Inclusion principle and stability-like properties of “partial” equilibrium position of a dynamical system

In numerous applications, dynamical mathematical models contain subsystems with common parts. In such cases, the extension of phase space of the studied system is used. This extension transforms an initial dynamical system to a system whose subsystems have no common parts. The considered dynamical systems can possess both equilibrium positions in the classical sense and so-called “partial” equilibrium positions. After investigation stability of extended systems equilibrium positions the obtained results are transferred to the initial system by means of phase space constriction. The important problem is that one of determinating conditions under which the realization of the extension-constriction process is possible. These conditions compose the basis of the inclusion principle. In the present paper the stability-like properties with respect to all or to a part of variables of a “partial” equilibrium position of a differential equations system are studied. The conditions are obtained under which these properties can be investigated by the use of the inclusion principle. On the example of the differential equations system with homogeneous right-hand sides of the order $\mu=3,5,\dots$ the technique of the inclusion principle application for the analysis of stability-like properties is demonstrated. The example of the system for which we failed to prove the asymptotic stability without using the inclusion principle is given.