Geodesic
Method of Lyapunov functions in the problem of stability of integral manifolds of a system of ordinary 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 2, Tome 186 (2020), pp. 74-82

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We consider the problem of stability of nonzero integral manifolds of a nonlinear finite-dimensional system of ordinary differential equations whose right-hand side is a periodic vector-valued function of the independent variable containing a parameter. We assume that the system has a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the property of exponential dichotomy. The aim of this work is to find sufficient conditions for stability, instability, and asymptotic stability of a local nonzero integral manifold. For this purpose, we use the method of Lyapunov functions modified to the problem considered and singularities of the right-hand sides of the system.
Keywords: method of Lyapunov functions, stability, asymptotic stability, instability, integral manifold, system of ordinary differential equations. 
@article{INTO_2020_186_a9,
     author = {M. I. kuptsov and V. A. Minaev and M. S. Maskina},
     title = {Method of {Lyapunov} functions in the problem of stability of integral manifolds of a system of ordinary differential equations},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {74--82},
     publisher = {mathdoc},
     volume = {186},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_186_a9/}
}
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M. I. kuptsov; V. A. Minaev; M. S. Maskina. Method of Lyapunov functions in the problem of stability of integral manifolds of a system of ordinary 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 2, Tome 186 (2020), pp. 74-82. http://geodesic.mathdoc.fr/item/INTO_2020_186_a9/