On the stability asymptotic characteristics of the differential equations.

Abstract

The aim is to study the central exponents of linear homogeneous systems of differential equations with continuous and bounded coefficients in critical cases and the development of stable characteristics of differential systems in the case of zero values of the central figures. The methodology of work amounted to methods of the qualitative theory of differential equations, methods of the first approximation of differential systems, methods of the central figures. The work in critical cases determined by central exponents of generalized upper and lower central exponents of linear homogeneous systems of differential equations with continuous coefficients tending to zero. A sufficient condition for the asymptotic stability of a linear system. The connection of the generalized upper function of the original system with the dual function of a generalized lower linear system. Using generalized upper central figure set top uniform estimate of solutions of nonlinear systems of differential equations. By using the generalized lower central figure installed below the uniform estimate of solutions of nonlinear systems of differential equations. An indication of the stability of the trivial solution of nonlinear differential equations of the first approximation.The results are the improvement of the method of the central figures in critical cases. The authors’ conclusions may be used in the process of theoretical problems in a number of applications of the theory of differential equations.