential equations in the critical case
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 61-70
%V 168
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_168_a7/
%G ru
%F INTO_2019_168_a7
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/