Asymptotic stability of homogeneous singular systems

Introduction. The paper provides an overview of singularly perturbed systems of ordinary differential equations with a homogeneous right-hand side of rational degree. The subject of the study is the asymptotic stability of the zero solution of these systems for sufficiently small values of the parameter. Materials and Methods. Decomposition of the perturbed system into a reduced and a boundary system of smaller dimension is used as the main method of investigation. For the stability analysis, Zubov's theorems on the stability of homogeneous systems are applied to Lyapunov second method. Results. In the course of research, the authors have obtained the conditions under which the asymptotic stability of the zero solution of a singularly perturbed system is a consequence of the analogous property of the reduced and boundary systems. This conclusion is valid for sufficiently small values of the perturbing parameter. To verify the hypothesis of the theorem, it is required to construct homogeneous Lyapunov functions. Discussion and Conclusions. The paper gives a numerical example showing the class of systems satisfying the obtained theorem is not empty. An upper bound for the variation of a small parameter has been obtained, within which the zero solution is guaranteed to be asymptotically stable.