nonlinear dynamic processes
boundedness of solutions
M. V. Zakharova, O. V. Druzhinina
The nonlinear dynamic processes described by the systems of the second order differential equations are studied in the paper. The sufficient conditions of boundedness of solutions have been obtained, being the generalization on vector case of the results achieved by H. Antosiewicz, R. Bellman, Z. Opial, B. Manfredi, J. Jiang. Boundedness conditions incoming in Theorems 3-5, 7, 9, 11-14 are effective and these conditions allow to carry out the analysis of boundedness of solutions of nonlinear differential systems in cases when the analysis of another qualitative properties is difficult (for example, when investigation of Lyapunov stability and orbital stability is difficult). The conditions of boundedness are based on the properties of the right parts of the studied systems and on the properties of corresponding Lyapunov functions. The differential systems under consideration can be used as models of some classes of mechanical systems. There is investigated important case when acting in the system exterior force is dependent from the time and it is the function of bounded variation on positive semiaxis. The obtained results can be used in qualitative investigation of the nonlinear dynamic processes and in solving problems of safe functioning of transport and engineering systems described by the multidimensional second order differential equations.