Stability integral manifold of the differential equations system in critical case

We consider the stability problem for non-zero integral manifolds of a non-linear finite-dimensional system of ordinary differential equations, where the right-hand side is a periodic vector function with respect to an independent variable and contains a parameter. It is assumed that the studied system has a trivial integral manifold for all values of the parameter, and the corresponding linear subsystem does not have the property of exponential dichotomy. The aim of the paper is to find sufficient conditions of existence in a neighborhood of the system equilibrium state for stable non-zero integral manifold to be lower dimension than the original phase space. For this purpose, based on the classical method of Lyapunov functions and the transforming matrix method operators are constructed, allowing solve the task by finding their fixed points. Due to the specific nature of the considered systems Lyapunov functions method is modified. (paper)