Lyapunov stability criteria for systems of ordinary differential equations in multiplicative and additive forms

Various Lyapunov stability criteria for systems of ordinary differential equations are presented in the form of necessary and sufficient conditions. The criteria are obtained under the conditions of existence and continuity of the solution on the semi-axis, continuity of the right part of the system and its continuous differentiability on the semi-axis. The criteria are constructed on the basis of recurrent transformations of difference schemes of numerical integration with a residual term at each step. The multiplicative and additive form of the criteria entails the possibility to computerize the stability analysis and perform it in real time.