Nonlinear Problems of Stability with Respect to Part of the Variables in the First (Linear and Nonlinear) Approximation

Abstract

Chapter 2 develops the method for investigating stability with respect to a given part of the variables of nonlinear systems. First we consider the problem of stability with respect to a given part of the variables in the linear approximation. To a much greater extent than in studies of stability with respect to all the variables, this problem depends on the structure of nonlinear terms. It is demonstrated that the class of nonlinear systems, for which the problem of stability with respect to part of the variables is solvable by using linear approximation, can be significantly extended. To this end, instead of the linear part of the original nonlinear system, we consider a linear approximating system constructed in a special way. The latter is “equivalent” to a certain nonlinear subsystem of the original system and can be obtained from it by special nonlinear transformations of variables. Then, from the analysis of stability of the linear system constructed with respect to all the variables, the conclusion is drawn about stability of the original nonlinear system with respect to a given part of the variables. Constructive procedures are described for constructing such linear approximating systems.