Method of Lyapunov Functions in the Problem of Stability of Integral Manifolds of a System of Ordinary Differential Equations

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 283; no. 3; pp. 419 - 427
• Main Authors: Kuptsov, M. I., Minaev, V. A., Maskina, M. S.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 10.08.2024; Springer; Springer Nature B.V
• Subjects: Analysis; Asymptotic methods; Differential equations; Independent variables; Liapunov functions; Linear systems; Manifolds; Mathematics; Mathematics and Statistics; Methods; Parameter modification; Singularity (mathematics); Stability; 34C25; 34D35; 34C45; method of Lyapunov functions; 34A34; instability; integral manifold; asymptotic stability; stability; system of ordinary differential equations
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-024-07270-2

We consider the problem of stability of nonzero integral manifolds of a nonlinear finitedimensional 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.