On the partial instability of the zero solution of nonlinear systems to the first approximation

Sufficient conditions for instability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation are obtained. The results are presented when the right-hand of the system under study is presented both in the most general form and in the form of a vector polynomial. The results are given for the cases when the right-hand of the system under study is presented both in the most general form and in the form of a vector polynomial. As a first approximation, a linear system of ordinary differential equations with a constant matrix is taken, whose eigenvalues may have zero real parts. Moreover, algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the system under study and its linear approximation. If such correspondence exists, solutions of such systems starting in a sufficiently small neighborhood of zero have some identical component-wise asymptotic properties. In particular, this article focuses on solution instability with respect to some variables, which is one of such properties. Conditions are given for the case when the instability properties of the zero solution of one system are preserved upon transition to another system. The paper gives an example of instability with respect to a part of variables of the zero solution of a nonlinear system, whose linear approximation matrix contains one positive, one negative and one zero eigenvalue, and algebraic and geometric multiplicities of the zero eigenvalue do not coincide.