Method of Lyapunov functions in the problem of stability of integral manifolds of a system of ordinary differential equations (Q6588111)

In the article the authors study the problem of stability of nonzero integral manifolds of a nonlinear finite dimensional system of ordinary differential equations under following three assumptions: (a) The right-hand side is a periodic vector-valued function of the independent variable containing a parameter; (b) The system has a trivial integral manifold for all values of the parameter; The corresponding linear subsystem does not possess the property of exponential dichotomy. Note that in this case the method of integral manifolds developed by \textit{N. Bogolyubov} [On some statistical methods in mathematical physics (О некоторых статистических методах в математической физике) (Russian). Kiev: Akad. Nauk Ukrainskiĭ SSR (1945; Zbl 0063.00496)], this approach cannot be implemented since the system has a zero integral manifold \(x = 0\) for all values of the parameter, and the system is a parametric family of periodic solutions. In this case, the problems of stability of the resulting manifold is solved due to the boundedness of the Green function. The results presented in the paper are obtained via a modification of the method of Lyapunov functions, which allows one to obtain new sufficient conditions for the existence of a stable, instable, and asymptotically stable local integral manifold of the studied system.