On stability with respect to part of variables in some critical cases

We consider the problem of stability with respect to a part of variables in critical cases, when it is necessary to take into account nonlinear summands in series expansions of the right-hand sides of equations. This problem is nonlocal because of presence of uncontrolled variables (the stability with respect to them is not analyzed), and has a number of features that complicate the study in comparison with the analogous problem of stability with respect to all variables. We discuss an analogue of the Lyapunov reduction principle as applied to this problem. Two situations, differing in the way for critical variables entering the equations for non-critical variables, are distinguished. We propose the signs of stability, asymptotic stability and instability with respect to a part of variables. They are established basing on similar properties of auxiliary systems of smaller dimension. For the case when the characteristic equation for the linear approximation system has several zero roots we obtain the conditions of asymptotic stability with respect to a part of variables, which are established on the basis of stable subsystems with homogeneous right-hand side. For the proof, the sign-constant scalar Lyapunov functions as well as vector Lyapunov – Matrosov functions and the comparison method are used. In order to compare our results with known ones, we present a number of examples that show the effectiveness of the application of the proved theorems.