Dragunov D. Numerical and qualitative analysis of the systems of ordinary differential equations

The analytical method for the stability investigations of the trivial solution of the systems of ordinary differential equations (SODE) of the second order was generalized and improved. The idea of the mentioned method was proposed by D.L. Mingori (1970). It is based on the combination of the structural transformations, which preserve Lyapunov's stability property, with the Lyapunov's II method, to be exact, with the Tomson-Tait-Chetayev theorems. The new concept of the L-equivalence of the k-th order (k=0,1,2) of the two (nonautonomous, in general) SODE of the second order was introduced. We have proved several theorems that give necessary and sufficient conditions for the given linear SODE of the second order to be L-equivalent to some linear SODE of the second order, that do not contain the nonconservative positional and/or gyroscopic forces. Using the idea of the FD-method for solving an operator equations of general form, we have developed the FD-method for solving the Cauchy problem for nonlinear SODE. The sufficient conditions that provide the convergence of the proposed method to the exact solution of the problem in the infinity interval were presented.