On stability in one critical case: PMM vol. 43, no. 6, 1979, pp. 963–969

Abstract

The problem on the stability of the trivial solution of an autonomous system of ordinary differential equations is solved in the critical case of one zero root, m pairs of pure imaginary roots, and q roots with negative real parts. It is proved that the presence of the zero root, as a rule, leads to instability, which can be detected already from the form of the second-order series expansion of the right hand sides of the equations. In the degenerate case necessary and sufficient stability conditions have been indicated for a model (simplified)system; it is shown that the absence of additional degeneracy the instability of the original system follows from that of the model. Sufficient conditions for the asymptotic stability and instability of the original system have been obtained under the fulfilment of the necessary stability conditions for the model system.