On the Stability of Integral Manifolds of a System of Ordinary Differential Equations in the Critical Case

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 262; no. 6; pp. 825 - 834
• Main Authors: Kuptsov, M. I., Minaev, V. A., Faddeev, A. O., Yablochnikov, S. L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 16.04.2022; Springer; Springer Nature B.V
• Subjects: Differential equations; Fixed points (mathematics); Independent variables; Linear systems; Manifolds (mathematics); Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Parameters; Stability; operator equation; 34C25; 34D35; method of transform matrices; 34C45; 34A34; Lyapunov method; stability of integral manifolds; system of ordinary differential equations
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-022-05861-5

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 vector-valued function containing a parameter and periodic in an independent variable. 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.